[Paper Review] Weak Differentiability of Solutions to SDEs With Semi-Monotone Drifts

Mahdieh Tahmasebi, Shiva Zamani|arXiv (Cornell University)|Sep 3, 2013

Mathematical and Theoretical Epidemiology and Ecology Models5 references3 citations

TL;DR

This paper establishes the infinite Malliavin differentiability of the solution to a stochastic differential equation (SDE) with locally Lipschitz and semi-monotone drift by constructing a sequence of SDEs with globally Lipschitz drifts. Using uniform bounds on the p-moments of Malliavin derivatives and convergence of solutions, it proves the original SDE's solution lies in the space $$\mathbb{D}^\infty$$, enabling advanced Malliavin calculus applications in finance and SDE theory.

ABSTRACT

In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To this end we construct a sequence of SDEs with globally Lipschitz drifts. We show that the solutions of these SDEs converge to the solution of the original SDE and the p-moments of their Malliavin derivatives are uniformly bounded.

Motivation & Objective

• To extend Malliavin calculus to SDEs with non-globally Lipschitz, semi-monotone drifts, which are common in finance and biology.
• To overcome the technical challenge that classical Malliavin calculus does not apply directly to non-globally Lipschitz coefficients.
• To construct a sequence of approximating SDEs with globally Lipschitz drifts whose solutions converge to the original SDE's solution.
• To establish uniform boundedness of p-moments of Malliavin derivatives across the approximating sequence.
• To prove the original SDE's solution is infinitely Malliavin differentiable ($$\mathbb{D}^\infty$$) via convergence and uniform integrability.

Proposed method

• Construct a sequence of SDEs with globally Lipschitz drifts by mollifying the original drift using smooth cutoff functions.
• Use Gronwall's inequality to derive uniform bounds on the p-moments of the k-th order Malliavin derivatives of the approximating solutions.
• Apply the classical Malliavin calculus framework to the approximating SDEs due to their globally Lipschitz coefficients.
• Prove almost sure and $L^p$-norm convergence of the approximating solutions to the solution of the original SDE.
• Leverage the uniform boundedness of derivative moments and convergence to infer infinite Malliavin differentiability of the limit solution.
• Use the representation of Malliavin derivatives via the derivative of the SDE flow and stochastic integration to derive the key moment bounds.

Experimental results

Research questions

• RQ1 Can the solution to an SDE with locally Lipschitz and semi-monotone drift be shown to be infinitely Malliavin differentiable?
• RQ2 Is it possible to construct a sequence of SDEs with globally Lipschitz drifts such that their solutions converge to the original SDE's solution?
• RQ3 Do the p-moments of the Malliavin derivatives of the approximating solutions remain uniformly bounded across the sequence?
• RQ4 Can the uniform boundedness of derivative moments and solution convergence be combined to infer infinite differentiability of the original solution?
• RQ5 What conditions on the drift and diffusion coefficients ensure the existence of a strong solution in the space $$\mathbb{D}^\infty$$?

Key findings

• The solution to the original SDE (2.1) with locally Lipschitz and semi-monotone drift is uniquely strongly Malliavin differentiable of all orders.
• The p-moments of the k-th order Malliavin derivatives of the approximating solutions are uniformly bounded for all $p \geq 1$ and all $k \geq 1$.
• The solutions of the approximating SDEs converge to the solution of the original SDE in $L^p$ and almost surely.
• The uniform boundedness of the derivative moments across the approximating sequence, combined with convergence, implies the original solution lies in $$\mathbb{D}^\infty$$.
• The key technical step is the application of Gronwall's inequality to control the growth of derivative moments in the approximating SDEs.
• The construction of the approximating drifts via mollification ensures global Lipschitz continuity while preserving the semi-monotonicity and local Lipschitz properties of the original drift.