Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.     

Sequential Monte Carlo Samplers: Error Bounds and Insensitivity to Initial Conditions

This article addresses finite sample stability properties of sequential Monte Carlo methods for approximating sequences of probability distributions. The results presented herein are applicable in the scenario where the start and end distributions in the sequence are fixed and the number of intermediate steps is a parameter of the algorithm. Under assumptions which hold on noncompact spaces, it is shown that the effect of the initial distribution decays exponentially fast in the number of intermediate steps and the corresponding stochastic error is stable in 𝕃 _p norm.     

An analytic approximation of solutions of stochastic differential delay equations with Markovian switching

In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the L^p-norm when the drift and diffusion coefficients are Taylor approximations.     

Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps

Abstract

In this article, we derive the sufficient conditions for the existence of mild solutions of Hilfer fractional stochastic integrodifferential equations with nonlocal conditions and Poisson jumps in Hilbert spaces. Results will be obtained in the pth mean square sense by using the fractional calculus, semigroup theory and stochastic analysis techniques. The article generalizes many of the existing results in the literature in terms of (1) Riemann–Liouville and Caputo derivatives are the special cases. (2) In the sense of pth mean square norm. (3) Stochastic integrodifferential with nonlocal conditions and Poisson jumps. A numerical example is provided to validate the obtained theoretical results.     

Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this article, we give new characterizations of the Buchsbaum and Cohen–Macaulay properties of the tangent cone gr _𝔪 (R), where (R, 𝔪) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr _𝔪 (R).     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Sequential Identification of Linear Dynamic Systems with Memory

This paper presents a sequential estimator for some unknown parameters in stochastic linear systems with memory. As examples stochastic differential equations with time delayed drift are considered. Based on the maximum likelihood method, we construct an estimation procedure with given accuracy in the sense of the L _p -norm (p 2). It is shown, that this procedure works also in certain cases, when the normalized information matrix of the observed process is asymptotically degenerated. The almost surely consistency of the proposed estimators and the asymptotic behavior of the length of observations are derived.     

Arithmetic Asian Options under Stochastic Delay Models

Abstract

Motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors such as Hobson and Rogers (1998 Hobson, D. and Rogers, L. C. G. 1998. Complete models with stochastic volatility. Mathematical Finance, 8(1): 27–48.  [Google Scholar], Complete models with stochastic volatility, Mathematical Finance, 8(1), pp. 27–48), we explore option pricing techniques for arithmetic Asian options under a stochastic delay differential equation approach. We obtain explicit closed-form expressions for a number of lower and upper bounds and compare their accuracy numerically.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

L p Solutions of One-Dimensional Backward Stochastic Differential Equations with Continuous Coefficients

In this article, we study one-dimensional backward stochastic differential equations with continuous coefficients. We show that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈L ^p (Ω, ? _T , P) with 1 < p ≤ 2, the backward stochastic differential equation has a unique L ^p solution.     

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ? ^q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ? ^q (𝒫) for some q ≥ 1.     

种群动力系统的数值解的振动性分析

本文主要研究下面动力系统的非线性延迟微分方程x't(t)+αVmx(t)p(t-τ)/βp+xp(t-τ)=λ,t≥0 数值解的振动性.这是由Mackey和Glass[1]提出来的关于动力系统疾病的方程.本文得到了数值方法振动的条件.同时对非振动的数值解的性质也做了研究,为了验证得到的结果,给出了数值算例.     

Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

In this paper, we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];R^d) which denotes the family of bounded continuous R^d-value functions defined on (-∞,0] with norm under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.     

Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral Xt（ω）=η（w）＋∫^t 0 asXs（ω）dWs＋∫^t 0 bsXs（ω）ds, t∈[0,1] where （Ws） is the canonical Wiener process defined on the standard Wiener space （W,H,u）, a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by （Ws）. The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

A numerical scheme for stochastic PDEs with Gevrey regularity

We consider strong approximations to parabolic stochastic PDEs.We assume the noise lies in a Gevrey space of analytic functions.This type of stochastic forcing includes the case of forcingin a finite number of Fourier modes. We show that with Gevreynoise our numerical scheme has solutions in a discrete equivalentof this space and prove a strong error estimate. Finally wepresent some numerical results for a stochastic PDE with a Ginzburg–Landaunonlinearity and compare this to the more standard implicitEuler–Maruyama scheme.