Stochastic Analysis on Groups of Diffeomorphisms

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.

The purpose of this chapter is to overview elements of the theory of stochastic differential equations, based on Wiener processes, for use in the subsequent chapters. This theory was founded by K. Itô in 1942 (Itô, Zenkoku Shijo Sugaku Danwakai 244:1352–1400, 1942; Itô, On stochastic differential equations. Memoirs of the American Mathematical Society, vol 4. AMS, New York City, 1951). His aim was to construct Markov processes, governed by Kolmogorov’s differential equations via Wiener processes, and to analyze their sample paths. After that, stochastic differential equations have been used to describe dynamical processes in random environments of various fields. Here we consider stochastic differential equations with random coefficients, because we aim at studying stochastic control problems.The chapter is organized as follows. Section 1.1 is preliminaries. The basic definitions and results on stochastic processes are collected for later use. Stochastic differential equations and stochastic analysis will be introduced in Sect. 1.2. Section 1.3 deals with asset pricing problems as an application of previous results.KeywordsBackward Stochastic Differential EquationsAsset Pricing ProblemsConstructing Markov ProcessesRandom CoefficientsBurkholder Davis Gundy InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper uses stochastic differential equation and risk theory to study the probability model of insurance company's bankruptcy function and the proportion model of financial insurance investment. It also studies the application of stochastic probability to the investment decision-making of institutional investors. Taking the four major insurance industries in China as an example, it gives the strategy of venture capital.

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

Adaptive stepsize based on control theory for stochastic differential equations

A theory of stochastic differential equations driven by predictable processes in Stratonovich sense is developed. These driving processes include a large class of discontinuous semimartingales. The theory of stochastic differential equations driven by continuous semimartingales in Stratonovich sense is extended without involving Lebesgue-Stieltjes integrals as done by Meyer. Moreover, a change of variables formula without extra terms involving the jumps of the processes holds for this theory. Results on approximation of driving processes are preserved.

The aim of this paper is to develop some theories of stochastic partial functional differential equations (SPFDEs) driven by infinite dimensional Wiener processes under the quasi-local Lipschitz condition and the restriction growth condition. Firstly, we establish the existence-uniqueness theorem of the global mild solutions for SPFDEs by using the intercept technique. Then, we discuss asymptotic behavior of the solutions. Furthermore, some criteria of exponential stability in the mean square are obtained by using Lyapunov method. An example is provided to show the effectiveness of the theoretical results.

The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type

In this article, we comment on the well known connection between noise and Q-Wiener processes. In physical applications, noise is usually given as a generalized Gaussian process and all assumptions are formulated in terms of the correlation functional. In contrast, the mathematical theory of stochastic partial differential equations on bounded domains is in many cases formulated with Q-Wiener processes. Our aim is to relate frequently made assumptions on the covariance operator Q to assumptions on the correlation function of the noise process, which is the kernel of Q. One of our main results gives necessary and sufficient conditions when the Wiener process is given as a series expansion in terms of eigenfunctions of the Laplacian.

This paper is focused on the development of a method to determine variances of the stochastic responses in transmission line models with random excitations. It is based on the theory of stochastic differential equations (SDE), namely a system of vector linear SDEs with additive noise is formulated to be prepared to simulate respective stochastic trajectories. One possibility to get the moments of the stochastic process is to determine sample statistics from a set of individual trajectories. This is, however, rather time consuming, and it's benefit when compared to common methods is not so obvious. In this paper differential equations for getting directly the first two stochastic response moments of transmission line models under randomly varying excitations are formulated and solved. A verification is performed via the above stated approach, while the Student's t-distribution can directly be used to determine respective confidence intervals in case of an additive noise. All simulations have been performed in a Matlab language environment.

The paper deals with a technique for the analysis of multiconductor transmission lines (MTL) with randomly varied parameters, that is based on the theory of stochastic differential equations (SDE). Sets of stochastic trajectories are computed as voltage or current responses, accompanied by relevant sample means and confidence intervals. The MTL model is based on a cascade connection of generalized RLGC networks, terminating circuits are replaced by their generalized Thévenin equivalents. To develop model equations a state-variable method is applied, and then a corresponding vector SDE is formulated. Finally, a stochastic implicit Euler numerical technique is used for the numerical solution being consistent with Itô stochastic calculus. All the computation were done in the MATLAB <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">®</sup> language and deterministic responses are also stated via a numerical inverse Laplace transforms (NILT) procedure to verify the results.

Nonlinear mechanisms of long-term variations in the Caspian Sea level are described. It is shown that with account taken of the dependence of the evaporation depth from the Volga basin surface on soil moisture content and the dependence of the evaporation depth from the sea surface on its level, we obtain a fundamentally new (chaotic) oscillation mechanism with several attraction levels. The stochastic differential equations describing the water budget of the sea basin and the sea proper and the respective solutions of the Fokker–Planck–Kolmogorov equation are shown to have stationary bimodal density of the level probability. The random process, characterizing the sea level variations at a nonlinear dependence between the evaporation rate and the level is found to be non-Gaussian. Noise-induced transitions, caused by nonlinear evaporation processes are described. A new nonlinear stochastic theory describing the Caspian Sea level variations and based on predicted physical effects is suggested.

Global mean square exponential stability of stochastic neural networks with retarded and advanced argument

Stochastic partial differential equations constitute a relatively new subject. There have been a number of interesting papers in the last few years, but the field is still in its infancy if compared with the highly developed theory of stochastic ordinary differential equations. In this paper we report on a study of a non linear heat equation in a finite interval of space subject to a white noise forcing term. Due to the non linearity the equation, without the forcing term, exhibits several equilibrium configurations two of which are stable, actually asymptotically stable. The solution of the complete forced equation is a stochastic process in space and time with continuous sample paths whose behaviour we study in the limit of small noise. We obtain lower and upper bounds for the probability of large fluctuations and then apply our estimates to the calculation of the transition probability between the stable configurations (tunneling).