SDE-based variance simulation in transmission line models with random excitations

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.

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 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.

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

Adaptive stepsize based on control theory for stochastic differential equations

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.

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 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.

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

Everywhere in this chapter we use the notions and notation introduced in Chapter 5. We describe a stochastic differential equation of a special sort on those groups of diffeomorphisms that arise in the applications to viscous hydrodynamics described in Section 16.4 below (see, e.g., Gliklikh (1989, 1994, 2003)). This class of equations is characterized by the fact that they involve finite-dimensional Wiener processes. It should be pointed out that a theory of stochastic differential equations on infinite-dimensional manifolds involving infinite-dimensional Wiener processes exists (see, e.g., Belopolskaya and Dalecky (1989), Brzezniak and Elworthy (2000), Elworthy (1982)) and is used in viscous hydrodynamics (see, e.g., Cipriano and Cruzeiro (2007)). However, the description of this theory requires a complicated functional-analytic machinery that is not included in our exposition. For simplicity of presentation, we restrict ourselves to the finite-dimensional version of the theory since, in applications, the theories yield very similar 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.

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).

The theory of backward stochastic differential equations (BSDEs) was pioneered by Pardoux and Peng [PaPe90]. It became now very popular, and is an important field of research due to its connections with stochastic control, mathematical finance, and partial differential equations. BSDEs provide a probabilistic representation of nonlinear PDEs, which extends the famous Feynman-Kac formula for linear PDEs. As a consequence, BSDEs can be used for designing numerical algorithms to nonlinear PDEs.

Natural systems are undeniably subject to random fluctuations, arising from either environmental variability or thermal effects. The consideration of those fluctuations supposes to deal with noisy quantities whose variance might at times be a sizable fraction of their mean levels. It is known that, under these conditions, noisy fluctuations can interact with the system's nonlinearities to render counterintuitive behavior, in which an increase in the noise level produces a more regular behavior. In systems with spatial degrees of freedom, this regularity takes the form of spatiotemporal order. An overview is presented of the mechanisms through which noise induces, enhances, and sustains ordered behavior in passive and active nonlinear media, and different spatiotemporal phenomena are described resulting from these effects. The general theoretical framework used in the analysis of these effects is reviewed, encompassing the theory of stochastic partial differential equations and coupled sets of ordinary stochastic differential equations. Experimental observations of self-organized behavior arising out of noise are also described, and details on the numerical algorithms needed in the computer simulation of these problems are given.

The theory of stochastic differential equations is used in various fields of science and engineering. This paper deals with vector-valued stochastic integral equations. We show some applications of the presented theory to the problem of modelling RLC electrical circuits by noisy parameters. From practical point of view, the second-order RLC circuits are of major importance, as they are the building blocks of more complex physical systems. The mathematical models of such circuits lead to the second order differential equations. We construct stochastic models of the RLC circuit by replacing a coefficient in the deterministic system with a noisy one. In this paper we present the analytic solution of these equations using the Itô calculus and compute confidence intervals for the stochastic solutions. Numerical simulations in the examples are performed using Matlab.