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Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 3, Pages 446–457
(Mi tvp2258)
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This article is cited in 35 scientific papers (total in 35 papers)
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
Citation:
N. V. Krylov, “On an inequality in the theory of stochastic integrals”, Teor. Veroyatnost. i Primenen., 16:3 (1971), 446–457; Theory Probab. Appl., 16:3 (1971), 438–448
Linking options:
https://www.mathnet.ru/eng/tvp2258 https://www.mathnet.ru/eng/tvp/v16/i3/p446
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