# One question about proof of martingale representation theorem

Does any one know which book I can find the proof of martingale representation theorem in detail? I.E. Any $F_B$ adapted local martingale M is continuous and can be written as a integration of Brownian.

There is a proof on my notes. It says there exists a sequence of stopping times {$T_n$} such that $M^{T_n}$ is a bounded martingale. But we don't know M is continuous, I don't know how to construct a well-defined sequence of stopping times to make $M^{T_n}$ be a bounded martingale.

Thanks!

-

## 1 Answer

There are many references for example Philip E. Protter's book Stochastic Integration and Differential Equations.

On the net George Lowther's Blog Almost sure where this post should do the trick.

Best regards

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9360478520393372, "perplexity_flag": "head"}