## 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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