Stochastic differential equations for Dirichlet processes |
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| Authors: | Richard F Bass Zhen-Qing Chens |
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| Institution: | (1) Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA. e-mail: bass@math.uconn.edu., US;(2) Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, USA. e-mail: zchen@math.washington.edu., US |
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| Abstract: | We consider the stochastic differential equation dX
t
= a(X
t
)dW
t
+ b(X
t
)dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness
results when a is in C
1/2 and b is only a generalized function, for example,the distributional derivative of a H?lder function or of a function of bounded
variation. When b = aa′, that is, when the generator of the SDE is the divergence form operator ℒ = , a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes
when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral.
Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001 |
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| Keywords: | |
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