This article is cited in 11 papers

Short Communications

Distributions of Itô processes: estimates for the density and for conditional expectations of integral functionals

N. G. Dokuchaev

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We obtain a priori estimates for the $L_2$-norms of solutions of parabolic Itô equations describing evolution of the distributions of solutions of ordinary Itô stochastic differential equations with random coefficients. In the case of nondegenerate equations, estimates for the $L_2$-norms of derivatives with respect to space variables are also obtained. As a consequence, we establish a generalisation of Itô's formula for functions that have only square-summable derivatives of the first and the second order (or even of the first order).

Keywords: distributions of Itô, processes, parabolic Itô, equations, Itô's formula.

Received: 25.07.1991

 English version: Theory of Probability and its Applications, 1994, 39:4, 662–670

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