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Belarus Companies
Listing Websites about Belarus Companies
Top Companies in Belarus Glassdoor
(3 days ago) Browse the top companies in Belarus. Salaries, reviews, and more - all posted by employees working at the top companies in Belarus. Apply to jobs near you.
https://www.glassdoor.com/Explore/top-companies-belarus_IL.14,21_IN33.htm
List of companies of Belarus - Wikipedia
(8 days ago) 29 rows · Belarus is a landlocked country in Eastern Europe bordered by Russia to the …
https://en.wikipedia.org/wiki/List_of_companies_of_Belarus
List of top Belarus Companies - Crunchbase Hub Profile
(8 days ago) Location Belarus, Europe. CB Rank (Hub) 7,239. Number of Founders 195. Average Founded Date Jan 14, 2009. Percentage Acquired 3%. Percentage Non-Profit 0%. Number of For-Profit Companies 378. Number of Non-profit Companies 1. Top Investor Types Accelerator , Co-Working Space , Incubator , Private Equity Firm.
https://www.crunchbase.com/hub/belarus-companies
Top 10+ IT Companies in Belarus (2021) - TechBehemoths
(3 days ago) As for companies, there is a relevant trust connection between Belarus-based and global markets, and the bigger the price/hour of services provided, the more responsible companies are. How Belarus IT infrastructure relates to its neighboring countries?
https://techbehemoths.com/companies/belarus
Belarusian Brands - Official website, Belarus Belarus.by
(2 days ago) The company, unique to Belarus, is an example of a private business based on the most advanced scientific achievements. In 1991 it was founded by Professor Vladimir Linev, a nuclear physics scientist, inventor, author of more than a hundred patents. Together with like-minded people, he turned his brainchild into a world-class innovative enterprise.
https://www.belarus.by/en/business/brands-of-belarus
10 Best Belarus Tour Companies & Reviews 2021-2022
(4 days ago) Trafalgar. 4.5 Great. 100+ Reviews 91%. Premier. USTOA. This operator is an Active Member in good standing in the United States Tour Operator Association and part of the $1 million traveler insurance program. #2 Pick of 13 Belarus Tour Companies.
https://www.travelstride.com/tc/belarus-tour-companies
Top 7 pharmaceutical companies in Belarus - …
(3 days ago) These companies are also subject to a variety of laws and regulations regarding but not limited to the testing and marketing of drugs.. Introduction of pharmaceutical companies in Belarus. From the start of the 19th century, the pharmaceutical company is now one of the most lucrative and influential in existence, attracting commendations and different discussions …
https://idealmedhealth.com/pharmaceutical-companies-in-belarus/
U.S. reimposes sanctions on Belarusian companies - …
(5 days ago) The Treasury Department revoked the Belarus-related General first issued the yearly renewed license authorizing U.S. citizens to conduct certain transactions with the nine companies in 2015
https://www.upi.com/Top_News/US/2021/04/20/United-States-revokes-license-Belarus-companies/2781618890041/
Are tech companies leaving Belarus? - CyberNews
(2 days ago) The US company EPAM Systems employs around 37,000 people, and 20,000 of them are situated in Belarus. Wargaming is a Belarusian company famous for the World of Tanks. World of Tanks. According to Maksimas Milta, the IT …
https://cybernews.com/news/are-tech-companies-leaving-belarus/
List of people and organizations sanctioned in relation to
(3 days ago) 14 rows · Since the 1990s, Belarus has been actively criticized by the United States, the …
https://en.wikipedia.org/wiki/List_of_people_and_organizations_sanctioned_in_relation_to_human_rights_violations_in_Belarus
List of top Private Belarus Companies - Crunchbase Hub Profile
(6 days ago) Location Belarus, Europe. CB Rank (Hub) 8,127. Number of Founders 195. Average Founded Date Mar 20, 2009. Percentage Acquired 3%. Percentage Non-Profit 0%. Number of For-Profit Companies 375. Number of Non-profit Companies 1. Top Investor Types Accelerator , Co-Working Space , Incubator , Private Equity Firm.
https://www.crunchbase.com/hub/private-belarus-companies
Find Manufacturing Companies in Belarus - Dun & Bradstreet
(Just Now) Find detailed information on Manufacturing companies in Belarus, including financial statements, sales and marketing contacts, top competitors, and firmographic insights. Dun & Bradstreet gathers Manufacturing business information from trusted sources to help you understand company performance, growth potential, and competitive pressures.
https://www.dnb.com/business-directory/company-information.manufacturing.by.html
Best Logistics Companies in Belarus - Reviews 2022 GoodFirms
(5 days ago) Company SOOO AgroSite started its activity in the market of the Republic of Belarus in 2005 and today has established itself as a reliable business partner. SOOO AgroSite is the first raw protein material supplier in the Republic of Belarus providing a constant supply of soybean meal, sunflower meal, fish meal, fodder yeast, and amino acids.
https://www.goodfirms.co/supply-chain-logistics-companies/belarus
Top Software Development Companies Belarus 2022 GoodFirms
(Just Now) SCAND is a custom software development company located in Minsk, Belarus - in the center of Eastern Europe. 20+ years in IT, more than 700 successful projects for hundreds of customers globally and over 250 highly experienced engineers are our main assets. The company is providing outsourcing services and primarily focusing on software
https://www.goodfirms.co/directory/country/top-software-development-companies/by
List of Pharmaceutical Companies in Belarus
(4 days ago) Read Also: List of Pharmaceutical Companies in Switzerland NatiVita. Minsk, Belarus Address: 220073, 20/11 Olshevskogo str, 5th floor, office 510 Tel: +375 17 33 65 585 Tel./fax : +375 17 33 65 686
https://www.pharmapproach.com/list-of-pharmaceutical-companies-in-belarus/
The Heroes 6 IT companies with offices in Belarus
(8 days ago) Zubr Capital is the first company in Belarus that manages direct investment funds and specializes in investments in capitals of Belarusian companies with their subsequent development. The size of ZCFI is $ 90 million, its portfolio includes 6 companies: SoftClub, 21vek, av.by, Uniflex, GC Softline (ActivePlatform, ActiveCloud), Target Process.
https://theheroes.media/en/news-list/6-it-companies-with-offices-in-belarus-entered-the/
Belarus Sanctions U.S. Department of the Treasury
(Just Now) Belarus General License 5- Authorizing the Wind Down of Transactions Involving Open Joint Stock Company Belarusian Potash Company or Agrorozkvit LLC (Effective December 02, 2021) Archive of Expired General Licenses Guidance on OFAC Licensing Policy. Certain activities related to the Belarus Sanctions may be allowed if they are licensed by OFAC.
https://home.treasury.gov/policy-issues/financial-sanctions/sanctions-programs-and-country-information/belarus-sanctions
Belarus decries sanctions US reimposed on 9 state companies
(5 days ago) Belarus' Foreign Ministry has expressed regret over sanctions the United States reimposed on nine state-owned companies in the ex …
https://abcnews.go.com/Business/wireStory/belarus-decries-sanctions-us-reimposed-state-companies-77194780
Belarus Company Search Info-clipper.com
(1 days ago) Get Registry and financial information about a business in Belarus. Get independent and reliable information reports about Belarusian firms on Info-clipper.com : Reduce credit risk and improve knowledge about your customers, suppliers, competitors in Belarus. Info-clipper.com is a pay-as-you-go, subscription-free service.
http://www.info-clipper.com/en/company/search/belarus.by.html
Belarus' Hi-Tech Park companies ranked among best
(2 days ago) 44 minutes ago · 19:15 Belarus' Hi-Tech Park companies ranked among best developers of apps, games Economy 18:44 Belarus' COVID-19 update: 1,387 new cases, 1,363 recoveries Society 18:15 Belarus' industrial output
https://eng.belta.by/economics/view/belarus-hi-tech-park-companies-ranked-among-best-developers-of-apps-games-147060-2022/
Belarus companies, individuals sanctioned after forced
(1 days ago) Belarus companies, individuals sanctioned after forced landing of jet. The sanctions are part of the EU response to the forced landing of a Ryanair passenger plane in Minsk on May 23 to arrest a dissident journalist. Also, following a Friday agreement, EU states are preparing to impose economic sanctions on Belarus' financial, oil, tobacco and
https://www.malaysiasun.com/news/270017945/belarus-companies-individuals-sanctioned-after-forced-landing-of-jet
Setting up a company in Belarus, Belarus Belarus.by
(4 days ago) Setting up a company in Belarus Starting a business in Belarus, types of businesses and the time needed to open a business Residents of the Republic of Belarus and foreign entities are provided with equal rights and opportunities to run a business by setting up legal entities.
https://www.belarus.by/en/business/companies
Belarusian National Reinsurance Organization
(5 days ago) About Belarus Re. We are happy to welcome you on the corporate website of Belarusian National Reinsurance Organization. State Unitary Enterprise "Belarusian National Reinsurance Organization" is the first and the only specialized reinsurance company in the Republic of Belarus.
http://belarus-re.com/en/
Belarus Medical Companies Archives - Medical Products
(7 days ago) Av. Rokossovski, 166, Premices 1N, 220101 Minsk Belarus Telephone +375 17 2145846, +375 29 6110228 [email protected] http://www.tahat.by
https://www.medica.mobi/medical/belarus-medical-companies
Find Wholesale Trade Companies in Belarus - Dun & Bradstreet
(2 days ago) Wholesale Trade Companies in Belarus. Find detailed information on Wholesale Trade companies in Belarus, including financial statements, sales and marketing contacts, top competitors, and firmographic insights. Dun & Bradstreet gathers Wholesale Trade business information from trusted sources to help you understand company performance, growth
https://www.dnb.com/business-directory/company-information.wholesale_trade.by.html
In Photos: What EU Sanctions Against Belarus Look Like
(8 days ago) The EU hit Belarus with a new round of sanctions on June 24. Here are some of the companies and industries with apparent links to Alyaksandr Lukashenka that are now restricted from trading with EU
https://www.rferl.org/a/eu-sanctions-belarus-companies/31326064.html
Timber company list in Belarus
(4 days ago) Gamarnika str. 30, Minsk, Belarus 220131 Minsk, Minsk. Parkway LLC is one of the leading sawn timber producers on market in Republic Belarus.Company was founded 2006 and has been exporting goods for more than 10 years to European Asian countries, especially UK, Germany, Lithuania, Latvia, Lebanon, China
https://www.company-list.org/belarus/timber/
US, EU and Britain Slap Sanctions on Belarus Officials and
(Just Now) EU states are also soon set to impose economic sanctions on Belarus' financial, oil, tobacco and potash sectors, after a provisional deal was agreed on Friday. The EU imported 1.2 billion euros
https://money.usnews.com/investing/news/articles/2021-06-21/eu-to-blacklist-86-belarus-officials-and-companies
EU Eyes Sanctions for Airlines, Tourism Companies Aiding
(2 days ago) Other companies named in the sanctions include Cham Wing Airlines, state tourism company Tsentrkurort, passport and visa distributor VIP Grub, and …
https://www.msn.com/en-us/news/world/eu-eyes-sanctions-for-airlines-tourism-companies-aiding-migrants-entrance-to-belarus/ar-AAR5Rr5
(5 days ago) Belarus decries sanctions US reimposed on 9 state companies April 20, 2021 GMT Belarus President Alexander Lukashenko plants young trees during a subbotnik, a Soviet-style Clean-up Day, in the village of Alexandria, Belarus, Saturday, April 17, 2021.
https://apnews.com/article/alexander-lukashenko-belarus-united-states-business-government-and-politics-53d15b25f1d6d9511399e75dee0721d4
EU to blacklist transport companies involved in human
(6 days ago) In the midst of the geopolitical confrontation unravelling at the Belarus-Poland border, the European Commission has unveiled new plans to punish transport companies that aid and abet the
https://www.euronews.com/2021/11/23/eu-to-blacklist-transport-companies-involved-in-human-trafficking-amid-belarus-border-cris
Category: Transport Show Companies
(6 days ago) Belarus' Foreign Ministry on Tuesday expressed regret over sanctions that the United States reimposed on nine state-owned companies in the ex-Soviet country, citing human rights violations. Belarus has become a target of Western sanctions after authoritarian President Alexander Lukashenko won his sixth term in office in a widely disputed election in August and …
https://news.yahoo.com/belarus-decries-sanctions-us-reimposed-194423649.html
Belarus - The World Factbook
(7 days ago) Belarus derives export revenue by refining Russian crude and selling it at market prices. Russia and Belarus have had serious disagreements over prices and quantities for Russian energy. Beginning in early 2016, Russia claimed Belarus began accumulating debt – reaching $740 million by April 2017 – for paying below the agreed price for
https://www.cia.gov/the-world-factbook/countries/belarus/
Top 10+ ERP Consulting Companies in Belarus (2021
(8 days ago) Discover Top IT Companies in Belarus specialized in ERP Consulting including branding, UX design, web design, web development, social media marketing, mobile apps. Featured Companies in Belarus. This month, the following ERP Consulting companies managed to provide an outstanding service and support. It's worth taking a look.
https://techbehemoths.com/companies/erp-consulting/belarus
Belarus withheld information from Ryanair diversion probe
(9 days ago) The EU has imposed travel bans and asset freezes on 86 Belarusian individuals and companies, including transport and defence ministers, in response to the grounding. read more Register now for
https://www.reuters.com/world/europe/belarus-withheld-information-ryanair-diversion-probe-un-says-2022-01-19/
Category: Transport, Travel Show Companies
Belarus theater company flees amid opposition crackdown
(3 days ago) Svetlana Sugako, managing director of Belarus Free Theater, fled to London in October. Sugako is one of 16 members of the theater company now in exile in the UK along with nine other family members.
https://www.msn.com/en-us/news/world/belarus-theater-company-flees-amid-opposition-crackdown/ar-AARMwyx
Belarus B2B and Companies Data, Phones, Emails, Addresses
(1 days ago) What is Belarus B2B and Companies Data, Phones, Emails, Addresses, Websites, Industries, Executives? Belarus B2B lead and contacts business based in Canada. B2B Data is the fuel for your customer acquisition and retention. It can help your business identify new sales leads and get in touch with them. It Spans entire Canada.
https://datarade.ai/data-products/belarus-b2b-and-companies-data-phones-emails-addresses-we-ipfind-io
Russia, Belarus to rehearse repelling external attack in
(Just Now) 1 day ago · Russia and Belarus will rehearse repelling an external attack when they hold joint military drills in Belarus next month, both sides said on Tuesday, at a time of acute tensions with the West over
https://www.nasdaq.com/articles/russia-belarus-to-rehearse-repelling-external-attack-in-joint-drills
EU sanctions to cut Belarus oil products from European
(4 days ago) The European Union has banned companies from doing business with Belarusian oil products as a part of wide-ranging economic sanctions on Belarus announced on Thursday, cutting one of the top fuel
https://www.nasdaq.com/articles/eu-sanctions-to-cut-belarus-oil-products-from-european-market-2021-06-25
Belarus News Belarusian news Belarus today news in
(1 days ago) 14.01.22 17:28 Belarus win three medals at World Para Snow Sports Championships Sport 14.01.22 17:03 Belarus' COVID-19 latest: 1,274 new cases, 1,312 recoveries Society
https://eng.belta.by/
Payment in Dollars, Euros Of Belarusian Banks, Companies
(7 days ago) Payment in Dollars, Euros Of Belarusian Banks, Companies Blocked 33. 5.12.2021, 22:54 ; 111,772; Sanctions against the financial sector in Belarus have begun to work.
https://charter97.org/en/news/2021/12/5/446521/
Russian Energy Companies Halt Oil Supplies to Naftan
(7 days ago) Belarus will most likely be able to receive the remaining contracted volumes from SOCAR as the contract predates the revocation of the moratorium. However, Azerbaijan's state-owned oil company announced that it too will avoid supplying oil to Naftan. In November 2020, Kazakhstan finally ratified the deal negotiated with Belarus in 2019.
https://newsforenergy.com/oil/russian-energy-companies-halt-oil-supplies-to-naftan-refinery-in-belarus-because-of-us-sanctions/
2 Best Tour Operators and Travel Agencies in Belarus
(8 days ago) Top Tour Operators and Travel Agencies in Belarus. Below you will find 2 of the best tour operators and travel agencies in Belarus offering in total 3 tours and holidays through-out Belarus. Combined they have received 1 customer reviews and …
https://www.bookmundi.com/belarus/tour-operators
U.S. Relations With Belarus - United States Department of
(9 days ago) More information about Belarus is available on the Belarus Page and from other Department of State publications and other sources listed at the end of this fact sheet. U.S.-BELARUS RELATIONS The United States supports a sovereign, independent Belarus which respects the rights and voices of the Belarusian people. On December 25, 1991, the United …
https://www.state.gov/u-s-relations-with-belarus/
Relocation: Big companies and international representative
(8 days ago) EPAM is the company engaged in the development of digital products and platforms, sophisticated design and software solutions. The Belarusian office employs over 10,000 professionals. In 2018, the company relocated 180 people to Belarus. Its customers are companies from the Fortune Global 2000.
https://minsk.page/en/it-belarus/big-companies-and-international-representative-offices/
Belarus opposition leader urges Austria to do more to
(5 days ago) Belarusian opposition leader in exile Svetlana Tikhanouskaya urged Austria's government and Austrian companies present in Belarus to do more to help her allies against President Alexander
https://www.reuters.com/markets/asia/belarus-opposition-leader-urges-austria-do-more-counter-lukashenko-2021-11-21/
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In pictures: 12 reasons why we love Susan Boyle
FIND out why we love Scotland's sweetheart, SuBo, in our picture gallery below.
SUSAN BOYLE stole the hearts of a nation with her rendition of Les Misérables's I Dreamed a Dream on Britain's Got Talent in 2009.
But now, with her unbelievable talent and loveable personality, Scots singing sensation SuBo has become a global icon.
Dressed in a sober frock with unstyled hair, she wowed the ITV show's judges with her performance of I Dreamed a Dream and became an overnight internet sensation.
And despite being worth an estimated £22million, SuBo, from Blackburn, West Lothian, has stayed close to her roots and become a national sweetheart.
Find out why we love SuBo in our picture gallery above.
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Visit our entertainment section here.
Did Susan Boyle make our list of sexy Scottish females? Find out by clicking on the gallery below.

redpajama_set_name: RedPajamaCommonCrawl


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FRANCHISOR CHALLENGES
WHY FRANCHISE FASTLANE
EMERGING TO THRIVING
From emerging to thriving: transform your franchise into The Next Big Thing
Partner with America's top producing Franchise Sales Organization to level up your brand without risky expenses or one-sided wins.
Franchisor Success
When Ryan Zink, Franchise FastLane's co-founder, walked in D1 Nashville, he immediately got goosebumps thinking about the potential of D1. With Peyton Manning, Tim Tebow, Chris Paul, Von Miller, and many other world-class athletes involved, it was obvious D1 was the place for the athlete. Unfortunately, D1's growth stalled early on due to the size and expense of their facilities. Together, D1 and FastLane create a much smaller option, significantly reducing expenses, and growth has not slowed down ever since.
When Franchise FastLane was introduced to Roger McGreal, Co-Founder of SPENGA, in 2018 they had five franchisees operating seven studios. SPENGA's leadership team was well-versed in fitness, with previous ownership of 40 big box gyms prior to beginning SPENGA, but was not familiar with growing and scaling as a franchise. FastLane quickly saw the huge potential of SPENGA's leadership team and their Spin, Strength and Yoga workout. In the first two years, FastLane was able to add over 250 franchise units for SPENGA.
Monster Tree and Franchise FastLane began working together in the winter of 2017. At the time Monster Tree had around 15 franchisees. Monster Tree's founder knew his franchise opportunity was positioned to grow into the Next Big Thing but was frustrated as he had gone through multiple franchise development leaders and team members trying to find the right strategy for growth. Using Franchise FastLane Franchise Growth Plan Monster Tree was able to quickly scale and grew to over 100 franchisees and over 250 territories. In Monster Tree 2021 sold the company to Authority Brands for a large exit value.
Michael Moorhouse, President of Mosquito Shield, says Franchise FastLane should re-name the company to Franchise Warp Speed. When Mosquito Shield joined the FastLane in the fall of 2020, they had been offering franchise opportunities for seven years and had approximately 50 franchise locations. In the first 8 months of using FastLane's Franchise Growth Plan, they added an additional 150 locations. Michael may be right, 'warp speed' may be more appropriate.
Smash My Trash was the fastest growing franchise in the country in 2020 during the COVID pandemic. In just 15 months, they grew 500% from 1 location to 500+. Smash brought a unique franchise opportunity with a big financial upside, and Franchise FastLane brought their growth plan. Together they smashed many franchise sales records!
Right away, Franchise FastLane knew Premier Martial Arts was the Next Big Thing in children's focused franchising. They had all the tools needed to be a successful franchisor, an experienced founder, an effective home office team, successful franchisees, and more. On their own, they grew to 40 locations over 20 years, and they were looking for expert help. Before joining Franchise FastLane, they hired another Franchise Sales Organization and for 12 months sold zero new franchise locations. They fired that firm and hired FastLane. Over the next two years, FastLane helped them add over 350 locations.
Koala Insulation was small but mighty when they first reached out to FastLane. With only a few franchise locations open, there was some concern if they were ready for the explosive franchise growth that Franchise FastLane would bring. After a phone call with Koala's founder and CEO, Scott Marr, those concerns quickly went away. He shared his vision, experience, and infrastructure he had already invested into for Koala franchisees. There was no doubt they had what it took to be the nation's largest Insulation franchise. FastLane and Koala teamed up when they were only 3 locations open and operating, and in the first year, they added over 150 new territories.
Color World was ready to make American homes look brighter through franchise growth. Since painting was their business and franchise development wasn't, they decided to hire another Franchise Sales Organization and only awarded 1 location after a year. The founder had a hunger to grow, and this just wasn't good enough for him. He decided to leave that sales company and join the FastLane. In the first 6 months, Color World added 30 new locations in just over 20 months and quickly met the 100 location mark.
3 keys to the success of FastLane's franchise brands
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A flow of highly qualified buyers: Get your franchise brand front of mind with the top consultants through our long-standing proven reputation.
ACCELERATE YOUR UNIT SALES WITH FRANCHISE FASTLANE
© Copyright 2022 Franchise FastLane . All rights reserved. Privacy policy


A long time previously, an individual who'd no concept he'd alter my entire life and determine my professional outlook gave me the best present. In my opinion that every individual you interact with is altering your lifetime around you're transforming theirs whilst not every nurse is likely to be luckily enough to possess this kind of specific connection with the result of the treatment they offer therefore early within their profession.
Breastfeeding has developed right into a highly technical occupation seated in medical proof, that works reduce hospital remains and to enhance individual results. Engineering and study assistance this work-in methods that are countless.
But while nurses must certainly be medication experts, specialized experts, and effectiveness experts, they have to additionally do their finest to ease the suffering of these within their cost. These several concurrent needs can lead to large burnout prices among fragmented in addition to nurses take care of sufferers.
Nurse provides amount should proceed together using care's quality. My very own description of quality treatment is concentrating on sufferers as significantly more than only a group of indicators, indicators, figures, and procedures looking for realignment and checking. Producing our period in the bedroom significant and realizing sufferers as people is usually as achieving our medical objectives as essential.
After I joined nursing, because of several individual restrictions till fundamentally acquiring my master's degree, I had been necessary to progress inside the occupation step from helper to certified professional nurse to RN, by step.
In the period I had been a healthcare facility, in my own authorized nursing plan wherever I labored being an LVN experienced from the nurses. The LVNs like me were requested to consider responsibilities that are diverse. Since I have near to graduating and was additionally within an RN plan, I had been placed directly under the immediate guidance of doctor and an RN to supply care. The in-patient to be a guy who had molds, available injuries had damaged just about any bone in his physique in a bike incident, and was comatose.
He was regarded a "safer" choice getting an RN whilst he absolutely required an ICU degree of treatment. His available injuries and he required treatment and normal rethinking, respectively every day. I'd nothing you've seen prior looked after an individual with a lot of complicated requirements on the standard schedule.


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\section{Introduction}
    The problem of valuation of a non-traded contingent claim has always been of central importance in mathematical finance.
One can distinguish between two fundamentally different notions of price: one inherent to the market, and one dependent on the specific investor.
  If a security was sold at an `unreasonable' price,  any investor could lock in a risk-less profit. Even worse, as long as an investor could be considered infinitesimally small with respect to the size of the market, he could create a `money pump' and make this risk-less profit arbitrarily big. These considerations lead to the fecund concept of \emph{arbitrage-free price}, which prescribes a necessary condition for a market model to be viable.

   Fruitful as it was, this notion is of little use to an investor when the mathematical idealization of a complete market falls short of accurately describing reality. Given a non-replicable security, the market mechanism is not sufficient to determine an  interval $I$  of `threshold' prices such that \emph{any} agent should buy at a price smaller than every $p \in I$, sell at a price greater than every $p\in I$, and \emph{do nothing at any price} $p\in I$. Indeed, since the buying or selling at any arbitrage-free price could lead both to a loss or to a gain, the attitude of the agent towards risk must be taken into consideration to decide what he should do at any such price. Intuitively, the interval of threshold prices should depend on the investor and his initial wealth, and it should be contained in the interval of arbitrage-free prices.

   The classical approach in mathematical finance is to assume that the preferences of the agent are determined by the maximal expected utility $u(x,q)$ that he can obtain by investing in the market an initial capital $x$ if holding an endowment consisting of $q$ illiquid contingent claims.
   Pricing rules derived from $u(x,q)$ are called utility-based, and they form the prevailing paradigm in the valuation of contingent claims: see for example  \citet*{Da:97},  \citet*{Frit:00},  \citet*{Fold:00},  \citet*{HodgNeuber:89},  \citet*{KaratKou:96}, \citet*{Kall:02}, \citet*{Hob05a},  \citet*{HeHo09}, \citet*{HugonKramSch:05},  \citet*{KramSirb:06b}.

   We will consider the notion of \emph{marginal (utility-based) price} at $(x,q)$, which is defined as a price $p$ such that the (utility-maximizing)  agent with initial endowment $(x,q)$ if given the opportunity to trade the contingent claim at price $p$  would neither buy nor sell any. So marginal prices are precisely the `threshold'  prices previously mentioned; moreover, they are defined also  when considering multiple contingent claims at once.
 The idea underlying this valuation principle is well known in economics: see for example  \citet*{Hick:56}.

  This paper is concerned with the investigation of whether all arbitrage-free prices are obtained by utility maximization.
More precisely, we ask if there is a `bad' subset of arbitrage-free prices which are never marginal prices, irrespectively of the investor and his endowment, or if all arbitrage-free prices are equally  `realistic'. And if they are, are they simply parameterized by the initial endowment, or does the preference structure of the investor matter, discriminating between which arbitrage-free prices are among his marginal prices and which arise only because of somebody else? To our knowledge these matters have not been previously investigated.

 What we find is that arbitrage-free prices always come from utility maximization; moreover, it is enough to consider any (one) agent.
Although  it may sound intuitive that there exists no arbitrage-free price at which every agent should buy (or sell), we recall that, in many occasions, the marginal price at $(x,0)$ is unique and does not depend on $x$ \emph{nor on the utility} function (see  \citet*{KaratKou:96}). Thus, to recover all arbitrage-free prices  it is sometimes not  enough to consider all possible risk aversions:
we must also take non-zero random endowments. If we do so, it then turns out that we do not need to consider many agents, nor special ones: any (one) utility would do.
In other words,  we show that there do not exist arbitrage-free prices at which an utility-maximizing agent would always want to buy (or sell) the derivative, irrespectively of his own initial endowment.

However, as we show with an example, there is a very delicate point: one can not a priori discard initial endowments on the boundary of the domain of the utility.
 This will force us to carefully reconsider the framework of  \citet*{HugonKram:04}, on which we rely, and to extend it by including \emph{all} initial endowments with finite utility $u$ (not just the ones in the interior).

    Another objective is to prove that marginal prices are compatible with the market mechanism, i.e., they never allow for an arbitrage. This is important since a negative answer would seriously undermine the legitimacy of this useful concept of price.  \citet*{KaratKou:96} also study this problem, although in a different setting.

 We also investigate a number of properties of marginal prices, with special emphasis  on the peculiarities which arise when dealing with endowments on the boundary of the domain of $u$, as these points are essential to answer positively our main question.
   As shown by  \citet*{KaratKou:96} (and, in our framework, by  \citet*{HugonKramSch:05}), under appropriate assumptions, the marginal price at $(x,0)$ is unique; so if an agent has no initial endowment of illiquid contingent claims,  the interval $I$ of marginal prices collapses to a point, a particularly pleasant situation.
 In stark contrast to this, we prove that marginal prices based at a point $(x,q)$   on the boundary, if they exist, are never  unique; we also characterize
  their existence, and show how to compute them.
   Moreover, we give an explicit example of non-uniqueness where the set of marginal prices contains an interval that has an \emph{arbitrage price} as an end-point.
We then prove that this considerably quirky behavior is a general fact; in particular,  given an investor with an  endowment on the boundary, there always is an  arbitrage price $p$ such that the agent  would  choose not  to trade in the derivative even if this one was being sold at a price arbitrarily close to $p$!
Thus, one is left with a choice: either  to restrict the attention to the well behaved marginal prices  based at endowments in the interior of the domain of $u$ (in which case however  arbitrage-free prices may not come from utility maximization),
or to consider also marginal prices with awkward properties.

  The plan of the rest of the paper is as follows:
in Section \ref{themodel} we introduce the model we work in, and
in Section \ref{Statement of the main theorem} we state our main theorems.
Then, in Section   \ref{counterexample} we show with an explicit example what can go
  wrong, and in Section \ref{Characterizations of Arbitrage-Free Prices} we present a useful   characterization of arbitrage-free prices. Finally, in Section \ref{u upper semi-continuous} we obtain that the maximal expected utility  $u(x,q)$ is an upper semi-continuous function,
  in Section \ref{Proof of the main theorem.} we prove our first main theorem, and  in Section \ref{Consequences and related results}
 we derive a formula for marginal utility-based prices and we prove our second main theorem.

     \section{The model}
     \label{themodel}
We use  the same model of an agent investing in a financial market as  \citet*{HugonKram:04}, and the same notations; however, we have to consider slightly more general random endowments, or our main result would not hold (see Section \ref{counterexample}). We thus present only a very brief introduction to the model, referring to  \citet*{HugonKram:04} for additional details and discussions, and pointing out the few instances where we differ from it.

    We consider a model of a financial market  composed of a savings account with zero interest rate, and $d$ stocks with price $S = (S^i)_{i=1}^d$. We assume that  $S$ is a  locally-bounded\footnote{This assumption is not really necessary, as the results in \citet*{KramSch:99}, \citet*{KramSch:03}, \citet*{HugonKram:04},  \citet*{DelbSch:97} on which our proofs hinge, although proved for a locally-bounded semi-martingale, are true also without the local boundedness assumption, if one replaces equivalent local-martingale measures with separating measures throughout. This observation is stated in \citet*[Remark 3.4]{HugonKramSch:05}.}
 semi-martingale on a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},P)$ whose filtration $\mathbb{F}=(\mathcal{F}_t)_{t\in [0,T]}$ satisfies the usual conditions.
  The wealth $ X$  of a (self-financing) portfolio $(x,H)$ evolves in time as the  stochastic integral
\begin{displaymath}
\label{wealth} \textstyle
X_t=x+(H\cdot S)_t=x+\int_0^t H_u dS_u , \qquad t\in [0,T] ,
\end{displaymath}
where $H$ is assumed to be a predictable $S$-integrable process.
We denote by $\mathcal{X}(x)$ the set of non-negative wealth processes whose
initial value is equal to $x\geq0$, and by $\mathcal{M}$ the
family of equivalent local martingale measures; we assume that
\begin{equation}
\label{na}
\mathcal{M}\ne \emptyset .
\end{equation}
 The  utility function $U$ is assumed to be strictly concave, strictly increasing and continuously differentiable on $(0,\infty)$ and to satisfy Inada conditions:
\begin{equation}
\label{inada} \textstyle
U'(0+):=\lim_{x \to 0+}U(x)=\infty  , \qquad
U'(\infty):=\lim_{x \to \infty}U'(x)=0 .
\end{equation}
Differently from \citet*{HugonKram:04}, it will be convenient for us to consider $U$ as defined on the whole real line. We want its extension to be concave and upper semi-continuous, and  (\ref{inada}) implies that there is only one possible choice: we define $U(x)$ to be $-\infty$ for $x$ in $(-\infty,0)$, and to equal $U(0+)$ at $x=0$.

In this paper we will be concerned with a family of non-traded European contingent claims whose payoff $f=(f_j)_{j=1}^n$ is  dominated  by the final value of a  non-negative wealth process $X'$, that is,
\begin{equation}
\label{domination} \textstyle
|f|:=\sqrt{\sum_{j=1}^n |f_j|^2 }\leq X'_T  \, .
\end{equation}
 To rule out doubling strategies in the model, one has to impose some sort of boundedness condition on the allowed wealth processes.  In the presence of contingent claims bounded with respect to some numéraire (i.e., satisfying \eqref{domination}), one should consider wealth processes that are admissible under some numéraire; these can be characterized as follows.
 A process $X$ in $\mathcal{X}(x)$ is said to be \emph{maximal} if its terminal value cannot be dominated by that of any other process in $\mathcal{X}(x)$.
A wealth process $X$ is called \emph{acceptable} if it admits a representation of the form
$X = X' - X''$, where $X'$
is a non-negative wealth process and $X''$ is a maximal wealth  process.

 Consider an agent with an initial endowment consisting of a cash amount $x$ and a quantity $q$ of contingent claims $f$. If the agent  followed the strategy $H$ his wealth process would be $X=x+H\cdot S$ and his final wealth  $X_T+qf$ (throughout this paper we will use the notation $vw$ for the dot product of the vectors $v$ and $w$).
   If the contingent claims $f$ cannot be traded and all the agent can do is to  invest in the stocks and the bond, then his maximal expected utility will be
 \begin{equation}
 \label{prrend}
 u(x,q):=\sup_{ \{X :\, X \textrm{ is acceptable}, \, X_0 = x\} } \mathbb{E}[U(X_T+qf)] ,   \qquad (x,q)\in \r\times \r^n
 \end{equation}
 where we define $\mathbb{E}[U(X_T+qf)]$ to be $-\infty$ when $\mathbb{E}[U^{-}(X_T+qf)]=-\infty$, whether or not $\mathbb{E}[U^{+}(X_T+qf)]$ is finite (so, unlike  \citet*{HugonKram:04}, we define $u$  for every $(x,q)\in \r\times \r^n$). Clearly, if the wealth process does not satisfy $X_T+qf\geq 0$,  the corresponding expected utility will be $-\infty$, so we set
\begin{displaymath}
\mathcal{X}(x,q):=\{ X : \,\, X \textrm{ is acceptable}, X_0 = x \textrm{ and } X_T+ qf  \geq 0 \} .
\end{displaymath}
Notice that $\mathcal{X}(x,0)=\mathcal{X}(x)$ for all $x\geq 0$. We will consider the convex cone
\begin{equation}
\label{defK}
\mathcal{\bar{K}}:=\{(x,q)\in \r \times \r^{n}: \mathcal{X}(x,q)\ne \emptyset \} .
\end{equation}
 which is closed and  contains $(1,0)$ in its interior $\mathcal{K}$  (see  \citet*[Lemma 6 and Lemma 1]{HugonKram:04}). Being convex, $\mathcal{\bar{K}}$ is then the closure of its interior  $\mathcal{K}$.

  We point out that, if conditions (\ref{na}), (\ref{inada}), and (\ref{domination}) hold and
$ u(x,0)<\infty$ for some $x>0 $,
 then the concave function $u$ defined on $\r^{n+1}$ by (\ref{prrend})
 never takes the value $\infty$ and $\mathcal{K}\subseteq \{u >-\infty
 \} \subseteq \mathcal{\bar{K}}$. In particular $\mathcal{K}$ is the interior of $\{u >-\infty \}$, and $-u$ is a proper convex function.

 Suppose now that it becomes possible to trade the contingent claims \emph{at time $0$} and at price $p$. Consider an  agent with random endowment $(x,q)$ who buys $q'$ contingent claims, spending $q' p:=\sum_{j=1}^n q'_jp_j$. If he then invests the remaining wealth $x-q'p$ dynamically into the stocks and bond following the strategy $H$, his wealth process will be $X=x-q'p+(H\cdot S)$ and his final wealth will be $X_T+(q+q')f$.

 Of course, we want to rule out arbitrage possibilities in the expanded market.
We will say that $p$ is an \emph{arbitrage-free price} for the European contingent claims $f$ if any portfolio with
zero initial capital and non-negative final wealth has identically zero final wealth, and we will denote by $\mathcal{P}$ the set of arbitrage-free prices; so, we set
 $$\mathcal{P}:=\{ p \in \r^n :  q\in\r^n, X\in\mathcal{X}(-pq,q) \text{ imply } X_T=-qf\}.$$

 Consider now an agent with utility $U$ and with corresponding maximal expected utility $u$ given by  (\ref{prrend}), and fix
 a point $(x,q)$ in $\{u >-\infty \}$.
We will say that, $p$ is a \emph{marginal (utility-based) price} at $(x,q)$ for  $f$ relative to $U$  if the agent with initial endowment $(x,q)$ given the opportunity to trade the contingent claims $f$ at time zero at price $p$ would neither buy nor sell any. We will denote by  $\mathcal{P}(x,q; U)$ (or simply $ \mathcal{P}(x,q)$) the set of marginal prices of $f$  at $(x,q)\in \{ u\in \r\}$; i.e., we set
 \begin{equation}
 \label{mubp}
 \mathcal{P}(x,q; U):=\{p\in \r^n :  u(x-q'p,q+q') \leq u(x,q)   \text{ for all } q' \in \r^{n}\}.
\end{equation}
  Intuition suggested that in the previous definition we exclude a priori the points $(x,q)$ in $\{u=-\infty \}$; our main result will show that indeed this is a good choice.

\section{Statement of the main theorem}
\label{Statement of the main theorem}
 In this section we state our main results; our first objective is to answer to the following questions:
  \begin{enumerate}
  \item
  Can one span all arbitrage-free prices using marginal utility-based prices? In particular, does one need to consider the marginal utility-based prices at all points $(x,q)$ in $\{ u>-\infty\}$, or is it enough to consider the `nicer' points in the interior? Does one have to take the prices relative to all possible utilities, or to some subfamily, or is it enough to consider any one utility function?
    \item
   It is true that all marginal utility-based prices are arbitrage-free?
  \end{enumerate}

We now need to introduce some more (standard) notation. We set
\begin{equation}
\label{Y}
\mathcal{Y}(y):=\{Y\geq0 : Y_0 = y , XY \textrm{ is a super-martingale for all } X\in \mathcal{X}(1)\},
\end{equation}
and we denote with $V$  the convex conjugate of $U$, i.e.,
\begin{equation}
\label{V} \textstyle
V(y):=\sup_{x \in \r }(U(x)-xy)=\max_{x >0 }(U(x)-xy), \quad y \in \r.
\end{equation}

 Following \citet*{HugonKram:04}, we will denote by $w$  the value function of the problem of optimal investment without the European contingent claims, and by $\tilde{w}$  its dual value function; in other words
 \begin{equation}
 \label{w}
 w(x):=\sup_{X \in \,\, \mathcal{X}(x)} \mathbb{E}[U(X_T)] , \quad \tilde{w}(y):=\inf_{Y\in\mathcal{Y}(y)}\mathbb{E}[V(Y_T)] .
\end{equation}

Following \citet*{DelbSch:97}, we will say that a wealth process $X$ is \emph{workable} if both $X$ and $-X$ are acceptable, and following \citet*{HugonKramSch:05} we will say that a random variable $g$ is \emph{replicable} if there is an workable process $X$ such that $X_T = g$.
 Provided that it exists, such a process $X$ is unique.
To simplify some proofs we will also  assume that
\begin{equation}
\label{wlog}
\textrm{ for any non-zero $q\in \r^n$ the random variable $qf$ is not replicable} ;
\end{equation}
we remark however that most of our result are clearly valid without this assumption (for reasons explained in \citet*[Remark 6]{HugonKram:04}).

The following theorem constitutes our first main result.

\begin{theorem}
\label{afp=mubp}
  Assume that conditions (\ref{na}), (\ref{inada}), (\ref{domination}) and (\ref{wlog})  hold and that
 \begin{displaymath}
 \tilde{w}(y) <\infty \textrm{  for all } y>0,
 \end{displaymath}
 where $\tilde{w}$ is the function defined by (\ref{Y}), (\ref{V}), (\ref{w}).
     Then the set of arbitrage-free prices coincides with the set of all marginal utility-based prices relative to the utility function $U$. In other words
    \begin{equation}
    \label{eqafp=mubp}
    \qquad  \mathcal{P}=\bigcup_{(x,q)\in \{ u >-\infty \}}  \mathcal{P}(x,q;U) ,
    \end{equation}
    where $u$ is the function defined in (\ref{prrend}).
\end{theorem}
Theorem \ref{afp=mubp} (which we will prove in Section \ref{Proof of the main theorem.}) says in particular that marginal prices are always arbitrage free, and to recover all arbitrage-free prices with marginal prices there is no need to consider multiple utilities, nor any special one: any one utility function will do!

   We will see in Section \ref{counterexample} that, to obtain a result like the above, it is \emph{not} sufficient to consider the points $(x,q)$ in the interior $\mathcal{K}$ of $\{ u >-\infty \}$; this is why we are forced to extend the framework of  \citet*{HugonKram:04}.
This leads us to study marginal prices based at boundary points, which turn out to behave quite unexpectedly. What is perhaps their most quirky  property is described in our second main theorem, where by
$[p_0,p)$ we denote the segment from $p_0$ to $p$, excluding $p$.

\begin{theorem}
   \label{Pnotclosed}
   Under the assumptions of Theorem \ref{afp=mubp}, if  $p_0 \in \mathcal{P}(x,q)$ for some non-zero $(x,q)$ on the
   boundary of $ \{ u >-\infty \} $, then there exists an
   \emph{arbitrage price} $p \in \r^n \setminus \mathcal{P}$ such
   that $[p_0,p) \subseteq \mathcal{P}(x,q)$.
\end{theorem}
Note how Theorem \ref{Pnotclosed} (which we will prove in  Section \ref{Consequences and related results}) also stands in stark contrast with the
fact that the marginal price at $(x,0)$ is unique (under appropriate assumptions), as shown by  Hugonnier, Kramkov and Schachermayer \citet*{HugonKramSch:05}. To the best of our knowledge, no assumptions are known under which the marginal prices based at the generic point  $(x,q)$ in the    interior of $ \{ u >-\infty \} $ are unique (or, equivalently, $u$ is differentiable).

 \section{An illuminating example}
 \label{counterexample}
We will now construct explicitly an illuminating example which
shows why we need all endowments with finite utility to prove Theorem \ref{afp=mubp}, and  illustrates how a number of intuitive statements about marginal prices fail to be true for prices based at  boundary points,  clarifying  in particular why Theorem \ref{Pnotclosed} holds.

  We will denote by $(1,p)^{\perp}$ the set of vectors in $\r^{n+1}$ orthogonal to $(1,p)$.
  Given any $x\in \r$ and the set $\mathcal{\bar{K}}$
  defined in (\ref{defK}), if we define
     \begin{equation}
     \label{optmset}
     \mathcal{\bar{K}}^p(x):=\mathcal{\bar{K}} \cap \{ (x-qp,q): q\in \r^n \},
     \end{equation}
     then $\mathcal{\bar{K}}^p(x)$ is the set of points $(a,b) \in \mathcal{\bar{K}}$ whose dot product with $(1,p)$ equals $x$,  i.e., the intersection of $\mathcal{\bar{K}}$ and the set $(x,0)+ (1,p)^{\perp}$.
   Since $u=-\infty$ outside $\mathcal{\bar{K}}$, for $(a,b)\in\{ u\in \r\}$
  it follows from the definition of marginal prices (\ref{mubp}) that $p$ \emph{is a marginal price at} $(a,b)$  \emph{if and only if} $(a,b)$ \emph{is a maximizer of} $u$ \emph{on} $\mathcal{\bar{K}}^p(a+bp)$.

 Let us now build an example of the following counter-intuitive
 situation: an arbitrage-free price $p$ and a maximal expected utility function $u$ that, for arbitrary $x\in \r$ s.t.  $\mathcal{\bar{K}}^p(x)\neq \emptyset$, when restricted to  $\mathcal{\bar{K}}^p(x)$ attains its maximum \emph{only} at some point $(x',q')$ on the boundary of $\mathcal{\bar{K}}$. This will give that $p$ belongs to $\mathcal{P}(x',q')$ but not to any $\mathcal{P}(\bar{x},\bar{q})$ with $(\bar{x},\bar{q})$ in $\mathcal{K}$ (since $(\bar{x},\bar{q})\in \mathcal{\bar{K}}^p(\bar{x}+\bar{q}p)$).

 Take $U(x)=\sqrt{x}$, and consider a market with the bond, no stocks and one contingent claim whose law has density
 \begin{displaymath}
 c(s+1)^{\frac{3}{2}} \, 1_{[-1,1]}(s)
 \end{displaymath}
 with respect to the Lebesgue measure, where $c$ is the normalization
 constant. In this case $\mathcal{\bar{K}}=\{(a,b)\in \r^2 : |b| \leq
 a \}$, and trivially $(-1,1)$ is the set of arbitrage-free prices
 $\mathcal{P}$. Fix $p=0$ and\footnote{The other cases are trivial: if $x<0$ then $\mathcal{\bar{K}}^p(x)=\emptyset$, and if $x=0$ then $\mathcal{\bar{K}}^p(x)=\{0\}$, which is a subset of the boundary of $\mathcal{K}$.}  $x>0$, then  $\mathcal{\bar{K}}^p(x)=\{x\} \times [-x,x]$ and we want to prove that the concave function
 \begin{displaymath} \textstyle
 h(q):=u(x,q)=c\int_{-1}^1 \sqrt{x+qs}\, \left(s+1\right)^{\frac{3}{2}} ds , \quad q\in [-x,x] ,
 \end{displaymath}
   does not attain a maximum at any point in the interior $(-x,x)$ of $[-x,x]$. By differentiating under the integral we compute the derivative of $h$ to be
 \begin{displaymath}
 h'(q)=c\int_{-1}^1 \frac{s(s+1)^{\frac{3}{2}}}{ 2\sqrt{x+qs}} ds .
 \end{displaymath}
 The function $h$ is concave, so its derivative is non-increasing. It follows that since
 \begin{displaymath} \textstyle
 h'(x)=\left( c\int_{-1}^1 s(s+1) ds \right) / 2 \sqrt{x}=c/3\sqrt{x}
 \end{displaymath}
  is strictly positive, $h$ is strictly increasing on $[-x,x]$, so its maximum is attained only at $q=x$.
 Thus $p=0 \in \mathcal{P}$ belongs to $\mathcal{P}(x,x)$ (for any $x>0$) but not to
 any $\mathcal{P}(\bar{x},\bar{q})$ with $(\bar{x},\bar{q})$ in
 $\mathcal{K}$, proving that it is not enough to consider endowments in the interior of $\{ u>-\infty \}$.

This example also illustrates how the uniqueness of
  marginal utility-based prices fails on the boundary; let us show
that indeed  $(-1,0) \subseteq \mathcal{P}(x,x)$.
 If $-1<r<0$ we have that (draw a picture!)
  \begin{equation}
\label{nonunique}
   \mathcal{\bar{K}}^r(x(1+r))= \{(a,b)\in \mathcal{\bar{K}}: a+br=x(1+r) \} \subseteq  \{(a,b)\in \mathcal{\bar{K}}: a\leq x \} \, .
  \end{equation}
It is clear\footnote{Indeed
$ u(\cdot,q)$ is obviously increasing, and we have seen above that the maximum of
$u$ on  $\{x\} \times [-x,x]$ is  attained at the point $(x,x)$.}  that the maximum of $u$ on the set $\{(a,b)\in
  \mathcal{\bar{K}}: a\leq x \}$ is attained at $(x,x)$. Thus  (\ref{nonunique}) implies that the vector $ (x,x)\in  \mathcal{\bar{K}}^r(x(1+r))$  is a maximizer of $u$ on
$\mathcal{\bar{K}}^r(x(1+r))$, i.e., $r\in \mathcal{P}(x,x)$.

In this example not only $\mathcal{P}(x,x)$  is not a singleton: it
even contains a whole open interval of prices with one extreme being
an arbitrage price. This is an example of the general behavior described in Theorem \ref{Pnotclosed}.

The inclusion $(-1,0] \subseteq \mathcal{P}(x,x)$, valid for all $x>0$, also shows that the same marginal price can correspond to multiple initial endowments.
 So marginal prices cannot be thought of as a parametrization---nor as a partition---of arbitrage-free prices, with the parameter being the initial endowment.

  Observe how this example yields a function $u$ which is finite and continuous on $\mathcal{\bar{K}}$. While this is not true in  general, we will see that $u$ is always upper semi-continuous.

 \section{Characterizations of Arbitrage-Free Prices}
   \label{Characterizations of Arbitrage-Free Prices}

We will need the following definition and technical lemma
\begin{align}
\label{primalSetC}
\mathcal{C}(x):=\{ g \in L_+^0(P) : g \leq X_T \textrm{ for some } X\in \mathcal{X}(x)\} .
\end{align}

\begin{lemma}
\label{existswork}
  Assume condition (\ref{na}). Then given any $X \in \mathcal{X}(x,q)$ there exists a workable $\tilde{X}\in \mathcal{X}(x,q)$ such that $\tilde{X}_T\geq X_T$.
\end{lemma}

  \textsf{ PROOF. }  Since $\mathcal{C}(x)$  is closed and bounded in $L^0(P)$ (see \citet*[Proposition 3.1]{KramSch:99}). Let $X=X'-X''$ be a decomposition of the acceptable process $X\in \mathcal{X}(x,q)$ into a nonnegative wealth process $X'$ and a maximal process $X''$, and define
\begin{displaymath}
B:=\mathcal{C}(x) \cap \{ g \in L^0(P) : g \geq X'_T \} .
\end{displaymath}
 The set $B$ is then closed and bounded in $L^0(P)$ and so, as stated and proved in the course of the proof of lemma 4.3 in \citet*{DelbSch:94}, $B$ contains a maximal element $\tilde{g}$. It follows that there exists a maximal $\tilde{X}'\in  \mathcal{X}(x)$ such that $\tilde{X}_T^{'}=\tilde{g}\geq X'_T$. Take $\tilde{X}:=\tilde{X}'-X''$, then $\tilde{X}$ is workable and $\tilde{X}_T\geq X_T$, so $\tilde{X}\in \mathcal{X}(x,q)$. $\Box$
 \newline

    We will denote by $\mathcal{\bar{L}}$ the polar of $-\mathcal{\bar{K}}$, i.e.,
\begin{equation}
\label{L}
\mathcal{\bar{L}}:=-\mathcal{\bar{K}}^o:=\{ v\in\r^{n+1} : vw \geq 0 \textrm{ for all } w \in \mathcal{\bar{K}} \} .
\end{equation}
 Clearly, $\mathcal{\bar{L}}$ is a closed convex cone; we will denote its relative interior by $\mathcal{L}$.
The following facts will be used in the proof of Lemma \ref{xacc0}.
 As shown in \citet*[Lemma 7]{HugonKram:04}, assumptions (\ref{na}),
 (\ref{domination}) and (\ref{wlog})  imply that  $\mathcal{L}$ is the interior of  $\mathcal{\bar{L}}$, and so
\begin{align}
\label{LK>0}
\mathcal{L}=\{(y,r)\in \r^{n+1} : xy+qr>0  \text{ for every non-zero } (x,q)\in \mathcal{\bar{K}} \}
\end{align}
and  $\mathcal{\bar{K}} \cap (-\mathcal{\bar{K}})=\{0\}$ (since   \citet*[Corollary 14.6.1]{Rock:70} implies that
 $\mathcal{\bar{K}} $ contains no line passing through the origin).
We recall that the recession cone\footnote{Also known as asymptotic cone.} rec(C) of a set $C\neq \emptyset$ is defined as the set of all $y$ such that $x+ty\in C$ for all $x\in C$ and $t\geq 0$; we refer to \citet*[Theorem 8.4]{Rock:70} for the following fundamental property of recession cones:
\begin{align}
\label{RecConeBdd}
\hspace{-0.2cm}\text{a closed convex set C$\neq \emptyset$ is unbounded iff there exists a non-zero } x\in \text{rec(C)}.
\end{align}

\begin{lemma}
 \label{xacc0}
Assume conditions (\ref{na}), (\ref{domination}) and (\ref{wlog}). Then the following are equivalent:
  \begin{enumerate}
\item  \label{Pagain} $p$ is an arbitrage-free price.
\item  \label{Lagain} $(1,p)\in \mathcal{L}$.
\item \label{E_Q(f)=p} There exists an equivalent local martingale measure $Q\in \mathcal{M}$ that  satisfies $\mathbb{E}_Q[f]=p$ and such that the maximal process $X'$ that appears in (\ref{domination}) is a uniformly integrable martingale under $Q$.
\item \label{perp=0} $ \mathcal{\bar{K}} \cap (1,p)^{\perp}=\{0\}$.
\end{enumerate}
Moreover, if $x\in \r$ is such that the set $\mathcal{\bar{K}}^p(x)$ defined in (\ref{optmset}) is non-empty, then its recession cone is  $ \mathcal{\bar{K}} \cap (1,p)^{\perp}$, and so another equivalent condition is
\begin{enumerate}  \setcounter{enumi}{4}
\item \label{Kpbdd}
The set $\mathcal{\bar{K}}^p(x)$ is bounded.
\end{enumerate}

 \end{lemma}

  The previous lemma shows that item \ref{E_Q(f)=p}, which would have been a plausible alternative definition of arbitrage-free price (and was implicitly used in \citet*{HugonKram:04}), is actually equivalent to our definition.    \newline

  \textsf{ PROOF. }
Item \ref{Lagain} implies item \ref{perp=0}, as it follows from \eqref{LK>0}.  To prove the opposite implication take $(1,p)\notin \mathcal{L}$, so that \eqref{LK>0} implies that there exists a non-zero $(\tilde{x},\tilde{q})\in \mathcal{\bar{K}}$ such that $\tilde{x}+p\tilde{q} \leq 0$.
We can then find a convex combination $(x',q')$  of $(\tilde{x},\tilde{q})$ and $(1,0)$ which satisfies $x'+pq'=0$, and so the non-zero\footnote{If the sum of two vectors $a,b\in \mathcal{\bar{K}}$ equals zero then $a=-b\in \mathcal{\bar{K}} \cap (-\mathcal{\bar{K}})=\{0\}$.} vector $(x',q')$ belongs to $ \mathcal{\bar{K}} \cap (1,p)^{\perp}$; thus, item \ref{perp=0} implies item \ref{Lagain}.
That item \ref{Lagain} implies item \ref{E_Q(f)=p} is part of \citet*[Lemma 8]{HugonKram:04}, and that item \ref{E_Q(f)=p} implies item \ref{Pagain} follows simply from the definition of $ \mathcal{X}(x,q)$. Finally, let us prove that item \ref{Pagain} implies item \ref{Lagain}.  Let $p$ be an arbitrage-free price and assume that $(x,q)\in \mathcal{\bar{K}}$ and $x+pq\leq 0$, let $X\in \mathcal{X}(x,q)$, $X=x+H\cdot S$ and define $X':=-qp+H\cdot S$. Then $X'_T+qf$ is the sum of the two non-negative quantities $-qp-x$ and $X_T+qf$. It follows that $X'$ belongs to  $\mathcal{X}(-qp,q)$, and so by no arbitrage $X'_T+qf=0$, which implies $x+qp=0$ and $X_T+qf=0$. This proves that if $(x,q)\in \mathcal{\bar{K}}$ then $x+qp \geq 0$ (so $(1,p)\in  -\mathcal{\bar{K}}^o=\mathcal{\bar{L}})$, and that if  $x+qp=0$ and $X\in \mathcal{X}(x,q)$ then $X_T+qf=0$. Since, we can assume that such an $X$ is workable (thanks to lemma \ref{existswork}), it follows that if $x+qp=0$ then $-X\in \mathcal{X}(-x,-q)$, proving that if $(x,q)$ is in $\mathcal{\bar{K}}$ and is orthogonal to $(1,p)$ then  $(x,q)$ belongs to $\mathcal{\bar{K}} \cap (-\mathcal{\bar{K}})=\{0\}$.

Finally, let us assume that  $\mathcal{\bar{K}}^p(x)$ it non-empty, and show that $ \mathcal{\bar{K}} \cap (1,p)^{\perp}$ is its recession cone, so that \eqref{RecConeBdd} will  conclude the proof.
Since $\mathcal{\bar{K}}^p(x)$ is the set of points $(a,b) \in \mathcal{\bar{K}}$ whose dot product with $(1,p)$ equals $x$, $ \mathcal{\bar{K}} \cap (1,p)^{\perp}$ is contained in the recession cone of $\mathcal{\bar{K}}^p(x)$. For the opposite inclusion assume that $(a,b)$ belongs to $\mathcal{\bar{K}}^p(x)$ and $(c,d)$  belongs to its recession cone, then for any $n$,   $(a,b)+n(c,d) $ belongs to $\mathcal{\bar{K}}^p(x)$, so $(c,d)$ is orthogonal to $(1,p)$ and $(c,d)= \lim_n (a+nc,b+nd)/n \in  \mathcal{\bar{K}}$. $\Box$
 \newline

      \section{Upper semicontinuity of $u$}
 \label{u upper semi-continuous}

    Given an arbitrary vector $(y, r)\in \r^{n+1}$ , we denote by $\mathcal{Y}(y,r)$ the set of
non-negative super-martingales $Y\in \mathcal{Y}(y)$ such that the inequality
\begin{equation}
\label{dualsetrend}
\mathbb{E}[Y_T (X_T + qf)] \leq xy + qr
\end{equation}
holds true whenever $(x, q)\in \mathcal{\bar{K}}$ and $X \in \mathcal{X}(x, q)$.
Clearly this set will be empty if $(y,r)\notin \mathcal{\bar{L}}$, and it coincides with the set defined in \citet*{HugonKram:04} since asking that (\ref{dualsetrend}) holds for all $(x, q)\in \mathcal{\bar{K}}$ is equivalent to asking that it holds for all $(x, q)\in \mathcal{K}$.
Let $\mathcal{D}(y,r)$ be the set of the positive random variables dominated by the final value of some element of  $\mathcal{Y}(y,r)$, i.e.,
 \begin{equation}
 \label{D}
\mathcal{D}(y,r):=\{h\in L_+^0(P) : h\leq Y_T \textrm{ for some } Y \in \mathcal{Y}(y,r) \} ,
\end{equation}
and define
\begin{equation}
\label{C}
\mathcal{C}(x,q):=\{g\in L_+^0(P) : g\leq X_T+qf \textrm{ for some } X \in \mathcal{X}(x,q)\} .
\end{equation}
 Analogously to \eqref{primalSetC} and following \citet*{KramSch:99} we define
 \begin{equation}
 \label{D1}
\mathcal{D}(y):=\{h\in L_+^0(P) : h\leq Y_T \textrm{ for some } Y \in \mathcal{Y}(y) \} .
\end{equation}

 Given two sequences of functions $(g_n)_{n\geq 1}$ and $(f_n)_{n\geq 1}$, we will say that  $(f_n)_{n\geq 1}$ is a \emph{forward convex combination} of $(g_n)_{n\geq 1}$ if, for every $n$, $f_n$ is a (finite) convex combination of $(g_k)_{k\geq n}$.

\begin{lemma}
\label{fatouconv}
Assume that conditions (\ref{na}) and (\ref{domination}) hold.
Let $g_n \in \mathcal{C}(x_n, q_n)$ for some sequence $(x_n, q_n)$ converging to $(x,q)$.
If a forward convex combination of $(g_n)_{n\geq 1}$ converges almost surely to a random variable $g$, then $g \in \mathcal{C}(x,q)$.
Analogously if a forward convex combination of  $h_n \in \mathcal{D}(y_n, r_n)$ is converging almost surely to  $h$ and $(y_n, r_n) \to (y,r)$, then $h \in \mathcal{D}(y,r)$.
\end{lemma}
   \textsf{ PROOF. } The first statement follows applying Fatou's lemma to the converging forward convex combination of $(g_n)_{n\geq 1}$ to obtain
\begin{align}
\label{ineqinC}
 \mathbb{E}[gh]\leq xy+qr \quad \text{ for all } (y,r)\in \mathcal{\bar{L}} \text{ and } h\in \mathcal{D}(y,r),
\end{align}
 so that $g\in \mathcal{C}(x,q)$ follows `from \citet*[Proposition 1, item 1]{HugonKram:04}' (although stated only for $(x,q)\in \mathcal{K}$, this automatically implies that it holds for any $(x,q)\in \mathcal{\bar{K}}$: indeed if  some function $g\in L_+^0(P)$ satisfies  \eqref{ineqinC} for some $(x,q)\in \mathcal{\bar{K}}$  then
$$ \mathbb{E}[(g+1)h]\leq xy+qr+ \mathbb{E}[h]\leq (x+1)y+qr \quad  \text{ for all } (y,r)\in \mathcal{\bar{L}} \text{ and } h\in \mathcal{D}(y,r),$$
 and thus, since\footnote{Recall that the convex cone $\mathcal{\bar{K}}$ contains $(1,0)$ in its interior (see  \citet*[Lemma 1]{HugonKram:04}).} $(x+1,q)\in \mathcal{K}$, from  \citet*[Proposition 1, item 1]{HugonKram:04}  we get  that $g+1\in \mathcal{C}(x+1,q)$, which trivially implies that  $g\in \mathcal{C}(x,q)$).

 The proof of the second statement follows analogously `from   \citet*[Proposition 1, item 2]{HugonKram:04}'; while this item is stated only for $(y,r)\in \mathcal{L}$, clearly\footnote{The proof of this item does not rely on any other lemma in \citet*{HugonKram:04}, and is short.} it is actually valid with the same  proof for any $(y,r)\in \mathcal{\bar{L}}$. $\Box$
 \newline

      \begin{remark}
      \label{Ynonempty}
          Assume that conditions (\ref{na}) and (\ref{domination}) hold. Then $\mathcal{Y}(y,r)\neq \emptyset$ if and only if $(y,r)\in \mathcal{\bar{L}}$.
      \end{remark}
          \textsf{ PROOF. } One implication is trivial. For the vice versa, observe first that $\mathcal{D}(y,r)$ is not empty if  $(y,r)\in \mathcal{L}$, as this follows from \citet*[Lemmas 8 and 9]{HugonKram:04}. If $(y,r)$ is an arbitrary point in $ \mathcal{\bar{L}}$ then take $(y_n,r_n)\in \mathcal{L}$ such that $(y_n,r_n) \to (y,r)$ and choose $h_n\in\mathcal{D}(y_n,r_n)\subseteq \mathcal{D}(\sup_n y_n)$. We can then apply Komlos' lemma to find a forward convex combination of $(h_n)_{n\geq 1}$ converging to some random variable $h$, and apply Lemma \ref{fatouconv} to show that $h\in \mathcal{D}(y,r)$. $\Box$
 \newline

 The following theorem allows us to control the behavior of the maximal expected utility function $u$ on the boundary of its domain.

      \begin{theorem}
      \label{uusc}
Under the assumption of Theorem \ref{afp=mubp} the function
 $u:\r^{n+1}\rightarrow [-\infty,\infty)$ is upper semi-continuous and for all $(x,q) \in \{u>-\infty\}$ there exists a unique  maximizer to (\ref{prrend}).
  \end{theorem}
To prove Theorem \ref{uusc} we will need the following fact, which was proved in the second half\footnote{Starting just after formula (25).} of the proof of \citet*[Lemma 1]{KramSch:03} under the assumption that $\lim_{x\to\infty} u(x)/x=0$, which is satisfied under our hyphotheses since\footnote{This simply follows from the biconjugacy relationship between $w$ and  $\tilde{w}$, proved in \citet*[Theorem 1]{KramSch:03}, and our assumption that $ \tilde{w}(y) <\infty \textrm{  for all } y>0 .$} $u'(\infty)=0$ (as stated in \citet*[Theorem 2]{KramSch:03}) and  $\lim_{x\to\infty} u(x)/x=u'(\infty)$ (by l'Hospital's rule).

 \begin{remark}
      \label{U(C)unifInt}
Assume that conditions (\ref{na}),(\ref{inada})  hold, and that
 \begin{displaymath}
 \tilde{w}(y) <\infty \textrm{  for all } y>0 .
 \end{displaymath}
Let  $(g^m)_{m \geq 1} \subseteq \mathcal{C}(x)$ for some $x>0$; then $(U(g^m)^+)_{m \geq 1}$ is uniformly integrable.

  \end{remark}

  \textsf{ PROOF OF THEOREM \ref{uusc}. }  To prove that $u$ is upper semi-continuous let $(x^k,q^k)$ be a sequence converging to $(x,q)$, assume without loss of generality that $u(x^k,q^k)>-\infty$ and take $X^k \in \mathcal{X}(x^k,q^k)$ such that $u(x^k,q^k)-1/k \leq \mathbb{E}[U(X_T^k+q^kf)]$. Define $g^k:=X_T^k+q^kf$ and $s:=\limsup_k u(x^k,q^k)$. Passing to a subsequence without re-labelling we can assume that $u(x^k,q^k)$ converges to $s$, and so
  \begin{equation}
  \label{ineq1}
  s= \lim_m \left( \inf_{k\geq m} u(x^k,q^k)-1/k \right) \leq \lim_m \left(\inf_{k\geq m} \mathbb{E}[U(g^k)]\right) .
  \end{equation}
  If $x_0^{'}$ is the initial value of the process $X^{'}$ appearing in (\ref{domination}) and $\bar{x}$ is the supremum of the bounded sequence
  \begin{displaymath} \textstyle
  \left(x^k+ x_0^{'} \max_{1 \leq j \leq n}|q_j^k| \right)_{k \geq 1} ,
  \end{displaymath}
   then assumption (\ref{domination}) implies that, for every $m$, $g^m \in \mathcal{C}(\bar{x})$.

  We can then apply Komlos' lemma to find  a forward convex combination $(\tilde{g}^k)_{k\geq 1}$ of $(g^k)_{k\geq 1}$ which is converging almost surely to some random variable $g$. Then Lemma \ref{fatouconv} gives that $g\in \mathcal{C}(x,q)$, and Jensen inequality yields
  \begin{equation}
  \label{ineqjen} \textstyle
  \inf_{k\geq m} \mathbb{E}[U(g^k)] \leq  \mathbb{E}[U(\tilde{g}^m)] .
  \end{equation}
   Since $(\tilde{g}^m)_{m \geq 1} \subseteq \mathcal{C}(\bar{x})$, Remark \ref{U(C)unifInt} says that the sequence $(U(\tilde{g}^m)^+)_{m \geq 1}$ is uniformly integrable.  Thus, Fatou's lemma and inequalities (\ref{ineq1}) and (\ref{ineqjen}) imply that
  \begin{equation}
  \label{ineq3} \textstyle
 \limsup_k u(x^k,q^k)=s\leq \limsup_m \mathbb{E}[U(\tilde{g}^m)] \leq \mathbb{E}[U(g)]\leq u(x,q),
  \end{equation}
  which says that $u$ is upper semi-continuous.
   If $u(x,q)>-\infty$ then one can take $(x^k,q^k)=(x,q)$ in the above, and so (\ref{ineq3}) gives that there exists a maximizer to (\ref{prrend}). Uniqueness follows from the strict concavity of $U$.  $\Box$
 \newline

\section{Proof of the first main theorem}
\label{Proof of the main theorem.}

We will repeatedly need the following simple observation.
 \begin{lemma}
\label{uincr}
 Under the assumptions of Theorem \ref{afp=mubp}, if  $(\tilde{x},\tilde{q}) \in \{u>-\infty\}$ and $  (x,q) \in
   \mathcal{\bar{K}} \setminus \{0\}$ then $u(\tilde{x},\tilde{q})<u(\tilde{x}+x,\tilde{q}+q)$.
\end{lemma}
        \textsf{ PROOF }
 Theorem \ref{uusc} yields the existence of a maximizer $X(x,q)$ to (\ref{prrend}), and \citet*[Lemma 7]{HugonKram:04} provides the existence of a $\tilde{X} \in
\mathcal{X}(\tilde{x},\tilde{q})$ such that $P(\tilde{X}_T+\tilde{q}f
>0)>0$.
     Thus the following inequality holds
     \begin{equation}
     \label{ineq}
     U(X_T(x,q)+\tilde{X}_T+(q+\tilde{q})f)\geq U(X_T(x,q)+qf) ,
     \end{equation}
    and it is a strict inequality with strictly positive
    probability.
The thesis follows integrating the two sides of
    (\ref{ineq}), since the right hand side is
    integrable and has integral $u(x,q)$, and the left hand side has
    integral at most $u(\tilde{x}+x,\tilde{q}+q)$.
$\Box$
 \newline

   In the next proof we will use \eqref{RecConeBdd} without further mention.
\newline

        \textsf{ PROOF. OF THEOREM \ref{afp=mubp}}
     Let $p$ be an arbitrage-free price.  Since $(1,0)\in \mathcal{K}$, the set $\mathcal{\bar{K}}^p(1)$ is non-empty, and Lemma \ref{xacc0} gives that it is bounded. Since \citet*[Lemma 6 ]{HugonKram:04} gives that $\mathcal{\bar{K}}^p(1)$  is closed, the function $u$, which  is  upper semi-continuous (Theorem \ref{uusc}), when restricted to the compact set $\mathcal{\bar{K}}^p(1)$  attains its maximum at some point $(x,q)$, so that $p\in \mathcal{P}(x,q)$. Since $u(x,q)\geq u(1,0)>-\infty$, it follows that
    \begin{displaymath}
    \qquad  \mathcal{P}\subseteq \bigcup_{(x,q)\in \{ u >-\infty \}}  \mathcal{P}(x,q;U)  .
    \end{displaymath}
     Vice versa, assume that $p$ is not an arbitrage-free price and that $u(\tilde{x},\tilde{q})\in \r$, set $y:=\tilde{x}+\tilde{q}p\in \r$ so that  $(\tilde{x},\tilde{q})\in \mathcal{\bar{K}}^p(y)$, and let's prove that $(\tilde{x},\tilde{q})$ is not a maximizer of $u$ on $\mathcal{\bar{K}}^p(y)$, so that $p \notin \mathcal{P}(\tilde{x},\tilde{q})$.
     By Lemma \ref{xacc0} there exists a non-zero
     $(x,q)$ in $\mathcal{\bar{K}} \cap    (1,p)^{\perp}$, the recession cone of $\mathcal{\bar{K}}^p(y)$.
Applying Lemma \ref{uincr} it follows that $u(\tilde{x},\tilde{q})<u(\tilde{x}+x,\tilde{q}+q)$ and so, since $(\tilde{x}+x,\tilde{q}+q)$ belongs to $ \mathcal{\bar{K}}^p(y)$, $(\tilde{x},\tilde{q})$ is not a maximizer of $u$ on $\mathcal{\bar{K}}^p(y)$.  $\Box$
 \newline

 \section{Consequences and related results}
 \label{Consequences and related results}
  As we have seen in Section \ref{counterexample}, to span all
 arbitrage-free prices  it is not  enough  in general to consider marginal  prices based at points in the interior of $ \{ u >-\infty \} $.  This warrants the study of
 $\mathcal{P}(x,q)$ in the case where $(x,q)$ belongs to the boundary
 of $ \{ u >-\infty \} $; in particular we ask which properties that
 hold in the case where $(x,q)$ belongs to the interior $\mathcal{K}$
 of $ \{ u >-\infty \} $ are still true when $(x,q)$ belongs to the
 boundary $\partial \mathcal{K}=\partial \{ u >-\infty \} $.

 First of all, if $(x,q)$ belongs to the interior of $ \{ u >-\infty
 \} $, $\mathcal{P}(x,q)$ is never empty, it is a singleton if and
 only if $u$ is differentiable at $(x,q)$, and can be `computed' using
the sub-differential $\partial u(x,q)$ (as mentioned in \citet*[Remark 1]{HugonKram:04}).

  If $(x,q)$ belongs  to the boundary of $ \{ u >-\infty \} $, $\mathcal{P}(x,q)$ can be
 empty and, if it is not, it is never a singleton.  Indeed the
 following corollary of Theorem \ref{afp=mubp} specifies for which
 points $(x,q)$ the set $\mathcal{P}(x,q)$ is empty, and allows us to
 `compute it' in the same way as if $(x,q)$ belonged to the interior
 of $ \{ u >-\infty \} $. This will allow us to prove
 Theorem \ref{Pnotclosed}, which in turn  implies that
 $\mathcal{P}(x,q)$ is never a singleton.

 \begin{corollary}
   \label{thformula}
   Under the assumptions of Theorem \ref{afp=mubp}, the set $\mathcal{P}(x,q;U)$ is non-empty if and only if
   $\partial u(x,q)$ is non-empty.  If $(y,r)\in \partial u(x,q)$ then
   $y >0$, and \begin{equation}
   \label{formula}
   \mathcal{P}(x,q;U)=\left\{ \, \frac{r}{y} : (y,r)\in \partial
     u(x,q)\right\}.  \end{equation}
 \end{corollary}
 \textsf{ PROOF. }  Given $p\in\r^n$ consider the linear function
 $A_p:\r^n \to \r\times \r^n$ given by $A_p(q'):=(-q'p,q')$ and its
 adjoint $A_p^{\star}$ given by $A_p^{\star}(y,r)=r-yp$. Now fix $x\in
 \r,q\in \r^n$, and consider the proper concave function $f_p$ given
 by $f_p(q')=u((x,q)+A_p(q'))$, so that by definition $p$ is a
 marginal price at $(x,q)$ if and only if $0$ is a maximizer of $f_p$
 , i.e., iff $0 \in \partial f_p(0)$

 If $(y,r)\in \partial u(x,q)$ then $y >0$, since $ f(t):=u(t,q)$ is
 strictly increasing on $[x,\infty)$ (by Lemma \ref{uincr}) and $y
 \in \partial f(0)$.  We can then define $p=r/y$ and
 \citet*[Theorem 23.9]{Rock:70} implies that \begin{displaymath} 0 \in A_p^{\star}( \partial u(x,q))
 \subseteq \partial f_p(0), \end{displaymath}  i.e., $p\in \mathcal{P}(x,q)$.

 For the opposite inclusion assume $p\in \mathcal{P}(x,q)$, so that by
 definition $(x,q)\in \mathcal{\bar{K}}$ and, by Theorem
 \ref{afp=mubp} and Lemma \ref{xacc0}, $(1,p)$ belongs to $
 \mathcal{L}$. It follows from \eqref{LK>0} that
 $y:=x+qp>0$, and so \citet*[Lemma 1]{HugonKram:04} implies that \begin{displaymath}
 (x,q)+A_p(-q)=(y,0)\in \mathcal{K} , \end{displaymath} and now another application
 of \citet*[Theorem 23.9]{Rock:70} yields \begin{displaymath} A_p^{\star} ( \partial
 u(x,q) )= \partial f_p(0) \ni 0 , \end{displaymath}
 i.e., $r-py=0$ for some
 $(y,r)\in \partial u(x,q)$.  $\Box$ \newline

 \textsf{ PROOF OF THEOREM \ref{Pnotclosed}} By Corollary \ref{thformula} there exists a $(y_0,r_0) \in \partial
u(x,q)$ such that $r_0/y_0=p_0$. The non-empty convex closed set $\partial u(x,q)$ is unbounded
 (see \citet*[Theorem 23.4]{Rock:70}), so \eqref{RecConeBdd} says that its recession cone contains a
 non-zero element $(y,r)$. For $t\geq 0$ define
\begin{equation}
\label{ytrt}
(y_t,r_t):= (y_0,r_0)+ t (y,r) \in \partial u(x,q) ,
\end{equation}
 and notice that Corollary \ref{thformula} implies that
$v_t:=r_t/y_t \in \mathcal{P}(x,q)$. Since  the function $(y,r) \mapsto r/y$, defined for $y>0$, sends injectively segments into segments  (see \citet*[Section 2.3.3]{BoydVan:04}) and $v_t\to r/y=:p$ as $t \to \infty$, $\{v_t: t \geq 0\}$ equals the segment $[p_0,p)$.
We now only need to prove that $p$ is not an arbitrage-free price.
From (\ref{ytrt}) it follows that, for every $(a,b) $ in $ \{ u >-\infty \} $,
\begin{displaymath}
u(a,b) \leq u(x,q) + (y_t,r_t)( (a,b) - (x,q)).
\end{displaymath}
Dividing times $t$ and taking the limit as $t \to \infty$ yields that
\begin{displaymath}
 0\leq (y,r)( (a,b) - (x,q)) .
\end{displaymath}
Thus, for any $\varepsilon>0$  choosing $(a,b):=(x/2,q/2)+\varepsilon (1,0) \in \mathcal{K}\subseteq \{ u >-\infty \} $  we get that $  (y,r) (x,q) \leq 2 \varepsilon y $.
Sending $\varepsilon$ to zero we obtain $  (y,r) (x,q) \leq 0$,
and so \eqref{LK>0} implies that  $ (y,r) $ does not belong to $\mathcal{L}$.
Lemma \ref{xacc0} now concludes the proof.
 $\Box$ \newline

Theorem \ref{Pnotclosed} implies that marginal prices based at a point $(x,q)$   on the boundary of $\mathcal{\bar{K}}$, if they exist, are not
  unique.  This raises the question of how to characterize
  the uniqueness of marginal prices based at all points, which is the content of the following corollary, where the function $v$ is the convex conjugate of $u$ (see \citet*{HugonKram:04}).

  \begin{corollary}
    \label{uniq}
   Under the assumptions of Theorem \ref{afp=mubp}, the following are equivalent:
    \begin{enumerate}
    \item
\label{1} At any $(x,q) \in \r \times \r^n$ the set $\mathcal{P}(x,q;U)$ is either empty or a singleton.
\item
\label{2} $\mathcal{P}(x,q;U)$ is a singleton at any $(x,q)\in    \mathcal{K}$, and is empty at any $(x,q) \notin \mathcal{K}$.
    \item
\label{3}
$u$ is continuously differentiable on $\mathcal{K}$ and the
      norm of its gradient $|\nabla u(x,q)|$ converges to $\infty$ as
      $(x,q)$ approaches the boundary of $\mathcal{K}$.
    \item
\label{4}
v is strictly convex on $\mathcal{L}$.
    \end{enumerate}

  \end{corollary}

\textsf{ PROOF }
In this proof we will use Corollary \ref{thformula} without further mention.
Theorem \ref{Pnotclosed} shows that item \ref{1} is equivalent to item \ref{2}.
Since  $\partial u(\mathcal{\bar{K}})=  \mathcal{L}$ (see \citet*[Theorem 10, item 3]{Sio12Merge}), item \ref{3} is equivalent to item \ref{4} (see \citet*[Theorem 26.3]{Rock:70}).
Finally, item \ref{2} is equivalent to item \ref{3}, as it follows from \citet*[Theorem 26.1]{Rock:70} and the fact that $\partial u$ is single-valued iff $\mathcal{P}$ is single-valued (because $y \mapsto v(y(1,p))$ is always strictly  convex when $(1,p) \in \mathcal{L}=\partial u(\mathcal{\bar{K}}))$. $\Box$ \newline

\bibliographystyle{apalike}

redpajama_set_name: RedPajamaArXiv


text:
package org.apache.shardingsphere.test.integration.framework.param.model;

import org.apache.shardingsphere.infra.database.type.DatabaseType;
import org.apache.shardingsphere.test.integration.cases.IntegrationTestCaseContext;

/**
 * Parameterized array.
 */
public interface ParameterizedArray {

    /**
     * Get test case context.
     *
     * @return test case context
     */
    IntegrationTestCaseContext getTestCaseContext();

    /**
     * Get scenario.
     *
     * @return scenario
     */
    String getScenario();

    /**
     * Get adapter.
     *
     * @return adapter
     */
    String getAdapter();

    /**
     * Get database type.
     *
     * @return database type
     */
    DatabaseType getDatabaseType();

    /**
     * Get key.
     *
     * @return key of parameterized array
     */
    default String getKey() {
        return String.join("-", getScenario(), getAdapter(), getDatabaseType().getName());
    }
}

redpajama_set_name: RedPajamaGithub


text:
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By Andrea Torres
MIAMI - Monica Lazaro has defied all odds. After graduating from Coral Gables High School, her 40-year-old mom died of cancer. Students organized a fundraiser and with the help of an anonymous donor she was able to study biology at Miami-Dade College and Florida International University.
Lazaro is working as a research associate at Nova Southeastern University. She is studying chronic fatigue syndrome. She made her dad proud when she received a security clearance to work at the Department of Veterans Affairs.
The future of the 24-year-old aspiring epidemiologist could soon be derailed. She is among the estimated 800,000 migrants waiting for President Donald Trump to make a decision that could send them back to the shadows of illegality and a life in fear -- now that the government knows everything about them.
"Keep your hopes up! Keep fighting! Let's be strong," Lazaro said to a crowd of students and activists during a recent conference at Miami Dade College.
Lazaro was born in Honduras, which has one of the highest murder rates in the world. She moved to Miami when she was 9 years old, and when Honduras already had a reputation for having more youth gang members than any other country in Central America.
Gang members threatened her dad's life. She and her brother traveled with their parents from San Pedro Sula to Miami on tourist visas that later expired. Her mom, who was a domestic worker and entrepreneur, and her dad, who was a construction worker, talked to her about the importance of not disclosing her immigration status.
Lazaro lived with the secret until President Barack Obama's five-year-old executive order for the Deferred Action for Childhood Arrivals changed her life. After paying a $495 application fee, she was able to get a temporary DACA card and she was no longer undocumented.
As of last year, DACA made it possible for her and tens of thousands of others in Florida to live without the fear of deportation, obtain a driver's license, qualify for in-state college tuition and secure better jobs. The program only applies to migrants who moved to the U.S. before their 16th birthday and have continuously lived in the U.S. since June 15, 2007.
California, Texas, Illinois and New York are the only other states in the country to have more DACA recipients than Florida. The majority are from Mexico, El Salvador, Guatemala and Honduras, the three countries where drug-trafficking operations continue to fuel the proliferation of gang activity.
Trump is under pressure to keep the Central American crisis out of the U.S., and conservatives believe this can be done with a border wall and tougher immigration enforcement. They want him to abolish DACA or refuse to defend it in court to put an end to it.
Executives from Apple, Google, Facebook, Microsoft and other tech giants joined more than 100 law professors who were asking Trump to keep the DACA program.
"All DACA recipients grew up in America, registered with our government, submitted to extensive background checks, and are diligently giving back to our communities and paying income taxes," their Friday letter said. "Our economy would lose $460.3 billion from the national GDP and $24.6 billion in Social Security and Medicare tax contributions."
Attorney General Jeff Sessions is among those in government who oppose DACA. Attorneys generals from 10 states demanded the suspension of DACA and threatened fo file suit unless Trump begins to dismantle the program by Sept. 5. Sessions would be responsible for defending DACA if there was a lawsuit.
Miami Rep. Ileana Ros-Lehtinen, a Cuban-American Republican, and Rep. Debbie Wasserman Schultz, a Democrat, are encouraging others to defend DACA and push for the Dream Act of 2017.
Sen. Lindsey Graham, a Republican from South Carolina, introduced the bill in the senate July 20. The legislation aims to protect minors who have continuously been in the U.S. for at least four years before the law's enactment. The new bill would create a path toward a green card and U.S. Citizenship.
In South Florida, the legislation has the support of Miami-Dade College President Eduardo Padrón, Miami-Dade Schools Superintendent Alberto Carvalho and Broward Schools Superintendent Robert Runcie. Miami Mayor Tomás Regalado is part of a group of mayors who are asking Trump to keep DACA.
An army of Immigration advocates such as the Immigration Partnership and Coalition Fund's Mike Fernandez, Americans for Immigrant Justice's Cheryl Little, Stand Up for Dreamers and The Florida Immigrant Coalition are doing what they can to help DACA recipients.
"We are just making sure they feel protected and supported," said Melissa Taveras with the Florida Immigrant Coalition.
Local authorities and immigration advocates were reminding parents who are afraid that U.S. Immigration and Customs Enforcement agents don't just come in and take children from schools. The prospect is highly unlikely.
During the Wednesday meeting in downtown Miami, Lazaro spoke for other DACA recipients from around the nation who are too afraid to stand in public. Records show some of the DACA recipients who consider the U.S. home were born in South Korea, Peru, Brazil, Ecuador, Colombia, Philippines, Argentina, Jamaica, India, Venezuela, Dominican Republic, Trinidad and Tobago, Uruguay, Costa Rica, Bolivia, Pakistan, Poland, Chile, Nicaragua and Nigeria.
While they wait, Lazaro asked the public to call their elected representatives and ask them to support DACA and the Dream Act. She was planning on attending a vigil from 7 to 9 p.m., Friday, at the Freedom Tower in downtown Miami.
White House press secretary Sarah Huckabee Sanders said when Trump makes his decision Local 10 News Ross Palombo will be informed and there will be a press conference to answer questions. A senior administration official told Reuters Trump was expected to rescind DACA as early as Friday.
"There are a lot of components that need to be looked at," Sanders said.
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MRSM: Nurturing Students In Rural Areas With The Best Education System
Posted on March 6, 2017	by admin
Sultan of Perak Sultan Nazrin Muizzuddin Shah and Raja Permaisuri of Perak Tuanku Zara Salim visiting an exhibition in Sultan Azlan Shah MRSM in Kuala Kangsar on Thursday. (Picture by Muhaizan Yahya).
SULTAN of Perak Sultan Nazrin Muizzuddin Shah officiated at the renaming ceremony of the Kuala Kangsar Mara Junior Science College (MRSM) here on Thursday, to the Sultan Azlan Shah MRSM.
It is named after his father and the 34th ruler of the state, Sultan Azlan Muhibbuddin Shah.
In a brief address, Sultan Nazrin proclaimed the change of name and signed a plaque to commemorate the occasion. occassion.
The MRSM colleges nationwide are significant for their contribution in empowering the Bumiputeras through education.
The Sultan Azlan Shah MRSM, located on 20ha of land hectares of land at Lot 2344, Liman Kati, Kuala Kangsar, was established on July 6, 2015, with 36 management and administration staff under its principal, Shahrome Daud.
After about more than two years in operation, Sultan Azlan Shah MRSM has grown to 725 students; 347 male and 378 female students, 51 teachers and 29 support staff.
The eighth MRSM in the state and 50th in the country has became a catalyst to the educational development at the Kota Lama Kiri area.
The strategic location, located 15km away from Kuala Kangsar town and a 10- minutes journey from the North South Expressway Kuala Kangsar toll exit, it offers easy access and mobility for parents and students. It also plays a vital role in developing the locals in economic and social sectors.
The MRSM education system focuses on four different domains that have been carefully planned — curriculum development, student growth, professional advancement, facilities and service improvement.
This is in accordance with the STREAM emphasis —, in Science; Technology; Religion; Engineering; Arts and Mathematics. ; in the MRSM education system.
Rural and Regional Minister Datuk Seri Ismail Sabri Yaakob said the students from rural areas should be nurtured with from the best educational system at Mara educational institutions, such as MRSM, to produce first class human capital at a global level, armed with integrity to face future challenges.
"In conjunction with the ministry''s transformational agenda, with Mara as an agency under it, we are committed to make Mara as an important agency to in carrying out its responsibility and to ensure that the rural communities nationwide do not miss out in the era of modernisation.
Mara prioritises rural students, the hardcore poor and urban poor with several forms of forms of assistance, such as financial and school equipment aid, based on the social and economic status level of the student.
This in done with the hope that the students would then become the agents of change to the rural communities, Ismail said. he added.
Ismail Sabri said there were many successful Malaysian icons in Malaysia who were former MRSM students. These include the nation''s first astronaut Dr Syed Sheikh Muzafffar Shukor and AirAsia chief executive officer Datuk Kamaruddin Meranun, a well-known Bumiputera corporate figure at international level.
The MRSM's history of MRSM can be traced back to when Mara requested for an special allocation for development expenditure over a five-year period in a paper, named the "Batik Paper" with the aim of building a Bumiputera secondary education institution, known as Mara Junior Science College. It usesd the New York Bronx High School of Science model.
The cost of first MRSM in Seremban was RM1 million, in which where two teachers from New York Bronx High School of Science . were brought in to write MRSM's framework as what is seen today.
At that time, the MRSM syllabus was the same as government schools. However, the curriculum implementation was different, where the learning session required students to search for the information themselves, based on the learning by discovery method.
In 1972, the first MRSM in Seremban started its operations with an the first intake of 150 students. The building was a renovated Negeri Sembilan Institute of Public Administration (Intan) Training Centre, which consisted of three laboratories, five classrooms, an office garage and a dining hall.
Now with 52 MRSM nationwide and more than 175,000 graduates, students having graduated from the college, most of the alumni have gone on to forge careers as political leaders, corporate and government leaders, government department directors, successful entrepreneurs and professionals.
Various innovations were executed by Mara at MRSM colleges nationwide, such as the implementation of two-certificate, a collaboration of national and international co-curriculum such as International General Certificate of Secondary Education (IGCSE) and the Ulul Albab programme.
Sumber dari : http://www.nst.com.my/news/2017/03/217577/mrsm-nurturing-students-rural-areas-best-education-system
This entry was posted in Pendidikan and tagged kklw, MARA, MRSM. Bookmark the permalink.


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<?php
namespace Kr\Bol\Http;

use Kr\Bol\Exception\UnauthorizedException;
use Kr\Bol\Exception\InvalidXmlException;
use Kr\Bol\Exception\ApiLimitException;
use Kr\Bol\Exception\BolException;
use GuzzleHttp\Exception\RequestException;

use Kr\Bol\Utils\XmlParser;

class ExceptionHandler
{
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    {
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     * Captch known exceptions
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     * @throws \Exception
     */
    public function handle(RequestException $e)
    {
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        $body       = $response->getBody(true);
        $message    = $this->xmlParser->parse($body);

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}


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Google rolls back Android Messages to old design and removes dark mode
11 monthes ago - By The Verge
On Thursday, it looked as though Google was starting to roll out a redesigned version of its vital Android Messages app. The update featured some visual changes like getting rid of the blue bar at the top and switching over to the company's Google Sans font. It also included a dark mode as an alternative to the default, so-much-white look.
But late last night, Google reverted back to the old Android Messages design - even for those who had already installed the most recent update. If you had the new Messages yesterday, open it up today and you'll see that it's gone. Yes, Google has the...
Google Goggles app gets updated to announce its death
If you're an Android old-timer that remembers when dinosaurs ruled the Earth (and by dinosaurs we mean the Motorola DROID and the Nexus One), when you opened what was called back then the Android Market, you would inevitably have your attention drawn to an app called Google Goggles. Maybe it was the alliteration that caught the eye. The app was launched in December 2009, and Google has now updated it for the first time since 2014 just to announce its death.
Google Goggles was developed to identify objects that users snapped with their camera. It also read QR and ...
Google Android News


The Yuma Community Center was built in 1991 as a place to rent a room for meetings, weddings, and birthday parties as well as have Community Events. Rent the building for an hour or as a Party Package for the whole day. We offer three different size rooms to accommodate almost every event. The Community Center is perfect for a board meeting of 10 or a wedding party of 350!
The mission of the City of Yuma Parks and Recreation Department is to provide diverse year-round leisure opportunities through park settings, recreational activities, recreational facilities and programs for the citizens of Yuma. The benefits of the leisure services are especially designed to meet the physical, mental, cultural, and social needs of our residents and visitors to our community, while enhancing the overall quality of life in Yuma.
In January of 1924 twelve women of the Methodist church met and organized the Yuma Woman's Club. The purpose of this group was to establish a public library. The Library collection was housed in closed cases at the town hall and began with just 40 books. This collection soon grew to 1200 and on January 31, 1925 the library was opened to the public. Mrs. Door Rising was the first librarian. The Library was open four hours per week.