Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

AbstractWe consider a process (Xt(α))t∈[0,T) given by the SDE dXt(α)=αb(t)Xt(α)dt+σ(t)dBt, t∈[0,T), with initial condition X0(α)=0, where T∈(0,∞], α∈R, (Bt)t∈[0,T) is a standard Wiener process, b:[0,T)→R∖{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming ddt(b(t)σ(t)2)=−2Kb(t)2σ(t)2, t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of ∫0tb(s)2σ(s)2(Xs(α))2ds and (Xt(α))2 for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) αˆt of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,IK(t)(αˆt−K)=L−sign(K)2∫01WsdWs∫01(Ws)2ds,∀t∈(0,T), where IK(t) denotes the Fisher information for α contained in the observation (Xs(K))s∈[0,t], (Ws)s∈[0,1] is a standard Wiener process and =L denotes equality in distribution. We also prove asymptotic normality of the MLE αˆt of α as t↑T for sign(α−K)=sign(K), K≠0. As an example, for all α∈R and T∈(0,∞), we study the process (Xt(α))t∈[0,T) given by the SDE dXt(α)=−αT−tXt(α)dt+dBt, t∈[0,T), with initial condition X0(α)=0. In case of α>0, this process is known as an α-Wiener bridge, and in case of α=1, this is the usual Wiener bridge