Adv. Math. Econ. 9, 33-48 (2006)

Asymptotic expansion for a filtering problem and a short term rate model

Hirotaka Fushiya

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan Received: January 11, 2005 Revised: December 1, 2005 JEL classification: C67, G12 Mathematical Subject Classification (2000): 60G35, 93E10 Abstract. We study the filtering problem in which a system process X_{t}(ε) and a observing process Y_{t}(ε) depend on the parameter ε, and X_{t}(ε) converges to a deterministic function X_{t}(0) as ε ↓ 0. We give an asymptotic expansion formula in L^{p} for the conditional expectation of a function of X_{t}(ε) under the σ-field generated by the process Y_{s}(ε), 0 ≤ s ≤ t. Key words: filtering, nonlinear, asymptotic expansion