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On some applications of Sobolev flows of SDEs with unbounded drift coefficients

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  • Menoukeu Pamen, Olivier

Abstract

We study two applications of spatial Sobolev smoothness of stochastic flows of unique strong solution to stochastic differential equations (SDEs) with irregular drift coefficients. First, we analyse the stochastic transport equation assuming that the drift coefficient is Borel measurable, with spatial linear growth and show that the above equation has a unique Sobolev differentiable weak coefficient for all t∈[0,T] for T small enough. Second, we consider the Kolmogorov equation and obtain a representation of the spatial derivative of its solution v. The latter result is obtained via the martingale representation theorem given in (Elliott and Kohlmann, 1988) and generalises the results in (Elworthy and Li, 1994; Menoukeu-Pamen et al., 2013).

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  • Menoukeu Pamen, Olivier, 2018. "On some applications of Sobolev flows of SDEs with unbounded drift coefficients," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 114-124.
    • Handle: RePEc:eee:stapro:v:141:y:2018:i:c:p:114-124
    DOI: 10.1016/j.spl.2018.05.029
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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    2. Elliott, Robert J. & Kohlmann, Michael, 1988. "A short proof of a martingale representation result," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 327-329, April.
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