Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it mean that there exists a process $H_s$ adapted to $\mathcal{F}_t^X$ and constant $x \in \mathbb{R} $ such that $$ \xi = x + \int_0^T H_sdX_s $$

The statement is true for Brownian Motion(Martingale representation theorem). However, for general semimartingales I am not able to proceed. Any references/help is highly appreciated. Thanks


1 Answer 1


The statement is false for general semimartingales. It is also false for smaller families of semimartingales, such as local martingales or even Lévy processes.

Consider the following definition. Let $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$ be a filtered probability space satisfying the usual conditions, and let $M$ be a local martingale with initial value zero. Write $\mathcal{L}(M)$ for the set of semimartingales $X$ given by $X_t = \int_0^t H_s dM_s$ where $H$ is some predictable process integrable with respect to $M$, that is, the space of processes representable as an integral of a predictable process with respect to $M$. We then say that $M$ has the strong property of predictable representation if $\mathcal{L}(M)$ is the space of all local martingales. Thus, $M$ has the strong property of predictable representation if all local martingales can be represented as stochastic integrals with respect to $M$.

The following then holds:

$\mathbf{Theorem.}$ Let $X$ be a Lévy process with initial value zero which is also a martingale. Then $X$ has the strong property of predictable representation if and only if $X$ is a scaled Brownian motion or a scaled compensated Poisson process.

Thus, even among Lévy processes, the only processes for which the representation theorem holds are Brownian motions and compensated Poisson processes. In particular, for Lévy processes not being Brownian motions or compensated Poisson processes, it will be possible to find $\xi$ as in your question (perhaps unbounded, though, but I doubt that'll make a difference) for which the representation does not hold.

You can find a statement and proof of the Theorem in the book "Semimartingale theory and stochastic calculus" by He, Wang and Yan. The statement of the result is Corollary 13.54.

The book "Stochastic integration and differential equations" by Philip Protter also contains some results in this direction.


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