[Paper Review] Convergence rate of stability problems of SDEs with (dis-)continuous coefficients

Hashimoto Hashimoto, Takahiro Tsuchiya|arXiv (Cornell University)|Jan 18, 2014

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper establishes explicit convergence rates for the stability of one-dimensional stochastic differential equations (SDEs) with (dis)continuous diffusion coefficients satisfying the Nakao-Le Gall condition. Using the Yamada-Watanabe method, it derives bounds of order $(\log n)^{-1/2}$ for $L^1$-error and $(\log n)^{-p/(4(p+1))}$ for $L^p$-supremum error, both invariant under drift removal, under $L^1$-convergence of coefficients at rate $n^{-1}$.

ABSTRACT

We consider the stability problems of one dimensional SDEs when the diffusion coefficients satisfy the so called Nakao-Le Gall condition. The explicit rate of convergence of the stability problems are given by the Yamada-Watanabe method without the drifts. We also discuss the convergence rate for the SDEs driven by the symmetric $α$ stable process. These stability rate problems are extended to the case where the drift coefficients are bounded and in $L^1$. It is shown that the convergence rate is invariant under the removal of drift method for the SDEs driven by the Wiener process.

Motivation & Objective

• To analyze the convergence rate of solutions to SDEs with (dis)continuous diffusion coefficients under the Nakao-Le Gall condition.
• To extend stability results to SDEs driven by symmetric $\alpha$-stable processes for $\alpha \in (1,2)$.
• To investigate the impact of bounded $L^1$ drift coefficients on convergence rates and show invariance under drift removal.
• To provide explicit quantitative bounds on the rate of convergence in $L^1$ and $L^p$ norms.

Proposed method

• Apply the Yamada-Watanabe method to derive pathwise estimates for SDEs with diffusion coefficients satisfying the Nakao-Le Gall condition.
• Use the transformation $s(x) = \int_0^x \sigma(u)^{-2} du$ to reduce the SDE to a time-changed Brownian motion framework.
• Establish bounds on the difference of scale functions $s_n^{-1}$ and $s^{-1}$ using uniform convergence of $\sigma_n$ to $\sigma$ at rate $n^{-1}$.
• Employ change-of-variable techniques and $L^1$-norm estimates to control the difference in the scale function derivatives and their compositions.
• Use the sequence $\{f_l\}$ of increasing functions approximating $f$ to handle the quadratic variation structure in the Nakao-Le Gall condition.
• Apply the removal of drift method to show that convergence rates are invariant under drift removal for Wiener-driven SDEs.

Experimental results

Research questions

• RQ1 What is the convergence rate of solutions to SDEs with (dis)continuous diffusion coefficients satisfying the Nakao-Le Gall condition?
• RQ2 How does the convergence rate behave when the SDE is driven by a symmetric $\alpha$-stable process instead of a Wiener process?
• RQ3 Does the convergence rate remain unchanged when the drift coefficient is removed from the SDE?
• RQ4 Can explicit $L^1$ and $L^p$ convergence rates be derived under $L^1$-convergence of coefficients at rate $n^{-1}$?

Key findings

• For driftless SDEs with $\sigma_n, \sigma \in \mathcal{C}_{NL}(\epsilon, \|f\|_\infty)$, the $L^1$-error satisfies $\mathbb{E}[|X(t) - X_n(t)|] \leq C_1 (\log n)^{-1/2}$ for $0 \leq t \leq T$.
• For $p > 1$, the $L^p$-supremum error satisfies $\mathbb{E}[\sup_{0 \leq t \leq T} |X(t) - X_n(t)|^p] \leq C_p (\log n)^{-p/(4(p+1))}$ for $n > 2$.
• The convergence rate is invariant under the removal of drift for Wiener-driven SDEs when the drift is bounded and in $L^1$.
• For SDEs driven by symmetric $\alpha$-stable processes with $\alpha \in (1,2)$, the same convergence rates are established under analogous coefficient convergence conditions.
• Under the $L^1$-convergence condition $\int_{\mathbb{R}} |\sigma_n(x) - \sigma(x)| dx \leq C_0 n^{-1}$, the convergence rates are preserved.
• The results extend to SDEs with bounded $L^1$ drifts, where the convergence rate remains unchanged after drift removal, as shown via the removal of drift method.