Discrete-time approximations of stochastic delay equations: The Milstein scheme

Abstract

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinitedimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel. Introduction.Discrete-time strong approximation schemes for stochastic ordinary differential equations (SODEs) are well developed. For an extensive study of these numerical schemes, one may refer to [17], [18] and [19], Chapters 5 and 6. Some basic ideas of strong and weak orders of convergence are illustrated in [13].If the rate of change of a physical system depends only on its present state and some noisy input, then the system can often be described by a stochastic ordinary differential equation (SODE). However, in many physical situations the rate of change of the state depends not only on the present but also on the past states of the system. In such cases, stochastic delay differential equations (SDDEs) or stochastic functional differential equations (SFDEs) provide important tools to describe and analyze these systems. For various aspects of the qualitative theory of SFDEs the reader may refer to [20,21] and the references therein.SDDEs and SFDEs arising in many applications cannot be solved explicitly. Hence, one needs to develop effective numerical techniques for such systems. Depending on the particular physical model, it may be necessary to design strong L p (or almost sure) numerical schemes for pathwise solutions of the underlying SFDE. Strong approximation schemes for SFDEs may be used to

Milstein schemes and antithetic multi-level Monte-Carlo sampling for delay McKean-Vlasov equations and interacting particle systems

“…In the sequel, we refer to these equations as point-delay McKean SDEs. In [24] the same type of delays was considered for classical SDEs.…”

“…The literature on higher-order numerical schemes (i.e., beyond the basic Euler-Maruyama method of order 1/2) for classical delay SDEs is sparse and restricted to Lipschitz continuous coefficients. In [24], the strong convergence of a Milstein scheme for point-delay SDEs was proven using an Itô formula for so-called tame functions and techniques from anticipative calculus. For the analysis of a Milstein scheme without employing methods from anticipative calculus, we refer to [29].…”

Nonlinear Dynamical Analysis on Four Semi-Active Dynamic Vibration Absorbers with Time Delay

“…In order to verify the correctness and precision of the first-order approximate analytical solutions, the system parameters are selected as Table 1 [21][22][23][24]. Although the value of time delay is influenced by the system characteristics [25][26][27][28]35], we omit this effect and consider it as a constant value for simplicity. Here the value for time delay is selected as 0.002 s or 2 ms because the response time for many controllable devices in vibration control system is on the order of millisecond.…”

“…As illustrated in many literatures [25][26][27][28]35], time delay will affect the dynamical response significantly, especially the system stability. Here we would analyze the stability for the steady-state responses of the four semi-active DVAs.…”

The truncated $θ$-Milstein method for nonautonomous and highly nonlinear stochastic differential delay equations

“…The convergence rate of the TMM for SDDEs was investigated in [42]. As for other papers about Milstein methods, we refer the readers to [7,16,17,21,33,38,39,45] for more detailed discussions.…”