A regression-based numerical scheme for backward stochastic differential equations

Based on Fourier cosine expansion, two approximations of conditional expectations are studied, and the local errors for these approximations are analyzed. Using these approximations and the theta-time discretization, a new and efficient numerical scheme, which is based on least-squares regression, for forward–backward stochastic differential equations is proposed. Numerical experiments are done to test the availability and stability of this new scheme for Black–Scholes call and calls combination under an empirical expression about volatility. Some conclusions are given.     

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.     

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated.     

On numerical solution of stochastic partial differential equations of elliptic type

We study lattice approximations of stochastic PDEs of elliptic type, driven by a white noise on a bounded domain in ? ^d , for d = 1, 2, 3. We obtain estimates for the rate of convergence of the approximations.     

Auxiliary equation method for solving nonlinear Wick-type partial differential equations

Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

Viability property for a backward stochastic differential equation and applications to partial differential equations

In the present paper, we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints. This property is the so-called “viability property”. In a separate section, this condition is translated to a class of partial differential equations. Received: 23 April 1998 / Published online: 14 February 2000     

Computational scheme for solving nonlinear fractional stochastic differential equations with delay

Abstract

This paper studies the numerical solution of fractional stochastic delay differential equations driven by Brownian motion. The proposed algorithm is based on linear B-spline interpolation. The convergence and the numerical performance of the method are analyzed. The technique is adopted for determining the statistical indicators of stochastic responses of fractional Langevin and Mackey-Glass models with stochastic excitations.     

Mean-field backward stochastic differential equations and related partial differential equations

In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. (2007) (in press). Available online: http://www.imstat.org/aop/future_papers.htm] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we study a mean-field BSDE in a Markovian framework, associated with a McKean–Vlasov forward equation. By combining classical BSDE methods, in particular that of “backward semigroups” introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations; Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (in Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.     

Stationary backward stochastic differential equations and associated partial differential equations

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the periodic solution of second order partial differential equations with periodic structure. Received: 11 October 1996 / Revised version: 15 February 1999     

Stability of stochastic partial differential equation

Stochastic partial differential equations such as occur in vibration problems for mechanical structures subjected to random loading are modelled as infinite dimensional stochastic Itô differential equations using a semigroup approach. Sufficient conditions for exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths of the displacement and velocity, are given. Under these same conditions it is shown that the zero solution is pathwise asymptotically stable relative to finite dimensional initial conditions. Illustrative examples are included.     

An explicit second-order numerical scheme for mean-field forward backward stochastic differential equations

Numerical Algorithms - In this paper, we propose an explicit second-order scheme for solving decoupled mean-field forward backward stochastic differential equations. Its stability is theoretically...     

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:

(SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      16. Mildly nonlinear elliptic partial differential equations and their numerical solutoin. II      Donald Greenspan  Seymour V. Parter 《Numerische Mathematik》1965,7(2):129-146       17. Mildly nonlinear elliptic partial differential equations and their numerical solution. I      Seymour V. Parter 《Numerische Mathematik》1965,7(2):113-128       18. An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations      Krystyna Twardowska 《随机分析与应用》2013,31(5):601-626 We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]      19. Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations      Min Niu 《Nonlinear Analysis: Real World Applications》2012,13(3):1346-1352 In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.      20. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II      《Stochastic Processes and their Applications》2001,93(2):205-228 This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]     

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)