Toward a stochastic calculus for several Markov processes

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Abstract

In this paper we investigate a class of harmonic functions associated with a pair xt = (xt11, xt22) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δx1Δx2f(x1 ,x2) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes xti-i obtained from xtii by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.

Original languageEnglish (US)
Pages (from-to)125-144
Number of pages20
JournalAdvances in Applied Mathematics
Volume4
Issue number2
DOIs
StatePublished - Jun 1983
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

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