Properties And Applications On Several Classical Kinds Of Stochastic Differential Equations

This doctoral dissertation consists of three parts to explore the properties and ap-plications of some classical stochastic (partial) differential equations (abbr. SDEs and SPDEs).In Chapter1, we mainly research a typical diffusion-skew diffusion processes. They, that are quickly attract many scholars since they are first proposed by K. It0and H.P. Mckean[1], are the generalization of diffusion processes. The difference between skew and ordinary diffusion is there exists an additional symmetric local time of one point. The skew diffusion processes satisfies the following SDE: dX1=μ(X1)dt+σ(Xt)dWt+βdLtX(a), where a is called the locus and|β|<1.The existence of symmetric local time makes the sample path of these processes interesting. When these processes haven’t hit the point a, their sample paths are just like the ordinary ones. They will go to upside with probability (1+β)/2and downside with probability (1-β)/2as long as they arrive at a. It is also worth noting that they will reduce to the ordinary diffusion processes if β=0and re-flected processes if|β|=1. We study the properties and applications of skew diffusion processes while taking skew OU as an example. In Section1.1, we get the properties of the processes themselves, such as the existence and uniqueness of the solution, the tran-sition densities and the distributions of the first hitting times, and also their applications in finance. In Section1.2, we compute the occupation time of stochastic diffusions. Occupation times are interested for stochastic processes. But except for certain well studied examples, explicit expressions are rarely known. Here we given the explicit expressions of Laplace transform of occupation time in interval [a,b] before they exit [c,d](c<a<b<d) for skew OU processes by using the method developed in [2].In Chapter2, we estimate the parameters in two common SDEs. Parameter es-timations in SDEs are also important research directions. In this chapter, we try to estimate some typical parameters. In Section2.1, we give the closed-form maximum likelihood estimation for one type of affine point processes which are widely used in credit risk pricing. In Section2.2, a kernel estimator in a parabolic stochastic partial differential equation driven by fractional noises is given.Chapter3mainly study the properties of stochastic partial differential equation-s. Stochastic partial differential equations is one of the most fast developing research fields in probability. Similar to the classification of partial differential equations, SPDEs are divided into three categories:parabolic type, elliptic type and hyperbolic type. In Section3.1, we first consider a familiar hyperbolic equations-wave equations. Based on the work done by Jiang et al.[3], in which the global weak and strong solutions of stochastic wave equations driven by compensated Poisson random measures arc es-tablished, we consider their long time behaviors. We obtain, under some appropriate conditions, the solutions are exponential stable. In Section3.2, we discuss the parabol-ic SPDE with reflection. It is known to us that, in the physical real world, it is ideal to assume the driven term to be white noise, sometimes it maybe time dependence. At the same time, the solutions to these equations have some boundary restrictions. Thus we propose stochastic partial differential equations driven by fractional noises with re-flection. We first obtain the existence and uniqueness of the solution. Moreover, we study the large deviation principle. In the last Section3.3of this chapter, we discuss a class of Cahn-Hilliard type stochastic interacting systems with stepping-stone noises. The diffusion coefficients are not Lipstchitz since the existence of stepping-stone noise, which presents considerable difficulties. We have to approximate them using SPDEs with Lipstchitz coefficients. Finally the existence of weak mild solution to this system by solving a martingale problem.