Vol. 97, No. 1, 1981

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Existence of strong solutions for stochastic differential equations in the plane

James Juei-Chin Yeh

Vol. 97 (1981), No. 1, 217–247
Abstract

Let B be the 2-parameter Brownian motion on D = [0,] × [0,) and Z be a 2-parameter stochastic process defined on the boundary ∂D of D. Consider the non-Markovian stochastic differential system in 2-parameter

{
dX (s,t) = α(s,t,X )dB(s,t) + β(s,t,X)dsdt   for (s,t) ∈ D,
X (s,t) = Z(s,t)                         for (s,t) ∈ ∂D.

Under the assumption that the coefficients α and β satisfy a Lipschitz condition and a growth condition and the assumption that Z has continuous sample functions and locally bounded second moment on ∂D, it is shown in this paper that the differential system has a strong solution. Pathwise uniqueness of solution is established under the assumption of the Lipschitz condition.

Mathematical Subject Classification 2000
Primary: 60H10
Milestones
Received: 2 September 1980
Published: 1 November 1981
Authors
James Juei-Chin Yeh