A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

Abstract

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.

Appendix

The following result which is due to [1, Theorem 1] provides a compactness criterion for subsets of $L^{2}(\mu ;\mathbb{R }^{d})$ using Malliavin calculus.

Theorem 5.1

Let $\left\{ \left( \Omega ,\mathcal A ,P\right) ;H\right\}$ be a Gaussian probability space, that is $\left( \Omega , \mathcal A ,P\right)$ is a probability space and $H$ a separable closed subspace of Gaussian random variables of $L^{2}(\Omega )$, which generate the $\sigma$-field $\mathcal A$. Denote by $\mathbf D$ the derivative operator acting on elementary smooth random variables in the sense that

$$\begin{aligned} \mathbf D (f(h_{1},\ldots ,h_{n}))=\sum _{i=1}^{n}\partial _{i}f(h_{1},\ldots ,h_{n})h_{i},\quad h_{i}\in H,f\in C_{b}^{\infty }(\mathbb R ^{n}). \end{aligned}$$

Further let $\mathbf D _{1,2}$ be the closure of the family of elementary smooth random variables with respect to the norm

$$\begin{aligned} \left\| {F}\right\| _{1,2}:=\left\| {F}\right\| _{L^{2}(\Omega )}+\left\| \mathbf{D }F\right\| _{L^{2}(\Omega ;H)}. \end{aligned}$$

Assume that $C$ is a self-adjoint compact operator on $H$ with dense image. Then for any $c>0$ the set

$$\begin{aligned} \mathcal G =\left\{ G\in \mathbf D _{1,2}:\left\| {G}\right\| _{L^{2}(\Omega )}+\left\| {C}^{-1} \mathbf D \,G\right\| _{L^{2}(\Omega ;H)}\le c\right\} \end{aligned}$$

is relatively compact in $L^{2}(\Omega )$.

In order to formulate compactness criteria useful for our purposes, we need the following technical result which also can be found in [1].

Lemma 5.2

Let $v_{s},s\ge 0$ be the Haar basis of $L^{2}([0,1])$. For any $0<\alpha <1/2$ define the operator $A_{\alpha }$ on $L^{2}([0,1])$ by

$$\begin{aligned} A_{\alpha }v_{s}=2^{k\alpha }v_{s}\text{, } \text{ if } s=2^{k}+j\quad \end{aligned}$$

for $k\ge 0,0\le j\le 2^{k}$ and

$$\begin{aligned} A_{\alpha }1=1. \end{aligned}$$

Then for all $\beta$ with $\alpha <\beta <(1/2),$ there exists a constant $c_{1}$ such that

$$\begin{aligned} \left\| {A}_{\alpha }f\right\| \le c_{1}\left\{ \left\| {f}\right\| _{L^{2}([0,1])}+\left( \int _{0}^{1}\int _{0}^{1}\frac{\left| {f}(t)-f(t^{\prime })\right| ^{2}}{\left| {t}-t^{\prime }\right| ^{1+2\beta }}dt\,dt^{\prime }\right) ^{1/2}\right\} \!. \end{aligned}$$

A direct consequence of Theorem 5.1 and Lemma 5.2 is now the following compactness criteria which is essential for the proof of Corollar 3.6:

Corollary 5.3

Let a sequence of $\mathcal F _1$-measurable random variables $X_n\in \mathbb{D }_{1,2}, \,n=1,2\ldots$, be such that there exist constants $\alpha > 0$ and $C>0$ with

$$\begin{aligned} \sup _n E[\Vert X_n\Vert ^2]&\le C ,\\ \sup _n E \left[ \Vert D_t X_n - D_{t^{\prime }} X_n \Vert ^2 \right]&\le C |t -t^{\prime }|^{\alpha } \end{aligned}$$

for $0 \le t^{\prime } \le t \le 1$ and

$$\begin{aligned} \sup _n\sup _{0 \le t \le 1} E \left[ \Vert D_t X_n \Vert ^2 \right] \le C. \end{aligned}$$

Then the sequence $X_n, \,n=1,2\ldots$, is relatively compact in $L^{2}(\Omega )$.