## Leap-frog method for stochastic functional wave equations

Henryk Leszczyński, Milena Matusik, and Monika Wrzosek

### Abstract

We perform a time-space discretisation, known as the leap-frog method, for nonlinear stochastic functional wave equations driven by multiplicative time-space white noise. To prove its stability we apply Cairoli's maximal inequalities for two-parameter martingales and provide a lemma for estimating solutions to a class of stochastic wave equations and a Gronwall-type inequality over cones. The method converges in $L^2$ at a rate of $O(\sqrt{h})$, where $h$ is a time-space step size.

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### Key words

leap-frog, stochastic wave equation

### AMS subject classifications

60H15, 35R60, 35R10

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