[Paper Review] On strong solutions of Itô's equations with a$\,\in W^{1}_{d}$ and b$\,\in L_{d}$

Н. В. Крылов|arXiv (Cornell University)|Jul 12, 2020

Advanced Mathematical Modeling in Engineering23 references4 citations

TL;DR

This paper establishes the existence and uniqueness of strong solutions for multidimensional Itô stochastic differential equations with diffusion coefficients in the Sobolev space $W^{1}_{d,{\rm loc}}$ and drift coefficients in $L_d$, under uniform nondegeneracy. The solutions are shown to be Hölder continuous in the initial condition with any exponent less than 1, and convergence of solutions under coefficient approximation is proven using a novel analytic criterion, bypassing Yamada-Watanabe or Zvonkin-type transformations.

ABSTRACT

We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. We prove the unique strong solvability for any starting point and prove that as a function of the starting point the solutions are Hölder continuous with any exponent $<1$. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.

Motivation & Objective

• To establish the existence and uniqueness of strong solutions for time-homogeneous Itô SDEs with $\sigma \in W^{1}_{d,{\rm loc}}$ and $b \in L_d$.
• To prove that solutions are Hölder continuous in the initial condition with any exponent $<1$.
• To demonstrate convergence of solutions under coefficient approximation in an appropriate topology.
• To develop a new analytic method for strong solution existence that avoids Yamada-Watanabe or noise transformation techniques.

Proposed method

• Uses an analytic criterion for strong solution existence first introduced in Krylov (2007), based on a priori estimates and regularity of the transition density.
• Employs a priori estimates on the derivative of the solution with respect to initial data via the derivative process $\xi_t(x,\eta)$.
• Applies a version of the Kolmogorov continuity theorem to derive Hölder continuity of the solution path in the initial condition.
• Utilizes the $L_d$-norm of the gradient of $\sigma$ and the $L_d$-norm of $b$ to control the growth of moments and derivatives.
• Applies Fatou’s lemma and approximation by smooth coefficients to extend results from the smooth case to the general $W^{1}_d \times L_d$ setting.
• Relies on moment estimates from [11] and a version of the maximal inequality for Itô processes to control convergence in sup-norm.

Experimental results

Research questions

• RQ1 Under what conditions on $\sigma$ and $b$ does the SDE (1.1) admit a unique strong solution?
• RQ2 Can Hölder continuity of the solution with respect to the initial condition be established when $\sigma \in W^{1}_d$ and $b \in L_d$?
• RQ3 Does convergence of coefficient sequences in $L_d$ imply convergence of the corresponding solutions in sup-norm?
• RQ4 Can strong solvability be proven without relying on Yamada-Watanabe theorem or Zvonkin-type transformations?

Key findings

• The SDE (1.1) has a unique strong solution for any starting point $x_0 \in \mathbb{R}^d$ when $\sigma \in W^{1}_{d,{\rm loc}}$ and $b \in L_d$, under uniform nondegeneracy.
• Solutions are Hölder continuous in the initial condition with any exponent $\alpha < 1$.
• For $\kappa > 2d$, the $\kappa$-th moment of the difference $|x_t(x) - x_t(y)|$ is bounded by $N|x-y|^{\kappa - 2d}$, implying Hölder continuity.
• If coefficient sequences $\sigma^{k}(n) \to \sigma^k$ and $b(n) \to b$ in $L_d$, then the solutions $x_t(n,x(n))$ converge to $x_t$ in $L^1$-sup norm.
• The convergence rate is exponential in $\rho$ for large $\rho$, with $\mathbb{P}(|x_t(x)| \geq \rho) \leq Ne^{-\mu \rho^2}$.
• The method avoids Yamada-Watanabe and Zvonkin-type approaches, relying instead on a novel analytic criterion and moment estimates.