Nonlinear Stochastic Partial Differential Equations Research Articles

A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING

AbstractWe study an optimal control problem related to swing option pricing in a general non‐Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first‐order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.

Insight into spontaneous recurrent calcium waves in a 3-D cardiac cell based on analysis of a 1-D deterministic model

Calcium dynamics in a cardiac cell are described by a system of 3-D non-linear stochastic partial differential equations. To obtain solutions that have biophysical properties, it is necessary to explore the model parameter space. To decrease the complexity of the parameter search, we reduce the 3-D stochastic model to a 1-D deterministic model. The reduction of the problem from 3-D to 1-D is done through an asymptotic approximation after non-dimensionalization and based on rational biophysical assumptions of the 3-D domain; the stochastic to deterministic transformation is based on the regular property of the 3-D solution. The result of the model reduction proves very effective in reducing the time required to get qualitative as well as quantitative information about parameter regions in the 3-D stochastic model including calcium dynamics (sparks, wave propagation, and recovery) observed in cardiac cells.

On a high-dimensional nonlinear stochastic partial differential equation

In this paper we investigate a nonlinear stochastic partial differential equation perturbed by a space-correlated Gaussian noise in arbitrary dimension , with a non-Lipschitz coefficient noisy term. The equation studied coincides in one dimension with the stochastic Burgers equation. Existence of a weak solution is established through an approximation procedure.

Invited review: KPZ. Recent developments via a variational formulation

Recently, a variational approach has been introduced for the paradigmatic Kardar-Parisi-Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation $\alpha + z = 2$, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation. Received: 25 June 2013, Accepted: 10 December 2013; Reviewed by: F. Reis, Instituto de Fisica, Univ. Fed. Fluminense, Brazil; Edited by: R. Dickman; DOI: http://dx.doi.org/10.4279/PIP.050010Cite as: H S Wio, R R Deza, C Escudero, J A Revelli, Papers in Physics 5, 050010 (2013)

Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method

In this paper, we present a new idea to approximate high-dimensional nonlinear stochastic partial differential equations (SPDEs) by a kernel-based collocation method, which is a meshfree approximation method. A reproducing kernel is used to construct an approximate basis. The kernel-based collocation solution is a linear combination of the reproducing kernel with the differential and boundary operators of SPDEs at the given collocation points placed in the related high-dimensional domains. Its random expansion coefficients are computed by a random optimisation problem with constraint conditions induced by the nonlinear SPDEs. For a fixed kernel function, the convergence of kernel-based collocation solutions only depends on the fill distance of the chosen collocation points for the bounded domain of SPDEs. The numerical experiments of Sobolev-spline kernels for Klein-Gordon SPDEs show that the kernel-based collocation method produces the well-behaved approximate probability distributions of the SPDE solutions.

Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ: dX(t) = A(X(t))dt+Φ(t)dB H(t), where A is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.

Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.

Some L p and Hölder Estimates for Divergence Type Nonlinear SPDEs on C 1-Domains

We prove uniqueness and existence of certain nonlinear stochastic partial differential equations (SPDEs) of divergence type defined on C 1-domains. Some L p and Holder estimates of the solution and its gradient are also obtained.

Identification of non‐linear stochastic spatiotemporal dynamical systems

A systematic identification method for non-linear stochastic spatiotemproal (SST) systems described by non-linear stochastic partial differential equations (SPDEs) is investigated in this study based on pointwise observation data. A theoretical framework for a semi-finite element model approximating to an infinite-dimensional system is established, and several fundamental issues are discussed including the approximation error between the underlying infinite-dimensional dynamics and the model to be identified, and its rationality etc. Based on the proposed theoretical framework, a general identification method with irregular observation data is provided. These results not only provide an effective method for the identification of non-linear SST systems using measurement data (both offline and online), but also demonstrate a potential solution for the analysis, design and control of non-linear SST systems from a numerical point of view.

Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

Approximating nonlinearities in stochastic partial differential equations (SPDEs) via the Wick product has often been used in quantum field theory and stochastic analysis. The main benefit is simplification of the equations but at the expense of introducing modelling errors. In this paper, we study the accuracy and computational efficiency of Wick-type approximations to SPDEs and demonstrate that the Wick propagator, i.e. the system of equations for the coefficients of the polynomial chaos expansion of the solution, has a sparse lower triangular structure that is seemingly universal, i.e. independent of the type of noise. We also introduce new higher-order stochastic approximations via Wick–Malliavin series expansions for Gaussian and uniformly distributed noises, and demonstrate convergence as the number of expansion terms increases. Our results are for diffusion, Burgers and Navier–Stokes equations, but the same approach can be readily adopted for other nonlinear SPDEs and more general noises.

A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation

We propose a convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation. The main difficulties in the convergence analysis for this nonlinear stochastic partial differential equation are to properly address the pointwise sphere condition and, in the fully discrete scheme, the Stratonovich noise. Approximations of the scheme proposed here satisfy a sphere constraint at nodal points of the spatial discretization and have finite energies, and their increments may be controlled uniformly with respect to discretization parameters. Sequences of corresponding continuified processes may then be generated which construct weak martingale solutions of the limiting equations.

Homogenization of Nonlinear Stochastic Partial Differential Equations in a General Ergodic Environment

In this article, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization of a stochastic nonlinear partial differential equation is addressed. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use the concept of sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem.

A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: Adaptivity and generalizations

This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/Np)3) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and Np is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than Np and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier–Stokes equations and the Boussinesq approximation with Brownian forcing.

Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm

This paper introduces a generalization and automation of the Wiener Hermite expansion with perturbation (WHEP) technique to solve a class of stochastic nonlinear partial differential equations with a perturbed nonlinearity. The automated algorithm generates the deterministic resultant linear equations according to the application of a general linear differential operator and the input parameters. Sample output with different nonlinearities, orders and corrections are presented. The resultant equations are solved numerically and the ensemble average and variance are computed and compared with previous research work. Higher order solutions with higher corrections are computed to show the importance of the generalization of the WHEP technique. The current work extends the use of WHEP for solving stochastic nonlinear differential equations.

The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \mathbb{R}^d$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in [Conus-Dalang, 2008] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at $(t,x)\in]0,T]\times\mathbb{R}^d$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.

Asymptotic behavior for neutral stochastic partial differential equations with infinite delays

This paper is concerned with the existence and asymptotic behavior of mild solutions to a class of non-linear neutral stochastic partial differential equations with infinite delays. By applying fixed point principle, we present sufficient conditions to ensure that the mild solutions are exponentially stable in $p$th-moment ($p\geq 2$) and almost surely exponentially stable. An example is provided to illustrate the effectiveness of the proposed result.

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

We introduce a new family of explicit integrators for stiff Ito stochastic differential equa- tions (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize re- duction faced by standard explicit methods. The family is based on the standard second or- der orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, and the mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

Numerical Analysis of the Stochastic Moving Boundary Problem

We consider a numerical solution of the stochastic moving boundary value problem, whose existence and uniqueness of solution are proved in [16]. Numerical approximations are based on the transformation, which transforms the stochastic moving boundary problem whose spatial domain is a priori unknown to a nonlinear stochastic partial differential equation which has a fixed spatial domain. We construct a numerical solution of the nonlinear stochastic partial differential equation and investigate the convergence theory.

LARGE PORTFOLIO ASYMPTOTICS FOR LOSS FROM DEFAULT

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogeneous portfolios. The density of the limiting measure is shown to solve a nonlinear stochastic partial differential equation, and certain moments of the limiting measure are shown to satisfy an infinite system of stochastic differential equations. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.

Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise

The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. These systems form the analytical core of the most advanced general circulation models. For this reason and due to their challenging nonlinear and anisotropic structure, the PEs have recently received considerable attention from the mathematical community.On the other hand, in view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the PEs and more generally.In this work we study a stochastic version of the PEs. We establish the global existence and uniqueness of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, estimates on the pressure and stopping time arguments.