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On estimates of moments of quasi-derivatives of solutions of stochastic equations with respect to initial data and their application. (Russian) Zbl 0658.60086

Let w be a \(d_ 1\)-dimensional Wiener process on a complete probability space. Denote by \(E^ d\) a d-dimensional Euclidean space. Suppose that \(b^ i,\quad \sigma^{ij}:\quad E^ d\to {\mathbb{R}}^ 1,\) \(i=1,...,d\); \(j=1,...,d_ 1\), are sufficiently smooth and consider the Ito equation \[ X_ t=x+\int^{t}_{0}\sigma (X_ s)dw_ s+\int^{t}_{0}b(X_ s)ds, \] where \(\sigma =(\sigma^{ij})\) and \(b=(b^ i)\). Finally, let c: \(E^ d\to (-\infty,0]\) and \(f:\quad E^ d\to {\mathbb{R}}^ 1\) be sufficiently smooth and introduce the mappings \(\phi\) and \(\nu\) by \[ \phi_ t=\int^{t}_{0}c(X_ s)ds,\quad \nu (x)={\mathbb{E}}_ x\int^{\infty}_{0}\exp (\phi_ t)f(X_ t)dt. \] The author considers the problem of giving conditions under which \(\nu\) is twice continuously differentiable. A typical condition is to require that for all \(x\in E^ d\) and \(\xi \in E^ d\) with \(| \xi |^ 2=1\) (\(|.|\) is the Euclidean norm) the inequality \[ (*)\quad 2(1+\delta)| \sigma^*_{(\xi)}\xi |^ 2+(1+\delta)\| \sigma_{(\xi)}\|^ 2+2(b_{(\xi)},\xi)\leq (1-\delta)| c| \] holds. Here \(\delta >0\) is a constant, \(\| \sigma \|^ 2=trace(\sigma \sigma^*)\) and \(u_{(\xi)}=u_{x^ i}(x)\xi^ i\). This requirement is quite sharp, which leads to the problem of finding out more flexible and weaker conditions. The author introduces the notion of the so-called quasiderivative, which allows to give conditions being essentially weaker than (*).
Reviewer: R.Manthey

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35J15 Second-order elliptic equations
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