3
$\begingroup$

Let $U,H$ be a separable $\mathbb R$-Hilbert spaces, $M$ be a $U$-valued square-integrable martingale on a filitered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ with martingale covariance $Q$, $\Phi:\Omega\times[0,T]\to\mathfrak L(U,H)$ be a predictable process with $$\operatorname E\left[\int_0^T\left\|\Phi_sQ_s^{1/2}\right\|_{\operatorname{HS}(U,\:H)}^2\:{\rm d}[M]_s\right]<\infty$$ and $\Phi\cdot M$ denote the Itō integral process.

Above $[M]$ denotes the (scalar) quadratic variation of $M$, i.e. the unique nondecreasing adapted process such at $\left\|M\right\|_H^2-[M]$ is a square-integrable martingale. Moreover, if $[\![M]\!]$ denotes the tensor-quadratic variation of $M$, i.e. the unique adapted process with values in the space of nuclear, nonnegative and self-adjoint operators on $U$ such that $M\otimes M-[\![M]\!]$ is a square-integrable martingale, then $$[\![M]\!]_t=\int_0^tQ_s\:{\rm d}[M]_s\;\;\;\text{for all }t\in[0,T]\;.$$

Now, if $N$ is another $U$-valued square-integrable martingale, can we find an expression for the covariation $[\Phi\cdot M,N]$?

I'm also interested in an expression for the tensor-covariation $[\![\Phi\cdot M,N]\!]$ where $N$ might take values in another separable $\mathbb R$-Hilbert space $\tilde H$.

All I was able to find in books are the classical results $$[\Phi\cdot M,\Psi\cdot M]_t=\int_0^t\langle\Phi_sQ_s^{1/2},\Phi_sQ_s^{1/2}\rangle_{\operatorname{HS}(U,\:H)}\:{\rm d}[M]_s$$ and $$[\![\Phi\cdot M]\!]_t=\int_0^t\Phi_sQ_s\Phi_s^\ast\:{\rm d}[M]_s\;.$$

Remark: Cross-posted on mathematics.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.