Stochastic Differential Equations and Applications

The expressions of solutions for general n × m matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential equations. As an application, some IR vector-valued inhomogeneous nonlinear stochastic differential equations are reduced to random differential equations, facilitating pathwise study of the solutions. 1 A Review of Stochastic Exponential Formulas We first review some existing results about solution formulas for linear stochastic differential equations (SDEs) or for their integral formulations. Let (Ω,F , (Ft),P) be a standard stochastic basis. For the following stochastic integral equation Xt = 1 + ∫ t 0 Xs−dZs, (1.1), Supported by the NSF Grant 0620539. Supported by the National Natural Science Foundation of China (No. 10571167)and the National Basic Research Program of China (973 Program) (No. 2007CB814902). 1 where Z is a semimartingale with Z0 = 0, Doléans-Dade (1970) proved that the unique solution of (1.1) is given by Xt = exp { Zt − 1 2 〈Z, Z〉t } ∏ 0<s≤t (1 + ∆Zs)e −∆Zs . (1.2) In the literature X is called Doléans exponential (or stochastic exponential) of Z, and is denoted by E(Z). The formula (1.2) is called the Doléans (or stochastic) exponential formula. In an unpublished paper, Yoeurp and Yor (1977) proved the following result for the solution formula of scalar SDEs (see also Revuz-Yor (1999) and Protter (2005) for the case where Z is a continuous semimartingale, and Melnikov-Shiryaev (1996)) for the general case). Theorem 1.1 (Yoeurp and Yor 1977 ) Let Z and H be semimartingales, and ∆Zs 6= −1 for all s ∈ [0,∞]. Then the unique solution of the inhomogeneous scalar linear SDE Xt = Ht + ∫ t 0 Xs−dZs, t ≥ 0, (1.3) is given by Xt = E(Z)t {