A recursive robust Bayesian estimation in partially observed financial market

Abstract

A RECURSIVE ROBUST BAYESIAN ESTIMATION IN PARTIALLY OBSERVED FINANCIAL MARKETAbstract. I propose a nonlinear Bayesian methodology to estimate the latent states which are partially observed in financial market. The distinguishable character of my methodology is that the recursive Bayesian estimation can be represented by some deterministic partial differential equation (PDE) (or evolution equation in the general case) parameterized by the underlying observation path. Unlike the traditional stochastic filtering equation, this dynamical representation is continuously dependent on the underlying observation path and thus it is robust to the modeling errors. Moreover, its advantages in financial econometrics are also discussed.1. Introduction. The cutting-edge works of Merton (1969Merton ( , 1971 and Black and Scholes (1973) open some frontiers of stochastic finance which models the financial states by stochastic processes such as Markov processes, or more specially, the Itô diffusion processes. Among them, the most canonical example is the Black-Scholes formula where the stock price is assumed to be some geometric Brownian motion (GBM). Through judicious choices of its parameters, this model provides enough flexibility to accommodate a wide range of dynamics of financial variables we are interested in. The motivation of my work stems from the real-world phenomenon that in many applications, the financial states of interest are often unobservable directly from the market and corrupted with some noise. Such states bear the name of "latent states". For this reason, the parametric specification of the underlying model is just the preliminary step while in the next step, it is of 2000 Mathematics Subject Classification: 60G35, 60J60, 62M05, 62P05, 91B28, 91B70.