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2019 Differentiability of SDEs with drifts of super-linear growth
Peter Imkeller, Gonçalo dos Reis, William Salkeld
Electron. J. Probab. 24: 1-43 (2019). DOI: 10.1214/18-EJP261

Abstract

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 [13, 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.

Citation

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Peter Imkeller. Gonçalo dos Reis. William Salkeld. "Differentiability of SDEs with drifts of super-linear growth." Electron. J. Probab. 24 1 - 43, 2019. https://doi.org/10.1214/18-EJP261

Information

Received: 31 August 2018; Accepted: 29 December 2018; Published: 2019
First available in Project Euclid: 8 February 2019

zbMATH: 1406.60084
MathSciNet: MR3916323
Digital Object Identifier: 10.1214/18-EJP261

Subjects:
Primary: 60H07
Secondary: 60H10, 60H30

JOURNAL ARTICLE
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