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

We consider a process (X (α) t )t∈[0,T ) given by the SDE dX (α) t = αb(t)X (α) t dt+σ(t) dBt, t ∈ [0, T ), with initial condition X 0 = 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 d dt ( b(t) σ(t)2 ) = −2K b(t) 2 σ(t)2 , t ∈ [0, T ), with some K ∈ R, we derive an explicit formula for the joint Laplace transform of ∫ t 0 b(s)2 σ(s)2 (X s ) 2 ds and (X (α) t ) 2 for all t ∈ [0, T ) and for all α ∈ R. Our motivation is that the MLE of α can be expressed in terms of these random variables. As an application, we prove asymptotic normality of the MLE α̂t of α as t ↑ T for sign(α−K) = sign(K), K 6= 0. We also show that in case of α = K, K 6= 0, √ IK(t) (α̂t −K) L = − sign(K) √ 2 ∫ 1 0 Ws dWs ∫ 1 0 (Ws) 2 ds , ∀ t ∈ (0, T ), where IK(t) denotes the Fisher information for α contained in the observation (X (K) s )s∈[0, t], (Ws)s∈[0,1] is a standard Wiener process and L = denotes equality in distribution. As an example, for all α ∈ R and T ∈ (0,∞), we study the process (X (α) t )t∈[0,T ) given by the SDE { dX (α) t = − α T−tX (α) t dt+ dBt, t ∈ [0, T ), X (α) 0 = 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. 2000 Mathematics Subject Classifications : 60E10, 60J60, 62F12.