Hyperbolic Stochastic Partial Differential Equations Research Articles

Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts

Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

We treat several classes of hyperbolic stochastic partial differential equations in the framework of white noise analysis, combined with Wiener–Itô chaos expansions and Fourier integral operator methods. The input data, boundary conditions and coefficients of the operators are assumed to be generalized stochastic processes that have both temporal and spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate Sobolev–Kondratiev or weighted-Sobolev–Kondratiev spaces. Moreover, an explicit chaos form of the solutions is obtained, useful for calculating expectations, variances and higher order moments of the solution.

Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients

We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result.

Random-field solutions of weakly hyperbolic stochastic partial differential equations with polynomially bounded coefficients

We study random-field solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider linear equations under suitable hyperbolicity hypotheses, and we provide conditions on the initial data and on the stochastic term, namely, on the associated spectral measure, so that these kind of solutions exist in suitably chosen functional classes. We also give a regularity result for the expected value of the solution.

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

English

We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small L2-perturbations.

Sigma-convergence of semilinear stochastic wave equations

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenomenon modelled by the underlying problem. We also provide an appropriate scheme for the approximation of the effective coefficients. To illustrate our approach, we work out some concrete problems such as the periodic homogenization problem, the almost periodic and the asymptotically almost periodic ones.

Dynamic Pricing for On-Demand Ride-Sharing: A Continuous Approach

We investigate the dynamic pricing problem in on-demand ride-sharing using a continuous-time continuous-space approach. A ride-sharing platform controls two sides of the market, supply (vacant vehicles) and demand (customers' trip requests) via setting dynamic prices and commissions with the objective of maximizing its expected revenue in the infinite horizon. The dynamic model of supply is described by the multi-population traffic flow with an intergroup transfer (a system of hyperbolic stochastic partial differential equations); the demand subjects to independent stochastic processes. This continuous setting allows solving the revenue maximization by optimal control without treating combinatorial explosions. The demand-supply-based pricing is smooth in space and the traffic congestion resulted from control is also considered. This work provides a macroscopic perspective in handling the complicated spatiotemporal pricing problem in ride-sharing and similar matching markets.

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.

English

english

A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation

This paper deals with the homogenization of a linear hyperbolic stochastic partial differential equation (SPDE) with highly oscillating periodic coefficients. We use Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized linear hyperbolic SPDE with constant coefficients. We also prove the convergence of the associated energies.

Simulation of volatility modulated Volterra processes using hyperbolic stochastic partial differential equations

We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and L\'{e}vy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we demonstrate our method with an example from the energy finance literature.

Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains

In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating � -periodic coefficients in a domain with small holes (of size-� ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established.

Newton’s method for first-order stochastic functional partial differential equations

We apply Newton’s method to hyperbolic stochastic functional partial differential equations of the first order driven by a multidimensional Brownian motion. We prove a first-order convergence and a second-order convergence in a probabilistic sense.

Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov's and Skorokhod's probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result. Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities at the microscopic level using prop- erties of the homogeneous media obtained by homogenizing these materials. These heterogeneous ma- terials consist of finely mixed different components like soil, paper, concrete for building, fibreglass, materials used in the manufacturing of high tech equipments such as planes, rockets and so on. This signifies that almost everything around us in real life is a heterogeneous material. The physical problems described on heterogeneous materials such as heat, mechanical constraints, flow of fluids in these me- dia leads to the study of PDEs with highly oscillating coefficients depending on macroscopic scales or boundary value problems for PDEs in domain with fine grained boundaries. The main obstacle in solv- ing these problems arises either from the character of the domain or the presence of high oscillations in the coefficients of the governing equation. To this end, it is expensive to compute solutions to these type of problems. Numerical methods have proved inefficient in solving such problems due to the fact that even the most advanced parallel computers are unable to simulate schemes related to the physically interesting such problems.

The forward dynamics in energy markets – infinite-dimensional modelling and simulation

In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.

Numerical methods for hyperbolic SPDEs: a Wiener chaos approach

In this paper we propose a novel numerical scheme based on the Wiener chaos expansion for solving hyperbolic stochastic partial differential equations (PDEs). Through the Wiener chaos expansion the stochastic PDE is reduced to an infinite hierarchy of deterministic PDEs which is then truncated to a finite system of PDEs, that can be addressed by standard techniques. A priori and a posteriori convergence results for the method are provided. The proposed method is applied to solve the stochastic wave equation with additive noise and the stochastic Klein–Gordon wave equation with multiplicative noise and the results are compared to those derived by the Monte Carlo method. The main advantages of the proposed scheme is that it provides almost identical results and is significantly faster than the Monte Carlo simulation method, providing a convenient way to compute numerically the statistical moments of the solution.

Multilevel Monte Carlo method with applications to stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h −2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.

On Solution Regularity of Linear Hyperbolic Stochastic PDE Using the Method of Characteristics

AbstractThe generalized Polynomial Chaos (gPC) method is one of the most widely used numerical methods for solving stochastic differential equations. Recently, attempts have been made to extend the the gPC to solve hyperbolic stochastic partial differential equations (SPDE). The convergence rate of the gPC depends on the regularity of the solution. It is shown that the characteristics technique can be used to derive general conditions for regularity of linear hyperbolic PDE, in a detailed case study of a linear wave equation with a random variable coefficient and random initial and boundary data.