Reflected Diffusion Processes And Some Applications

As we all know, reflected diffusion processes with one or two reflected boundary con-ditions, especially reflected Brownian motion and reflected Ornstein-Uhlenbeck process, have played an important role in many applications such as economics, finance, queueing, bioinformatics, and electrical engineering. Among economics and mathematical finance applications, we mention the target zones and exchange rate dynamics models which is originated in a seminal paper by Paul R. Krugman. In exchange rate target-zone theory, the spot exchange rate has two lower and upper barriers and is allowed to freely float inside the two boundaries without any interventions. But the monetary authority or the central banks should enforce some interventions in the foreign exchange market as soon as the spot log-exchange rate hits the pre-setted boundaries in order to prevent the ex-change rate from touching the upper boundary, and to prohibit the exchange rate from fall bellowing the minimum boundary. This characteristic stimulate us to employ reflecting diffusion process with two barriers to characterize the dynamics of the log-exchange rate in a target zone. This thesis is organized with two important applications:Firstly, we will exploit the reflecting diffusion process with two barriers to model the target-zone exchange rate model in this thesis. In the following sections, we study the properties of our target-zone model in details. We compute the steady-state distribution of the log-exchange rate and attain the explicit expression. By simulation and comparison with the stationary distribution of the two barriers reflecting Brownian motion, we can obviously find that the probability of the reflected Brownian motion at high levels in steady state is much greater than that for our exchange rate model. In my thesis, we also provide a rigorous verification for this conclusion. Subsequently, we apply the Ito formula and Girsanov theorem to construct the relations of the transition density function between the log-exchange rate process and the reflected Brownian motion. Therefore, we attain the Kolmogrov backward equation and an approximation to the transition density function of the log-exchange rate process. In the fifth section of the third chapter, we discuss the properties of the first arrival time of the log-exchange rate dynamic process. We get the mean time which the log-exchange rate dynamic process spend in traveling from the original point to another point. At last, we offer the European call currency option pricing formulas about the without realignment model and realignment model, respectively. Secondly, we resort to the reflecting Brownian motion to resolve a system with diffus-ing and reacting chemicals in bioinformatics. It is interesting to note that the molecules of proteins in an aqueous solution are in everlasting motion as a result of the thermal energy and these molecules will collide with water molecules or other molecules in their neigh-borhood. On the one hand, owing to such collisions the trajectory of a molecule is not straight any more. In turn, it practises a random walk. At the same time, these molecules slowly diffuse around. On the other hand, some appropriate molecular collisions may also cause chemical reactions. Consequently, diffusion and chemical reactions are viewed as two fundamental processes in molecular biology. In this thesis, we will use reflected and absorbed Brownian motion and stochastic differential equations to construct a closed form solution to one dimensional Robin boundary problems. Meanwhile, we will give a reason-able explanation to the closed form solution from a stochastic point of view. Finally, we will extend the problem to Robin boundary problem with two boundary conditions and give a specific solution by resorting to a stopping time.