Mean value theorems for the noncausal stochastic integral

Abstract

We are concerned with the validity of the mean value theorem for the noncausal stochastic integral $\int _s^t f(X_r)d_*W_r$ with respect to Brownian motion $W_t(\omega )$, where $X_t$ is an Itô process. We establish first a mean value theorem for the noncausal stochastic integral $\int _s^t f(X_r)dX_r$ and based on the result we show the corresponding formulae for the noncausal integral $\int _s^t f(X_r)d_*W_r$ or for Itô integral $\int _s^t f(X_r)d_0W_r$ as well. We also study the problem for such a genuin noncausal case where the process $X_t$ is noncausal, that is, not adapted to the natural filtration associated to Brownian motion. The discussions are developed in the framework of the noncausal calculus. Hence some materials and basic facts in the theory of noncausal stochastic calculus are briefly reviewed as preliminary.

Appendix—List of classes of random functions

By a random function $f(t,\omega )$ we understand a real function such that $P\{\int _0^T f^2(t,\omega )dt<\infty \}=1$. Here are the symbols for various classes of random functions.

$\mathbf{H}$: totality of random functions, $\mathbf{M}$: totality of causal functons,

$\mathbf{H}_{bv}$: totality of random functions whose sample functions are almost surely of bounded variation on [0, T], $\mathbf{M}_{bv}=\mathbf{M}\cap \mathbf{H}_{bv}$.

By an Itô process we mean such a random function $f(t,\omega )$ that admits the following decomposition; $df_t=a(t,\omega )d_0W_t + db(t,\omega ), \ \ a(t,\omega ) \in \mathbf{M} \ \text{ and } \ b(\cdot ) \in \mathbf{M}_{bv}$, When $b(\cdot ) \in \mathbf{H}_{bv}$ the $f(t,\omega )$ is called quasi Itô process,

$\mathbf{H}_Q$: totality of quasi Itô processes and $\mathbf{M}_Q=\mathbf{H}_Q \cap \mathbf{M}$.