Asymptotic expansions and distribution properties for diffusion processes

• Main Author: She, Qihao
• Other Authors: Nicolas Privault
• Format: Theses and Dissertations
• Language: English
• Published: 2017
• Subjects: DRNTU::Engineering::Mathematics and analysis::Simulations; DRNTU::Engineering::Computer science and engineering::Mathematics of computing::Probability and statistics
• Online Access: http://hdl.handle.net/10356/69543
• Institution: Nanyang Technological University

For this thesis, we derive new applications and theoretical results for some multidimensional mean-reverting stochastic processes via series expansion. We use Malliavin calculus and moment cumulant formula to specify the conditions on the stochastic process, under which, we are able to obtain some conditional Gaussian identities for Brownian stochastic integrals. In particular, if we have the matrix condition ATA2=0, then from a characterization of Yor, we have the identity for the quadratic Brownian integral. Afterwhich, we derive a conditional Edgeworth-type expansions. We then consider multiple stochastic integrals, where we obtain the Stein approximation bounds using the derived conditional Edgeworth-type expansions. For application, we derive a closed-form analytical approximations formula in terms of series expansion for the option prices, implied volatility and delta under the 2-Hypergeometric stochastic volatility model with correlated Brownian motions. Most notably, the approximation formula we derive for the implied volatility is capable of recovering the well-known smiles and skew phenomenon on implied volatility surfaces, depending on the correlation. Lastly, we consider a multidimensional ergodic Ornstein-Uhlenbeck process, X and let Y be a multidimensional stochastic process such that its stochastic differential equation is written as a drift-perturbation of X and µY be the stationary distribution of Y. We derive a first and second order expansion of µY in terms of X and their respective error estimates. Thereafter, we turn these approximations into a simulation scheme to sample µY approximately.