Abstract
We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, 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.
Highlights
In this manuscript we work with the class of Stochastic Differential Equations (SDEs) with drifts satisfying a super-linear growth and a monotonicity condition
We prove Malliavin Differentiability through a less well-known method developed by Sugita [Sug85] which uses the concepts of Ray absolute continuity and Stochastic Gâteaux Differentiability see [MPR17, IMPR16]
While studying different examples of processes with monotone growth, we became interested in the particular example where the drift term has polynomial growth of order q but only finite moments up to p < q − 2
Summary
In this manuscript we work with the class of Stochastic Differential Equations (SDEs) with drifts satisfying a super-linear growth (locally Lipschitz) and a monotonicity condition ( called onesided Lipschitz condition). To establish Malliavin differentiability for an SDE with solution X and with monotone drifts the most natural path to follow is to try to apply [Nua, Lemma 1.2.3] by employing a truncation procedure. The latter, yields a sequence Xn of SDEs with Lipschitz coefficients converging to X. We prove Malliavin Differentiability through a less well-known method developed by Sugita [Sug85] which uses the concepts of Ray absolute continuity and Stochastic Gâteaux Differentiability see [MPR17, IMPR16]
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