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

Thomas Skov Knudsen

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    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 is an element of C-5, 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 c. 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.
    Original languageEnglish
    JournalStochastic Processes and Their Applications
    Volume68
    Issue number2
    Pages (from-to)155-179
    ISSN0304-4149
    DOIs
    Publication statusPublished - 16 Jun 1997

    Keywords

    • stochastic differential equations
    • random field of solutions
    • pathwise continuity wrt coefficients
    • p-step nilpotent Lie algebras
    • shuffle product

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