On Time-changed Processes And Their Applications In Financial Market

This dissertation consists of two parts:theoretical research and applied research. In the first part, we define several stochastic processes and study some properties of the corresponding processes, such as Laplace transform, martingale property and fractional diffusion equations. In the second part, we use the relevant processes as the underly-ing price model to describe the market data and study option pricing under different assumptions.The main content of this paper is divided into five chapters.In the first chapter, the background of our research work and research significance as well as the main results are introduced.In the second one, we define a subordinator Uv(t) determined by a Borel prob-ability measure v on (0,1). In some sense, this subordinator can be considered as the mixture of independent a-stable subordinators under the given measure v. We prove the existence of this subordinator and give some properties of the process and its inverse. Then we apply the inverse process defined as a random time of Brownian motion to generate a time-changed Brownian Motion. We discuss the diffusion property of the process and prove that it is a martingale.In chapter3, we study the connection between time-changed geometric Brown-ian motion and fractional differential equations. We prove that the probability density function of time-changed geometric Brownian motion satisfies a fractional differential equation of distributed order.In chaper4, we discuss the martingale property of time-changed geometric Brow-nian motion. We prove the existence of equivalent martingale measure, with respect to which the time-changed geometric Brownian motion is a martingale. Moreover, this kind of measure is not unique, so we can show that the market is arbitrage free but incomplete under the corresponding time-changed BS model. In the end of this chap-ter, we respectively give the option pricing formulae for European call options under different hypotheses of the risk-less interest rate.In the last chapter, we give another extension form of BS model named time-changed fractional BS model by combining subdiffusive and fractional BS models. The underlying asset of this model follows a time-changed geometric fractional Brownian motion, We study the related properties of the process and derive the pricing formula for the European call option in discrete time setting.