Article2005

One-Dimensional Stochastic Differential Equations with Singular and Degenerate Coefficients

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Stochastic processes and financial applications Nonlinear Differential Equations Analysis Stochastic processes and statistical mechanics

Abstract

Xt = Xo 4/ a(Xt)dWt + / V?(X) p(dw) for t > 0, Jq JR where W is a one-dimensional Brownian motion, a is a function that is bounded between two positive constants, \x is a finite measure with |/x({w}) | 0 for all t. Although this second equation does not have a unique solution, it does have a unique solution if one restricts attention to those solutions that spend zero time at 0.

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2007Brownian Motion and Stochastic Calculus 1999Continuous Martingales and Brownian Motion 1971On the uniqueness of solutions of stochastic differential equations 1984One — dimensional stochastic differential equations involving the local times of the unknown process 1972On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations 1983Applications du temps local aux equations differentielles stochastiques unidimensionnelles 1985On solutions of one-dimensional stochastic differential equations without drift 1982One Dimensional Stochastic Differential Equations with No Strong Solution

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