Limit Theorems On Stochastic Differential Equations With Jumps And Applications In Finance

Stochastic differential equation (SDE) plays an important role in mathematical finance. We often use the continuous diffusion process which is the solution of SDE driven by Brownian motion to describe the behavior of asset prices. However, many important events, such as new inventions, wars, economic polices and other news can cause the stock price to make a sudden shift. To model this, the asset price descirbed by a SDE with jumps is more suitable. The main objective of this work is to consider the limit theorems on stochastic equations of non-negative processes with jumps and applications in finance. First, we get the small noise asymptotic results for stochastic differential equation with jumps. We prove that the sequence of SDEs with jumps satisfies the uniform large deviation principle when the coefficients are Lipschitz under some assumptions. Moreover, we establish the large deviation principle in the Sko-rokhod space D([0,∞), [0,∞)) endowed with the Skorokhod-Lindvall topology d. We use these results to explain the probabilistic behavior of the meaning reverting asset process such as the stock price process. Second, we get some stability results of optimal investment in a simple Levy market, where the stock price is driven by a Brownian motion plus a Poisson process. The optimal investment portfolios are given explicitly under the hypotheses that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solutions for the HARA utility are stable in the sense of weak convergence when the parameters vary in a suitable way.