On an inequality in the theory of stochastic integrals

N. V. Krylov

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 3, pp. 446-457

Résumé

Let $$ x_t=\int_0^t\sigma_s\,d\xi_s+\int_0^tb_s\,ds $$ be an $n$-dimensional stochastic integral, $U$ be a bounded domain in the $n$-dimensional Euclidean space, $x'\in U$, $\tau$ be the first exit time of $x'+x_t$ out of $U$. Let $|b_t|\le M\cdot\sqrt[n]{\det\sigma_t^2}$ for all $t$, $\omega$. In the paper, a constant $N$ is proved to exist that depends only on $n$ and the diameter of $U$ such that, for all Borel functions $f$ $$ \mathbf M\int_0^\tau|f(x'+x_t)|\sqrt[n]{\det\sigma_t^2}\,dt\le N\|f\|_{L_n,U}. $$ The proof is based on the theory of convex polyhedrons.