Fractional smoothness of functionals of diffusion processes under a change of measure ∗

Let v : [0, T ] × R → R be the solution of the parabolic backward equation ∂tv + (1/2) ∑ i,j [σσ ]i,j∂xi∂xjv + ∑ i bi∂xiv + kv = 0 with terminal condition g, where the coefficients are timeand state-dependent, and satisfy certain regularity assumptions. Let X = (Xt)t∈[0,T ] be the associated R -valued diffusion process on some appropriate (Ω,F ,Q). For p ∈ [2,∞) and a measure dP = λT dQ, where λT satisfies the Muckenhoupt condition Ap, we relate the behavior of ‖g(XT )−EP(g(XT )|Ft)‖Lp(P), ‖∇v(t,Xt)‖Lp(P), ‖D 2v(t,Xt)‖Lp(P) to each other, where Dv := (∂xi∂xjv)i,j is the Hessian matrix.