Some Stochastic Models And Their Applications In Finance

In this doctorial dissertation, the main content is made up of three types of stochastic models. The first one is a pair of interacting stochastic partial differential equations (abbr. SPDEs). We proved the week (mild) solution of the system can be approximated by a pair of Markov chains with specified birth, death and transition rates. Based on this approximation, the solution obtained. The second model is a storage model fed by a Markov modulated Brownian motion (abbr. BM). The main effort is devoted to proving some limit properties of its loads as time goes to infinity. For the third model, as an application of stochastic models in finance, we establish a new term structure of forward rate modeled by a hyperbolic SPDE, and then this term structure model is applied to pricing CDS.More specially, the dissertation is made up of the following three chapters.In Chapter 1, we prove the existence of weak (mild) solutions to a pair of inter-acting SPDEs with branching noises, which is also known as a competitive stochastic Lotka-Volterra system, by approximating the equations with a pair of space-time rescaled particle systems. More accurately, we start with a pair of Markov chains with specified birth-death rates. Then by appropriate space-time rescaling, we apply Dynkin's formula to arrive at a pair of HN-valued stochastic differential equations in mild form, where HN is the space of all step functions taking constants on each interval [κ-1/N,κ/N).Note that the martingale components of the system can be de-composed into three parts according to the reaction, diffusion and branching jumps, respectively. We first present some up bound estimations and convergence results concerning the components of the system. Then the tightness for these processes is proved under some appropriate Sobolev spaces. Based on the tightness results and employing Prohorov's theorem and Skorohod representation theorem, we can prove that there is a subsequence which converges to the weak (mild) solution to the system.In Chapter 2, we explore-a storage model fed by a Markov modulated Brownian motion (abbr. MMBM). In recent years, MMBMs have been becoming increasingly popular in describing various financial and queue models, see, e.g., Asmussen (2003). Two indicative properties about the process are studied in this chapter. Firstly, we prove the existence of the stationary distribution under some mild conditions. When we assume that the initial distribution of the process is this stationary law, we give the growth rate of the running maximum process, which is proved to grow like log t when the time t goes to infinity.In Chapter 3, a new term structure of the instantaneous forward rate is proposed. Rather than describing the dynamics of short rates, Heath et al (1992) proposed a new model to formulate the term structure of forward rate directly. The model had been extensively studied in the literature. While the two time arguments are treated by different means:one as a variable and the other as a parameter. In this chapter, a wave-typed SPDE is used to formulate the term structure of forward rate. The inconsistent treatment of the arguments can be overcome under this model. Another advantage to use the wave-typed equation is that they can capture the stochastic shocks. Further we derive the conditions such that the model is arbitrage-free. The conditions are expressed by the drift term of the forward rate similar to that of H JM conditions. Finally applying the bonds corresponding to this forward rate, several defaultable derivatives are priced, such as credit default swaps.