Series Expansions for Stochastic Partial Differential Equations with Symmetric $$\alpha $$-Stable Lévy Noise

Understanding phenomena ranging from biological processes to financial markets involves uncertainty. Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs) serve as robust mathematical frameworks for modelling such systems. Given the stochastic influences within these models, comprehending the dynamics of complex systems becomes pivotal for accurately predicting system behaviour. However, traditional numerical techniques frequently encounter challenges in effectively addressing the intricacies and stochastic properties inherent in these equations. This chapter explores several numerical methods that offer streamlined and dependable solutions capable of handling the complexities inherent in stochastic differential and partial differential equations. Also, numerical challenges associated with these methods are discussed and the solution strategies are also suggested.

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

On Stochastic Processes Defined by Differential Equations with a Small Parameter

Neural Jump SDEs (Jump Diffusions) and Neural PDEs

Stochastic differential equations (SDEs) are playing a growing role in financial mathematics, actuarial sciences, physics, biology and engineering. For example, in financial mathematics, fluctuating stock prices and option prices can be modeled by SDEs. In this chapter, we focus on a numerical simulation of systems of SDEs based on the stochastic Taylor series expansions. At first, we apply the vector-valued Ito formula to the systems of SDEs, then, the stochastic Taylor formula is used to get the numerical schemes. In the case of higher dimensional stochastic processes and equations, the numerical schemes may be expensive and take more time to compute. We deal with systems with standard n-dimensional systems of SDEs having correlated Brownian motions. One the main issue is to transform the systems of SDEs with correlated Brownian motions to the ones having standard Brownian motion, and then, to apply the Ito formula to the transformed systems. As an application, we consider stochastic partial differential equations (SPDEs). We first use finite difference method to approximate the space variable. Then, by using the stochastic Taylor series expansions we obtain the discrete problem. Numerical examples are presented to show the efficiency of the approach. The chapter ends with a conclusion and an outlook to future studies.

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

Stochastic partial differential equations are derived for the reaction-diffusion process in one, two, and three dimensions. Specifically, stochastic partial differential equations are derived for the random dynamics of particles that are reacting and diffusing in a medium. In the derivation, a discrete stochastic reaction-diffusion equation is first constructed from basic principles, that is, from the changes that occur in a small time interval. As the time interval goes to zero, the discrete stochastic model leads to a system of It stochastic differential equation. As the spatial intervals approach zero, a stochastic partial differential equation is derived for the reaction-diffusion process. The stochastic reaction-diffusion equation can be solved computationally using numerical methods for systems of It stochastic differential equations. Comparisons between numerical solutions of stochastic reaction-diffusion equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.

In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion.

Probability theory is the study of random phenomena. Many dynamical systems with random influence, in nature or artificial complex systems, are better modeled by equations incorporating the intrinsic stochasticity involved. In probability theory, stochastic partial differential equations (SPDEs) generalize partial differential equations through random force terms and coefficients, while stochastic differential equations (SDEs) generalize ordinary differential equations. They are both abound in models involving Brownian motion throughout science, engineering and economics. However, Brownian motion is just one example of a random noisy input. The goal of this thesis is to make contributions in the study and applications of stochastic dynamical systems involving a wider variety of stochastic processes and noises. This is achieved by considering different models arising out of applications in thermal engineering, population dynamics and mathematical finance.1. Power-type non-linearities in SDEs with rough noise: We consider a noisy differential equation driven by a rough noise that could be a fractional Brownian motion, a generalization of Brownian motion, while the equation's coefficient behaves like a power function. These coefficients are interesting because of their relation to classical population dynamics models, while their analysis is particularly challenging because of the intrinsic singularities. Two different methods are used to construct solutions: (i) In the one-dimensional case, a well-known transformation is used; (ii) For multidimensional situations, we find and quantify an improved regularity structure of the solution as it approaches the origin. Our research is the first successful analysis of the system described under a truly rough noise context. We find that the system is well-defined and yields non-unique solutions. In addition, the solutions possess the same roughness as that of the noise.2. Parabolic Anderson model in rough environment: The parabolic Anderson model is one of the most interesting and challenging SPDEs used to model varied physical phenomena. Its original motivation involved bound states for electrons in crystals with impurities. It also provides a model for the growth of magnetic field in young stars and has an interpretation as a population growth model. The model can be expressed as a stochastic heat equation with additional multiplicative noise. This noise is traditionally a generalized derivative of Brownian motion. Here we consider a one dimensional parabolic Anderson model which is continuous in space and includes a more general rough noise. We first show that the equation admits a solution and that it is unique under some regularity assumptions on the initial condition. In addition, we show that it can be represented using the Feynman-Kac formula, thus providing a connection with the SPDE and a stochastic process, in this case a Brownian motion. The bulk of our study is devoted to explore the large time behavior of the solution, and we provide an explicit formula for the asymptotic behavior of the logarithm of the solution.3. Heat conduction in semiconductors: Standard heat flow, at a macroscopic level, is modeled by the random erratic movements of Brownian motions starting at the source of heat. However, this diffusive nature of heat flow predicted by Brownian motion is not observed in certain materials (semiconductors, dielectric solids) over short length and time scales. The thermal transport in these materials is more akin to a super-diffusive heat flow, and necessitates the need for processes beyond Brownian motion to capture this heavy tailed behavior. In this context, we propose the use of a well-defined Levy process, the so-called relativistic stable process to better model the observed phenomenon. This process captures the observed heat dynamics at short length-time scales and is also closely related to the relativistic Schrodinger operator. In addition, it serves as a good candidate for explaining the usual diffusive nature of heat flow under large length-time regimes. The goal is to verify our model against experimental data, retrieve the best parameters of the process and discuss their connections to material thermal properties.4. Bond-pricing under partial information: We study an information asymmetry problem in a bond market. Especially we derive bond price dynamics of traders with different levels of information. We allow all information processes as well as the short rate to have jumps in their sample paths, thus representing more dramatic movements. In addition we allow the short rate to be modulated by all information processes in addition to having instantaneous feedbacks from the current levels of itself. A fully informed trader observes all information which affects the bond price while a partially informed trader observes only a part of it. We first obtain the bond price dynamic under the full information, and also derive the bond price of the partially informed trader using Bayesian filtering method. The key step is to perform a change of measure so that the dynamic under the new measure becomes computationally efficient.

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

Physics-informed variational inference for uncertainty quantification of stochastic differential equations

Contaminated sites are recognized as the “hotspot” for the development and spread of antibiotic resistance in environmental bacteria. It is very challenging to understand mechanism of development of antibiotic resistance in polluted environment in the presence of different anthropogenic pollutants. Uncertainties in the environmental processes adds complexity to the development of resistance. This study attempts to develop mathematical model by using stochastic partial differential equations for the transport of fluoroquinolone and its resistant bacteria in riverine environment. Poisson’s process is assumed for the diffusion approximation in the stochastic partial differential equations (SPDE). Sensitive analysis is performed to evaluate the parameters and variables for their influence over the model outcome. Based on their sensitivity, the model parameters and variables are chosen and classified into environmental, demographic, and anthropogenic categories to investigate the sources of stochasticity. Stochastic partial differential equations are formulated for the state variables in the model. This SPDE model is then applied to the 100 km stretch of river Musi (South India) and simulations are carried out to assess the impact of stochasticity in model variables on the resistant bacteria population in sediments. By employing the stochasticity in model variables and parameters we came to know that environmental and anthropogenic variations are not able to affect the resistance dynamics at all. Demographic variations are able to affect the distribution of resistant bacteria population uniformly with standard deviation between 0.087 and 0.084, however, is not significant to have any biological relevance to it. The outcome of the present study is helpful in simplifying the model for practical applications. This study is an ongoing effort to improve the model for the transport of antibiotics and transport of antibiotic resistant bacteria in polluted river. There is a wide gap between the knowledge of stochastic resistant bacterial growth dynamics and the knowledge of transport of antibiotic resistance in polluted aquatic environment, this study is one step towards filling up that gap.

Stochastic partial differential equations for the one-dimensional telegraph equation and the two-dimensional linear transport equation are derived from basic principles. The telegraph equation and the linear transport equation are well-known correlated random walk (CRW) models, that is, transport models characterized by correlated successive-step orientations. In the present investigation, these deterministic CRW equations are generalized to stochastic CRW equations. To derive the stochastic CRW equations, the possible changes in direction and particle movement for a small time interval are carefully determined. As the time interval decreases, the discrete stochastic models lead to systems of It stochastic differential equations. As the position intervals decrease, stochastic partial differential equations are derived for the telegraph and transport equations. Comparisons between numerical solutions of the stochastic partial differential equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.