Multiscale integration schemes for jump-diffusion systems - eScholarship
Dror Givon · 2009
MULTISCALE INTEGRATION SCHEMES FOR JUMP-DIFFUSION SYSTEMS DROR GIVON ∗ AND IOANNIS G. KEVREKIDIS Abstract. We study a two-time-scale system of jump-diffusion stochastic differential equations. We analyze a class of multiscale integration methods for these systems, which, in the spirit of [1], consist of a hybridization between a standard solver for the slow components and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain explicit bounds for the discrepancy between the results of the multiscale integration method and the slow components of the original system. 1. Introduction. A wide variety of problems in the natural sciences give rise to singularly perturbed systems of stochastic differential equations (SDEs). In many cases, one is only interested in predicting the time evolution of some “slow compo- nent”, yet this cannot be done, in a direct approach, without solving the full system of equations. No computer can deal with such a disparity of scales. In the past four decades, singularly perturbed systems have been the focus of extensive research, within the framework of averaging methods. The separation of scales is then taken to advantage to derive a reduced equation, which approximates the evolution of the slow components. Conditions under which the averaging principle can be applied to this kind of systems are well known in the classical literature. While the averaging principle and its resulting effective dynamics provide a substantial simplification of the original system, it is often impossible, or impractical to obtain the reduced equa- tions in closed form. This has motivated the development of multiscale integration algorithms [2, 1]. Multiscale integration schemes along the lines described in [1] have been studied for different systems of SDEs [3, 4]. However, similar questions for jump- diffusion processes are not yet fully addressed. We consider two-time scale systems of jump-diffusion SDEs, of the form dx t = a(x t , y t ) dt + b(x t , y t ) dB t + c(x t , y t ) dP t dy t = f (x t , y t ) dt + √ g(x t , y t ) dW t + h(x t , y t ) dN t x 0 = x 0 y 0 = y 0 , (1.1a) (1.1b) where x t is an n-dimensional jump-diffusion process and y t is an m-dimensional jump- diffusion process. The functions a(x, y) ∈ R n and f (x, y) ∈ R m are the drifts, the functions b(x, y) ∈ R n×d 1 and g(x, y) ∈ R m×d 2 are the diffusion coefficients, and the functions c(x, y) ∈ R n and h(x, y) ∈ R m are the jump coefficients; B t and W t are d 1 , d 2 -dimensional independent Wiener processes, P t is a scalar simple Poisson process with intensity λ 1 , and N t is a scalar simple Poisson process with intensity λ 2 . The parameter represents the ratio between the natural time scales of the x t and y t variables. We are concerned with situations where 1, i.e. where a separation of time scales prevails; in such case the vector x t is called the “slow component” of the system, and the vector y t is called the “fast component” of the system. Within the framework of averaging methods, the separation of scales is taken to advantage to derive, in the limit → 0, a reduced equation for an n-dimensional process x t , which approximates the slow component x t [5, 6, 7, 8, 9]. Define L the ∗ Department † Department of Mathematics, UCB and LBNL, Berkeley CA 94720 of Chemical Engineering, PACM and Mathematics, Princeton University