Asymptotic expansion for a filtering problem and a short term rate model资料.pdfVIP

Adv. Math. Econ. 9, 33^ 8 (2006) Advances in MATHEMATICAL ECONOMICS ©Springer-Verlag 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 (e-mail: fushiya@mail4.alpha-net.ne.jp) Received: January 11, 2005 Revised: December 1, 2005 JEL classification: C67, G12 Mathematics Subject Classification (2000): 60G35, 93E10 Abstract. We study the filtering problem in which a system process Xt (s) and a ob- serving process Yt{£) depend on the parameter £, and Xt{e) converges to a determin- istic function Xt{0) as £ j 0. We give an asymptotic expansion formula in L^ for the conditional expectation of a function of Xt (s) under the a-field generated by the process Ys{£),0 s t. Key words: filtering, nonlinear, asymptotic expansion 1. Introduction The Filtering problem has been studied by many authors (c.f. [5] and its refer- ences). It is sometimes used for pricing in models of finance (e.g. [6]), and the asymptotic approach can also be used (e.g. [9, 10]). In the present paper, we consider an asymptotic approach to the filtering problem, which may be useful for pricing in financial models. [3] considered a similar problem. Now let us state our result. Let (1^,^ , {J^t}teT,^) be a filtered proba- bility space, {(VF/, Wt)}teT be an / + Z-dimensional .^i-Brownian motion, T = [ 0 , r ] , r 0 , a n d : T x R^- R^ and a i : T x R ^ - R^(8)R^ be con- tinuous functions. For each e G [0, oo), we consider the following stochastic differential equation Xt{e)=xo^ f b{s,Xs{e))ds-^e