Electronic Journal of Probability

Differentiability of SDEs with drifts of super-linear growth

Peter Imkeller, Gonçalo dos Reis, and William Salkeld

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 3, 43 pp.

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

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Digital Object Identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Malliavin calculus parametric differentiability monotone growth SDE one-sided Lipschitz Bismut-Elworthy-Li formula

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Imkeller, Peter; dos Reis, Gonçalo; Salkeld, William. Differentiability of SDEs with drifts of super-linear growth. Electron. J. Probab. 24 (2019), paper no. 3, 43 pp. doi:10.1214/18-EJP261. https://projecteuclid.org/euclid.ejp/1549616424

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