Abstract
Numerical analysis of stochastic delay differential equations has been widely developed but frequently for the cases where the delay term has a simple feature. In this paper, we aim to study a more general case of delay term which has not been much discussed so far. We mean the case where the delay term takes random values. For this purpose, a new continuous split-step scheme is introduced to approximate the solution and then convergence in the mean-square sense is investigated. Moreover, given a test equation, the mean-square asymptotic stability of the scheme is presented. Numerical examples are provided to further illustrate the obtained theoretical results.
Highlights
In many physical phenomena with random nature, the state future of a system depends on the current state and depends on the whole past history of the system over a finite time interval, and certainly the mathematical modelling describing the system leads to a stochastic delay differential equation (SDDE) and not a stochastic ordinary differential equation (SODE)
The inaccessibility of the closed-form of the solutions or their distributions of these mathematical modelings, which arise in diverse areas of applications, reveal the significance of addressing numerical methods, because they play an important role to educe a realistic view of the solution behaviour of such equations
The main contribution of this paper is to investigate the numerical solution of Eq (1.1), under sufficient conditions which will be mentioned later, with three cases of lag process as follows: (L1) τ is a constant, (L2) τ is time-dependent as τ (t), (L3) τ is state-dependent as τ (t, X(t))
Summary
In many physical phenomena with random nature, the state future of a system depends on the current state and depends on the whole past history of the system over a finite time interval, and certainly the mathematical modelling describing the system leads to a stochastic delay differential equation (SDDE) and not a stochastic ordinary differential equation (SODE). In addition to the time, it depends on the Akhtari Advances in Difference Equations (2019) 2019:396 solution process, it is named state-dependent. The authors [13, 14] considered the continuous-time GARCH(1, 1) model for stochastic volatility involving state-dependent delayed response and applied the Euler–Maruyama discrete-time approximation in the strong convergence sense to simulate. There exist some papers which extend some types of stochastic functional (evolution, fractional, neutral) differential equations with state-dependent delay and study some theoretical aspects the existence and uniqueness of (mild) solution and controllability results, see, e.g. The main contribution of this paper is to investigate the numerical solution of Eq (1.1), under sufficient conditions which will be mentioned later, with three cases of lag process as follows:
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