Global asymptotic stability of solutions of cubic stochastic difference equations
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Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in R1 . As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed. 1. Introduction Suppose that a filtered probability space (Ω,Ᏺ, {Ᏺn }n∈N , P) is given as a stochastic basis with filtrations {Ᏺn }n∈N . Let {ξn }n∈N be a one-dimensional real-valued {Ᏺn }n∈N martingale difference (for details, see [2, 14]) and let Ꮾ(S) denote the set of all Borel sets of the set S. Furthermore, let a = {an }n∈N be a nonincreasing sequence of strictly positive real numbers an and let κ = {κn }n∈N be a sequence of real numbers κn . We use “a.s.” as the abbreviation for wordings “P-almost sure” or “P-almost surely”. In this paper, we consider discrete-time stochastic difference equations (DSDEs)
3 + f n xl xn+1 − xn = κn xn3 − an xn+1
0≤l≤n
+ σn xl
0≤l≤n
ξn+1
(1.1)
with cubic-type main part of their drift in R1 , real parameters an ,κn ∈ R1 , driven by the square-integrable martingale difference ξ = {ξn+1 }n∈N of independent random variables ξn+1 with E[ξn+1 ] = 0 and E[ξn+1 ]2 < +∞. We are especially interested in conditions ensuring the almost sure global asymptotic stability of solutions of these DSDEs (1.1). The main result should be such that it can be applied to numerical methods for related continuous-time stochastic differential equations (CSDEs) as its potential limits. For example, consider
dXt = a1 t,Xt + a2 t,Xt dt + b t,Xt dWt Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 249–260 2000 Mathematics Subject Classification: 39A11, 37H10, 60H10, 65C30 URL: http://dx.doi.org/10.1155/S1687183904309015
(1.2)
250
Asymptotic stability of cubic SDEs
driven by standard Wiener process W = {Wt }t≥0 and interpreted in the Itoˆ sense, where a1 ,a2 ,b : [0,+∞) × R → R are smooth vector fields. Such CSDEs (1.2) with additive drift splitting can be discretized in many ways; for example, see [13] for an overview. However, only few of those discretization methods are appropriate to tackle the problem of almost sure asymptotic stability of their trivial solutions. One of the successful classes is that of partially drift-implicit θ-methods with the schemes
xn+1 = xn + θn a1 tn+1 ,xn+1 + 1 − θn a1 tn ,xn + a2 tn ,xn ∆n
+ b tn ,xn ∆Wn
(1.3)
applied to equation (1.2), where ∆n = tn+1 − tn and ∆Wn = Wtn+1 − Wtn , along any discretizations 0 = t0 ≤ t1 ≤ · · · ≤ tN = T of time intervals [0, T]. These methods with uniformly bounded θn (with supn∈N |θn | < +∞) provide L2 -converging approximations to (1.2) with rate 0.5 in the worst case under appropriate conditions on a1 ,a2 ,b. For d
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