White Noise Analysis For Lévy Processes And The Applications

Levy processes are the most important parts in stochastic analysis, which have abundant mathematic structure. They have been used in a broad range of applications, such as physics and finance. A Levy process can be written as a linear combination of time t. a Brownian motion and a pure jump Levy process. And a Levy white noises are regarded as the derivative of a Levy process. In this thesis, we construct the framework of Levy white noises which are combined by Gaussian white noises and pure jump Levy white noises. Furthermore we study its applications in stochastic differential equations, finance, stochastic population and principal-agent problem. The main results, obtained in this thesis, are summarized as follows:1. We construct the framework of Levy white noises by combining the framework of Gaussian white noises and that of pure jump Levy white noises. The white noise approach to stochastic differential equation (SDE) driven by a Levy process is based on the characterization theorem of stochastic distribution space in this framework. The main idea of white noise approach can be concluded as follows:the SDE is firstly reduced to the deterministic differential equation (DDE) by Hermite transform, which can be solved. Then the solution of the SDE is obtained by the characterization theorem of stochastic distribution space, converting the solution of the DDE to a distribution. In the framework of pure jump Levy white noises, we give the explicit solution of the stochastic Schrodinger equation (SSE) driven by pure jump Levy white noises. And we prove that the solution of SSE is in L1(v) in sense of weak distribution under some conditions. In the framework of Levy white noises, we apply white noise approach to stochastic transport equation (STE) driven by Levy white noises. The explicit solution of the STE is obtained in stochastic distribution space. Moreover, we get the explicit solution of the stochastic heat equation by the solution of the STE.2. In the framework of Levy white noises, we give the white noise generalization of the Clark-Haussmann-Ocone theorem for Levy processes. As an application, in a financial market driven by Levy processes, the optimal replicating portfolios for a European option are represented by the explicit functional of Malliavin derivatives under full in-formation and under partial information. On the other hand, we set up by white noise analysis the stochastic population equation in a crowded environment perturbed by Levy processes. And the stochastic singular control is introduced to study the optimal harvesting problem. Based on the verification theorem of Integrovariational Inequali-ties, we reduce the optimal harvesting problem to solving a deterministic differential equations. The solution gives the optimality conditions.3. We introduce stochastic control, combined optimal stopping-stochastic control and stochastic differential game to study a principal-agent problem with Lump-sum pay-ments. Firstly, for a given linear contract in continuous diffusion setting, we develop the classic verification theorem of HJB equations in weak formulation and present the optimality conditions for the agent's problem in hidden actions. Secondly, the agent is allowed to exercise the contract prior the terminal time. For a given general contract, the agent's problem is formulated as an combined optimal stopping-stochastic control problem. In Levy diffusion setting, the classic verification theorem of Variational-Inequality-HJB equations is generalized in weak formulation to give optimality con-ditions for the agent's optimal exercise time in the case of hidden actions. Finally, when the principal is allowed to choose exercise time in Levy diffusion setting, we formulate the principal-agent problem as a nonzero-sum optimal stopping-stochastic control differential game between the principal and the agent. The principal controls stopping and the agent controls stochastic control. We prove a verification theorem in term of Variational-Inequality-HJB equations to provide the optimality conditions for Nash equilibrium of the game. The interesting feature of Nash equilibrium is that, the principal can induce the agent to make the best efforts, by an appropriate stopping rule design. Conversely, the agent can force the principal to exercise at a time of the agent's choosing, by applying suitable efforts. With the help of the verification the-orem, the characterization of Nash equilibrium is reduced to the solutions of a family of nonlinear variational-integro inequalities. The theorem for the game is applicable to more general nonzero-sum optimal stopping-stochastic control differential game than the specified principal-agent problem studied in this thesis.