Numerical solution of stochastic state-dependent delay differential equations: convergence and stability

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.Assumption 1 The functions a : (R d ) 2 → R d and b : (R d ) 2 → R d×m are globally Lipschitz continuous, i.e. there is a positive constant K 1 such that a(Assumption 2 In case (L2), let τ : [t 0 , T] → (0, r] be of Lipschitz continuous aswhere K 2 is a positive constant. In case (L3), let τ : [t 0 , T] × R d → (0, r] be of Lipschitz continuous, i.e. there exist two positive constants K 2 and K 3 such thatfor all t, s ∈ [t 0 , T] and X, X 1 , X 2 ∈ R d . Note that in each of the two cases above, the fact that τ is positive guarantees the measurability and then the existence of the Itô integral.

Computational solution of advanced stochastic time-delay differential equations using hybrid block extended Adams Moulton methods for customer’s satisfaction when using electronic payment systems in Nigerian banks

“…In order to determine the accuracy, efficiency and advantage of our method HEBAMM, we compared the absolute maximum errors of our method with other existing methods in Evelyn (2000); Bahar (2019) and Osu et al (2021).…”

“…The applications of SDDEs can be seen in applied sciences, economics and engineering. Several authorssuch as Evelyn (2000), Zhang et al (2009), Bahar (2019), Wang et al (2011), Kazmerchuk (2005), Kazmerchuk et al (2004), Akhtari et al (2015) used Euler-Maruyama scheme to formulate continuous split-step schemes of SDDE on a continuous interval for the numerical solutions and encountered some challenges in the use of interpolation techniques in evaluating their delay terms as studied by Majid et al (2013). In order to circumvent these challenges, we applied Hybrid Block Extended Adams Moulton Methods (HBEAMM) as a linear multistep collocation method to discretize ASTDDEs on a discrete interval [ ) 0 , a t t .…”

A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

“…We selectively mention results from An incomplete selection of approximation results in the huge literature on SFDEs is [17,6,22,23,2,12,18,1,29,24,30,19,11,19]…”

“…In addition, (28) and the fact that p ∈ [2, ∞) show for all a, x ∈ H, b ∈ HS(U, H) that 2 a, x H + 1 2 trace (bb * 2Id H ) + 0.5p−1…”

Stability of Stochastic Differential Equations with Distributed and State-Dependent Delays

“…Differential equations with delays that are depending on the system state, so called, state-dependent delays, are enough new but at the same time are enough popular in research both in the deterministic case [1, 5-9, 11, 12, 14, 15, 17-20, 22-29, 32-34] and in the stochastic case [2,3,21,30,36,37]. However, it is necessary to note that research for stochastic differential equations with state-dependent delays, in particular, with problems of stability for equations of such type, one can meet much less common.…”