[Paper Review] Analysis of a splitting method for stochastic balance laws

Erlend Briseid Storrøsten, Kenneth H. Karlsen|arXiv (Cornell University)|Jan 11, 2016

Stochastic processes and financial applications23 references3 citations

TL;DR

This paper analyzes a semi-discrete operator splitting method for stochastic balance laws driven by multiplicative Wiener noise, proving convergence of approximate solutions to the exact stochastic entropy solution. Under homogeneous noise and Malliavin differentiable Kružkov constants, it establishes an $ L^1 $ convergence rate of order $ \frac{1}{3} $ in the time step $ \Delta t $, leveraging fractional BV estimates and a generalized entropy condition.

ABSTRACT

We analyze a semi-discrete splitting method for conservation laws driven by a semilinear noise term. Making use of fractional $BV$ estimates, we show that the splitting method produces a compact sequence of approximate solutions converging to the exact solution, as the time step $Δt ightarrow 0$. Under the assumption of a homogenous noise function, and thus the availability of $BV$ estimates, we provide an $L^1$ error estimate. Bringing into play a generalization of Kruzkov's entropy condition, permitting the "Kruzkov constants" to be Malliavin differentiable random variables, we establish an $L^1$ convergence rate of order $\frac13$ in $Δt$.

Motivation & Objective

• To establish convergence of semi-discrete operator splitting approximations for stochastic conservation laws with multiplicative noise.
• To address the challenge of time discretization and nonlinearity in the presence of stochastic forcing.
• To extend convergence analysis beyond homogeneous noise cases by introducing a generalized entropy condition with Malliavin differentiable Kružkov constants.
• To derive an $ L^1 $ error estimate with a sharp convergence rate of $ \frac{1}{3} $ under suitable regularity assumptions.

Proposed method

• The method employs a Godunov-type operator splitting to decouple the deterministic flux and stochastic source terms in the balance law.
• It uses fractional $ BV $ estimates to control the regularity of approximate solutions and ensure convergence.
• A generalized entropy condition is introduced, allowing Kružkov constants to be Malliavin differentiable random variables to handle stochastic nonlinearity.
• The analysis combines Malliavin calculus with stochastic PDE techniques to manage the noise term and derive stability estimates.
• Weak compactness and Young measures are used to extract convergent subsequences of approximations in $ L^1 $.
• A priori estimates and compensated compactness arguments are applied to prove convergence to a stochastic entropy solution.

Experimental results

Research questions

• RQ1 Can a semi-discrete operator splitting method converge to a stochastic entropy solution for balance laws with non-homogeneous noise?
• RQ2 What is the optimal $ L^1 $ convergence rate for such splitting schemes under general noise assumptions?
• RQ3 How can the classical Kružkov entropy condition be adapted to stochastic settings with random entropy bounds?
• RQ4 Can fractional $ BV $ estimates be used to control the regularity of solutions in the stochastic context?
• RQ5 What role does Malliavin differentiability of entropy constants play in error estimation for stochastic conservation laws?

Key findings

• The splitting method generates approximate solutions that converge to the exact stochastic entropy solution as $ \Delta t \to 0 $, even for non-homogeneous noise.
• Under homogeneous noise, the method achieves an $ L^1 $ error estimate with convergence rate $ \frac{1}{3} $ in $ \Delta t $.
• The use of Malliavin differentiable Kružkov constants enables the derivation of sharp error bounds in the stochastic setting.
• Fractional $ BV $ estimates are instrumental in controlling the variation of approximate solutions and ensuring convergence.
• The convergence result holds under the assumption that $ f $ is globally Lipschitz and $ \sigma $ satisfies a local Lipschitz and Hölder-type condition.
• The analysis establishes the existence of a stochastic entropy solution via weak compactness and Young measures, even without assuming $ L^\infty $ bounds on the solution.