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

We 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)t∈[0,T), with initial condition X0(α)=0, where T∈(0,∞]T∈(0,∞], α∈Rα∈R, (Bt)t∈[0,T)(Bt)t∈[0,T) is a standard Wiener process, b:[0,T)→R∖{0}b:[0,T)→R∖{0} and σ:[0,T)→(0,∞)σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming ddt(b(t)σ(t)2)=−2Kb(t)2σ(t)2, t∈[0,T)t∈[0,T), with some K∈RK∈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)t∈[0,T) and for all α∈Rα∈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=Kα=K, K≠0K≠0,IK(t)(αˆt−K)=L−sign(K)2∫01WsdWs∫01(Ws)2ds,∀t∈(0,T), where IK(t)IK(t) denotes the Fisher information for α   contained in the observation (Xs(K))s∈[0,t], (Ws)s∈[0,1](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↑Tt↑T for sign(α−K)=sign(K)sign(α−K)=sign(K), K≠0K≠0. As an example, for all α∈Rα∈R and T∈(0,∞)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)t∈[0,T), with initial condition X0(α)=0. In case of α>0α>0, this process is known as an α  -Wiener bridge, and in case of α=1α=1, this is the usual Wiener bridge.