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* keywords * : model uncertainty ; stochastic differential equations ( sdes ) ; probability measures ; hellinger distance ; stationary probability density ; parameter estimation ; numerical simulation          mathematical models for scientific and engineering systems often involve with some uncertainties .
we may roughly classify such uncertainties into two kinds .
the first kind of uncertainties may be called _ model
uncertainty_. they are due to physical processes that are not well understood or not well - observed , and thus are not or not well represented in the mathematical models .
the second kind of uncertainties may be called _ simulation uncertainty_. this arises in numerical simulations of multiscale systems that display a wide range of spatial and temporal scales , with no clear scale separation . due to the limitations of computer power , not all scales of variability
can be explicitly simulated or resolved .
these uncertainties are sometimes also called _ unresolved scales _ , as they are not represented ( i.e. , not _ resolved _ ) in modeling or simulation .
although these unresolved scales may be very small or very fast , their long time impact on the resolved simulation may be delicate ( i.e. , may be negligible or may have significant effects @xcite , or in other words , uncertain ) .
thus , to take the effects of unresolved scales on the resolved scales into account , representations or parameterizations of these effects are desirable .
model uncertainties have been considered in , for example , @xcite .
research works relevant for parameterizing unresolved scales include , @xcite among others .
stochastically representing unresolved scales in fluid dynamics has considered as well @xcite .    in this chapter , we only consider model uncertainties .
specifically , we consider dynamical systems containing uncertain parameters or unknown system functions , and examine how to estimate these parameters , using observed probability distributions of the system evolution .
after briefly comment on estimating uncertain parameters based on observed sample paths for the system evolution in  [ paths ] , we then , in  [ measures ] , propose a method of estimating uncertain parameters based on observed probability distributions ( i.e. , probability measures ) and present a few examples to demonstrate this method , analytically or numerically .
since random fluctuations are common in the real world , mathematical models for complex systems are often subject to uncertainties , such as fluctuating forces , uncertain parameters , or random boundary conditions @xcite .
stochastic differential equations ( sdes ) such as @xmath0 are appropriate models for many of these systems @xcite . here
@xmath1 is a brownian motion or wiener process , the drift @xmath2 and diffusion @xmath3 contain uncertain parameters ( or @xmath4 and @xmath5 are unknown ) , to be estimated based on observations .
for example , the langevin type models are stochastic differential equations describing various phenomena in physics , biology , and other fields .
sdes are used to model various price processes , exchange rates , and interest rates , among others , in finance .
noises in these sdes may be modeled as a generalized time derivative of some distinguished stochastic processes , such as brownian motion ( bm ) or other processes .
theoretical results on parameter estimations for sdes driven by brownian motion are relatively well developed @xcite , and various numerical simulations for these parameter estimations @xcite are also implemented .
see @xcite for a more recent review about estimating and computing uncertain parameters , when dynamical systems are submit to colored or non - gaussian noises
.    these research works on estimating uncertain parameters in dynamical systems are based on observations of sample paths . in the next section ,
we devise a method to estimate uncertain parameters based on observations of probability distributions of the system evolution .
consider a dynamical system with model uncertainty , modeled by a scalar sde @xmath6 where the drift @xmath2 and diffusion @xmath3 contain uncertain parameters , to be estimated based on observations of probability distributions ( i.e. , probability measures ) of the system paths @xmath7 .    to this end , we need to introduce the hellinger distance @xcite between two probability measures .
it is used to quantify the similarity between two probability distributions .
this is a metric in the space of probability measures .    for our purpose here
, we define the hellinger distance @xmath8 between two probability density functions @xmath9 and @xmath10 as follows @xmath11 the hellinger distance @xmath12 satisfies the property : @xmath13 .
we estimate uncertain parameters by minimizing the hellinger distance between the true probability density @xmath14 for the solution process @xmath15 and its observed probability density @xmath16 . in reality , the probability density @xmath14 has to be numerically formulated or discretized .
but in order to demonstrate the method , we consider two examples for which the true probability density @xmath14 can be analytically formulated . in the first example , we minimize the hellinger distance between the true stationary probability density for the solution process @xmath15 and its observed stationary probability density , while in the second example , we do this for time - dependent probability densities .      under appropriate conditions on @xmath17 and @xmath18 ( see @xcite , p.170 ) , such as , @xmath19 and @xmath20 as well as some smoothness requirements , there exists a stationary probability density @xmath21 for the sde , as a solution of the steady fokker - planck equation @xmath22 , @xmath23 where the positive normalization constant @xmath24 is chosen so that @xmath25 and @xmath26 , i.e. , @xmath27 note that @xmath28 here may be an arbitrary point so that the integral @xmath29 exists ( say , take @xmath30 if that is possible ) .
\(i ) a special case : langevin equation + @xmath31 with parameter @xmath32 . given an  observation " of the stationary probability density @xmath33 . find a @xmath17 so that the hellinger distance @xmath34 ^ 2 dx$ ] is minimized .
\(ii ) a more general case : + @xmath35 with function @xmath36 . given an  observation " of the stationary probability density @xmath37 ( the cauchy distribution ) . find a function @xmath36 so that the hellinger distance @xmath38 ^ 2 dx$ ] is minimized .
insert @xmath40 into the hellinger distance @xmath41 , which is now an algebraic function of parameter @xmath32 .
thus we use deterministic calculus to find a minimizer @xmath17 ( possibly by hand , or matlab if needed ) .
note : @xmath42 .
insert @xmath40 into the hellinger distance @xmath47 , which is now a functional of @xmath48 and thus we use calculus of variations ( on @xmath47 ) to find a minimizer @xmath48 .
we then derive the euler - lagrange equation to be satisfied by @xmath48 , together with appropriate boundary conditions ( needed for @xmath49 and @xmath50 ) .
consider a scalar sde @xmath70 the fokker - planck equation @xcite for the probability density @xmath71 for the solution @xmath72 is @xmath73 with an observation of @xmath74 , we can estimate parameters , or @xmath75 , or @xmath5 , by examining the inverse problem of the fokker - planck equation . for more information about inverse problems of partial differential equations ,
see @xcite .
* solution : * + the fokker - planck equation for is @xmath81 in this case , we define the hellinger distance : @xmath82}\int_{-\infty}^{\infty } ( \sqrt{q_i(x , t)}-\sqrt{p(x , t , b)})^2 dx\ ] ] where @xmath83 , and @xmath84 is the time period when @xmath85 are observed .
we numerical find @xmath17 by minimizing @xmath86 .        by the definition @xmath90}\int_{-\infty}^{\infty } ( \sqrt{q_1(x , t)}-\sqrt{p(x , t , b)})^2 dx$ ]
, we have the plot of @xmath91 in figure [ pq1 ] . and
whatever @xmath84 is , @xmath91 is always minimized when @xmath92 .
this gives us the parameter value @xmath92 .     and
the observation @xmath87 : @xmath93 ( top ) and @xmath94 ( bottom).,title="fig:",width=245,height=188 ]   and the observation @xmath87 : @xmath93 ( top ) and @xmath94 ( bottom).,title="fig:",width=245,height=188 ]        by the definition @xmath96}\int_{-\infty}^{\infty } ( \sqrt{q_2(x , t)}-\sqrt{p(x , t , b)})^2 dx$ ] , we have the plot of @xmath97 in figure [ pq2 ] .
so we see that if @xmath93 , @xmath97 is minimized when @xmath98 and if @xmath94 , @xmath97 is minimized when @xmath99 .     and the observation @xmath95 : @xmath93 ( top ) and @xmath94 ( top).,title="fig:",width=245,height=188 ]   and the observation @xmath95 : @xmath93 ( top ) and @xmath94 ( top).,title="fig:",width=245,height=188 ]                  l.  arnold , hasselmann s program visited : the analysis of stochasticity in deterministic climate models . in j .- s .
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arxiv : 0912.0280
due to lack of scientific understanding , some mechanisms may be missing in mathematical modeling of complex phenomena in science and engineering . 
 these mathematical models thus contain some uncertainties such as uncertain parameters . 
 one method to estimate these parameters is based on pathwise observations , i.e. , quantifying model uncertainty in the space of sample paths for system evolution . 
 another method is devised here to estimate uncertain parameters , or unknown system functions , based on experimental observations of probability distributions for system evolution . 
 this is called the quantification of model uncertainties in the space of probability measures . 
 a few examples are presented to demonstrate this method , analytically or numerically .