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Differentiability of SDEs with drifts of super-linear growth. (English) Zbl 1406.60084
Summary: We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth and with random coefficients, namely, we prove Malliavin and parametric differentiability of such SDEs. The former is shown by proving stochastic Gâteaux differentiability and Ray absolute continuity. This method enables one to take limits in probability rather than mean square or almost surely bypassing the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology of D. Nualart [The Malliavin calculus and related topics. 2nd ed. Berlin: Springer (2006; Zbl 1099.60003), Lemma 1.2.3] for this setting. Several examples illustrating the range and scope of our results are presented.
We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.
MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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