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Distributions of Itô processes: Estimates for the density and for conditional expectations of integral functionals. (English. Russian original) Zbl 0844.60030
Theory Probab. Appl. 39, No. 4, 662-670 (1994); translation from Teor. Veroyatn. Primen. 39, No. 4, 825-833 (1994).
Let for the finite-dimensional equation $dy(t,\omega) = f(y(t,\omega), t,\omega) dt + \beta(y(t,\omega), t,\omega)dw(t)$ $$y^{a,s} (t)$$ be the solution with the initial condition $$y(s) = a$$, which is independent from $$w(t) - w(s)$$, $$t \geq s$$. The article is devoted to the investigation of the functionals $V(x,s,\omega) = E \Biggl\{ \int^t_s \varphi(y^{x,s}(t,\omega), t, \omega) dt/{\mathcal F}_s \Biggr\},$ where $${\mathcal F}_s = \sigma\{w(\tau) : \tau \leq s\}$$. Estimations for the different functional norms of $$V$$ are obtained. The main instrument is a stochastic parabolic equation for the conditional density of $$y^{a,s}(t)$$ with respect to $${\mathcal F}_t$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H15 Stochastic partial differential equations (aspects of stochastic analysis)