# martingale decomposition problem

Let $G_{t}$ be a filtration and $M_{t}$ a $G_{t}$-martingale. Why do we have this decomposition: $H_{t}=\mathbb{E}[H|G_t]=\int_{0}^{t}h_{s}dM_{s}+R_{t}$ where $R_{t}$ is a martingale orthogonal with M

Thank you.

• Is the expectation without the conditioning on $G_t$? May 5, 2015 at 9:05
• God sorry for that . Yes it is $H_{t}=\mathbb{E}[H|G_{t}]$ May 5, 2015 at 9:25
• You may need some more details, such as what is $h$ etc. May 5, 2015 at 13:23
• To be helpful, you may provide a reference where you have this decomposition. May 5, 2015 at 13:38
• in fact it's not precised what is $h$ ...it's just written: Let $H_{t}=\mathbb{E}[H|G_{t}]=\int_{0}^{t}h_{s}dM_{s}+R_{t}$ with R martingale orthogonal to M. I don't know how to upload a pdf file where you could see the context in which it's written May 5, 2015 at 16:57

## 1 Answer

As per your comments, this is the Kunita Watanabe decomposition. See the post at https://math.stackexchange.com/questions/413103/kunita-watanabe-decomposition and the presentation http://www.eurandom.nl/events/workshops/2011/ISI_MRM/Presentation/Vanmaele.pdf

• thank you very much for the link you provided. I can now put a name on this decomposition ^^ May 5, 2015 at 17:55