Stochastic Evolution Systems by Boris L. Rozovsky & Sergey V. Lototsky

Author:Boris L. Rozovsky & Sergey V. Lototsky
Language: eng
Format: epub
ISBN: 9783319948935
Publisher: Springer International Publishing

(5.1.3)

where

(5.1.4)

as usual,

If we further assume that c ≡ σ il  ≡ h l  ≡ 0, then (4.​4.​3) becomes the backward Kolmogorov equation (1.​5.​8) and representation (5.1.3) turns into (1.​5.​7).

Representation (5.1.3) implies that (4.​4.​3), (4.​4.​4), including the corresponding backward Kolmogorov equation, can be solved by the method of random characteristics, the diffusion process being the characteristic, although, unlike (1.​5.​7), representation (5.1.4) is a conditional averaging over the characteristic, relative to the σ-algebra .

The representation of the type (5.1.3) for problem (4.​1.​1), (4.​2.​2) or (4.​4.​3), (4.​4.​4) will be called the averaging over the characteristics (AOC) formula.

An important corollary of the AOC formula is the maximum principle for the Itô parabolic equation, to be proved in Sect. 5.2.

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