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Continuity in a pathwise sense with respect to the coefficients of solutions of stochastic differential equations

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  • Knudsen, Thomas Skov

Abstract

For stochastic differential equations (SDEs) of the form dX(t) = b(X)(t)) dt + [sigma] (X(t))dW(t) where b and [sigma] are Lipschitz continuous, it is shown that if we consider a fixed [sigma] [epsilon] C5, bounded and with bounded derivatives, the random field of solutions is pathwise locally Lipschitz continuous with respect to b when the space of drift coefficients is the set of Lipschitz continuous functions of sublinear growth endowed with the sup-norm. Furthermore, it is shown that this result does not hold if we interchange the role of b and [sigma]. However for SDEs where the coefficient vector fields commute suitably we show continuity with respect to the sup-norm on the coefficients and a number of their derivatives.

Suggested Citation

  • Knudsen, Thomas Skov, 1997. "Continuity in a pathwise sense with respect to the coefficients of solutions of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 155-179, June.
  • Handle: RePEc:eee:spapps:v:68:y:1997:i:2:p:155-179
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