Abstract
We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous. Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 − ε)/2, for ε ∈ (0,1), where ε becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.
Highlights
Consider the d-dimensional semi-linear stochastic differential equation (SDE) of Itotype dX(t) = [AX(t) + f (X(t)]dt + g(X(t))dW (t), t ∈ [0, T ]; X(0) ∈ Rd, (1)where T > 0, A ∈ Rd×d, f : Rd → Rd, g : Rd → Rd×m, and W is an mdimensional Wiener process
We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous
We provide a comparative illustration of the combined effect of semi-implicitness and adaptivity using five examples ranging from geometric Brownian motion to a system of SDEs arising from the spatial discretization of an SPDE
Summary
The literature already contains numerical schemes with fixed stepsizes that converge strongly to solutions of SDEs with coefficients that satisfy local Lipschitz and monotone conditions Several of these extend the idea of taming as introduced in [12], which rescales the functional response of the drift coefficient in the scheme; they do so by allowing the entire stochastic Euler map to be rescaled by some combination of drift and diffusion responses. The projected Euler method of [2] handles runaway trajectories by projecting them back onto a ball of radius inversely proportional to the step size; the authors control moments of the numerical solution It was shown in [24] that a drift-implicit discretization could ensure strong convergence in our setting.
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