Abstract
We consider a stochastic differential equation, driven by a Brownian motion, with Lipschitz coefficients. We prove that the corresponding flow is, almost surely, almost everywhere derivable with respect to the initial data for any time, and the process defined by the Jacobian matrices is a GLn(ℝ)-valued continuous solution of a linear stochastic differential equation. In dimension one, this process is given by an explicit formula.
These results partially extend those which are known when the coefficients are C1,α-Hölder continuous. Dirichlet forms are involved in the proofs.
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© 1989 Springer-Verlag
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Bouleau, N., Hirsch, F. (1989). On the derivability, with respect to the initial data, of the solution of a stochastic differential equation with Lipschitz coefficients. In: Bouleau, N., Feyel, D., Mokobodzki, G., Hirsch, F. (eds) Séminaire de Théorie du Potentiel Paris, No. 9. Lecture Notes in Mathematics, vol 1393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085770
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DOI: https://doi.org/10.1007/BFb0085770
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