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On an inequality in the theory of stochastic integrals
N. V. Krylov Moscow
Abstract:
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.
Received: 13.01.1970
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