Session I.7 - Stochastic Computation

Monday, June 12, 14:30 ~ 15:00

The robustness of the Euler scheme for scalar SDEs with non-Lipschitz diffusion coefficient

Andreas Neuenkirch

University of Mannheim, Germany   -   neuenkirch@uni-mannheim.de

We consider stochastic differential equations (SDEs) that are given by $$dV_t = a(V_t)dt + \left(b(V_t)\right)^{1-\gamma}dW_t, \qquad t\in[0,T],$$ where $V_0=v_0 \in \mathbf{R}$ is deterministic, $W=(W_t)_{t \in [0,T]}$ is a Brownian motion and $\gamma\in\left(0,\frac{1}{2}\right]$. We assume that $a:\mathbf{R}\rightarrow \mathbf{R}$ and $b:\mathbf{R}\rightarrow [0,\infty)$ are globally Lipschitz continuous. Well-known examples that fall into this class of SDEs are the CIR process, the CEV process or the Wright-Fisher diffusion. We analyze the equidistant Euler scheme for the above SDE and, among other results, we show $L^1$-convergence order $1/2-\varepsilon$ in the discretization points (for $\varepsilon \gt 0$ arbitrarily small) if $$\int_0^T \mathbf{E} \left[ \frac{1}{b(V_t)^{2 \gamma}} \right] dt \lt \infty.$$ Thus, the loss of Lipschitzness, i.e. $\gamma \gt 0$, for the diffusion coefficient can be compensated by an appropriate inverse moment condition. This result yields in particular a unifying framework for the above mentioned SDEs: for the CIR or Wright-Fisher process, the above condition corresponds to the non-attainability of the boundaries of their support, while for the CEV process this inverse moment condition is always fulfilled.

Joint work with Annalena Mickel (University of Mannheim, Germany).

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