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 Hölder 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.
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Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001
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Bass, R., Chens, ZQ. Stochastic differential equations for Dirichlet processes. Probab Theory Relat Fields 121, 422–446 (2001). https://doi.org/10.1007/s004400100151
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DOI: https://doi.org/10.1007/s004400100151
Keywords
- Differential Equation
- Stochastic Differential Equation
- Dirichlet Process