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Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:

$$V_t=V_0+\int_0^tH_sdX_s-K_t$$

where $H$ is a $X$-integrable predictable process and $K$ is an adapted increasing process with $K_0=0$. How can we characterize $H$ by our observations $V$ and $X$, i.e. how to express $H$ by $V$ and $X$? Thanks a lot for the reply!

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  • $\begingroup$ In fact, by the uniqueness of Doob-meyer decompositoin we know $\int_0^tH_sdX_s$ is the local martingale part. Thus intuitively, this part should be uniquely determined by $V$. Combined with our assumption, I believe that $H$ should be characterized uniquely by $V$ and $X$, but I do not know how to express explicitly $\endgroup$
    – CodeGolf
    Mar 28, 2014 at 14:06
  • $\begingroup$ If $V$ and $X$ are continuous, it is easy to find (formally) that $H_t=\frac{d<V,X>_t}{d<X,X>_t}$, but i do not know for the cadlag case $\endgroup$
    – CodeGolf
    Mar 28, 2014 at 14:07
  • $\begingroup$ @ CodeGolf : Well in the càdlàg case I believe that there is no uniqueness in the decomposition of $V$ unless you add some assumptions on $H$ for example predictability for special semimartingale might work for example. Best regards. $\endgroup$
    – The Bridge
    Nov 25, 2014 at 21:45

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