I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect something like that:
Assume that $L(t)$ is Levy process and $X_n(t)$ a sequence of stochastic processes such that $\lim_{n \to \infty}X_n(t)=X(t)$ in a monotonous way (limit defined in some sense, for example in distribution). Then $\lim_{n \to \infty} \int_{-\infty}^t X_n(s-)dL_s=\int_{-\infty}^t X(s-)dL_s$.
Is it true? If it is, do you have any hints how to prove it and/or where I could find a proof?
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$\begingroup$ How do you define $\int\dots dL_s$? $\endgroup$– zhorasterCommented Feb 3, 2017 at 8:40
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$\begingroup$ In a "convenient way". So can be Ito integral, can be Stratonovich integral. I'd be happy to find a result for any of these definitions. $\endgroup$– PaulaCommented Feb 3, 2017 at 11:46
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$\begingroup$ And what do you need these results for? Anyway, the chances are minimal to get something making use of monotonicity and not included in existing general results. Maybe only when $L$ is a subordinator: the integral is defined pathwise in Lebesgue-Stieltjes sense, so you can use deterministic monotone convergence theorem. $\endgroup$– zhorasterCommented Feb 3, 2017 at 13:09
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$\begingroup$ I'm actually still trying to solve the question I asked before: mathoverflow.net/questions/259991/…. Could you please explain the subordinator case? $\endgroup$– PaulaCommented Feb 3, 2017 at 13:37
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2 Answers
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Firstly, to give the formal definition of integral wrt Lévy process is necessary for prove the "MCT"(monotone convergence theorem).
Secondary, for the most definitions of stochastic integrals(in path, $L^1$, $L^2$ or locally etc.), the MCT could be deduced by DCT(dominated convergence theorem), if the $X_1$ and $X=\lim\limits_{n\to\infty}X_n$ are supposed integrable. Since $$ \sup_{n,m\ge 1}|X_n-X_m|\le 2\max(|X_1|,|X|).$$
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To add to the answer by JGWang: The MCT is useful beyond the DCT only when we do not know that the limit is integrable. But in this context, the MCT depends crucially on some form of nonnegativity so that we can define the putative integral of the limit. We will not have this for stochastic integrals.
