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

We consider a process ( X t ( α ) ) t ∈ [ 0 , T ) given by the SDE d X t ( α ) = α b ( t ) X t ( α ) d t + σ ( t ) d B t , t ∈ [ 0 , T ) , with initial condition X 0 ( α ) = 0 , where T ∈ ( 0 , ∞ ] , α ∈ R , ( B t ) t ∈ [ 0 , T ) is a standard Wiener process, b : [ 0 , T ) → R ∖ { 0 } and σ : [ 0 , T ) → ( 0 , ∞ ) are continuously differentiable functions. Assuming d d t ( b ( t ) σ ( t ) 2 ) = − 2 K b ( t ) 2 σ ( t ) 2 , t ∈ [ 0 , T ) , with some K ∈ R , we derive an explicit formula for the joint Laplace transform of ∫ 0 t b ( s ) 2 σ ( s ) 2 ( X s ( α ) ) 2 d s and ( X t ( α ) ) 2 for all t ∈ [ 0 , T ) and for all α ∈ R . Our motivation is that the maximum likelihood estimator (MLE) α ˆ t of α can be expressed in terms of these random variables. As an application, we show that in case of α = K , K ≠ 0 , I K ( t ) ( α ˆ t − K ) = L − sign ( K ) 2 ∫ 0 1 W s d W s ∫ 0 1 ( W s ) 2 d s , ∀ t ∈ ( 0 , T ) , where I K ( t ) denotes the Fisher information for α contained in the observation ( X s ( K ) ) s ∈ [ 0 , t ] , ( W s ) s ∈ [ 0 , 1 ] is a standard Wiener process and = L denotes equality in distribution. We also prove asymptotic normality of the MLE α ˆ t 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 ( X t ( α ) ) t ∈ [ 0 , T ) given by the SDE d X t ( α ) = − α T − t X t ( α ) d t + d B t , t ∈ [ 0 , T ) , with initial condition X 0 ( 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.