| This dissertation proposes a methodology for inference in the context of diffusion processes with jumps. There are many applications. For example, in finance, this methodology can be used to study asset pricing. My dissertation consists of two chapters which are closely related. They reveal the relationship between the power of a test, jump height and jump frequency. In the first chapter I construct a likelihood ratio test to test whether a diffusion process has jumps. This test statistic is independent of the distribution of jump height. I show the test is asymptotically optimal when the jump height is O(1/nalpha) , the jump frequency is O(1/ nbeta) where n is sample size, 3alpha + beta = 2, alpha > 1/2, beta > 0. By constructing this optimal test, I derive the asymptotic power envelopes for testing continuous diffusion process against diffusion processes with asymmetric jumps. In recent years, many tests for this problem were proposed. I compare the power of these tests with the envelopes using simulations.;In chapter two I test the continuous diffusion process against a diffusion process with symmetric jumps. I show my test statistic is optimal when the jump height is O(1/nalpha), the jump frequency is O(1/n.;beta) where n is sample size, 4alpha + beta= 5/2, alpha > 2/3,0 In chapter two I test the continuous diffusion process against a diffusion process with symmetric jumps. I show my test statistic is optimal when the jump height is O(1/n.;alpha), the jump frequency is O(1/n.;beta) where n issample size, 4alpha + beta = 5/2, alpha > 2/3, 0. |
