A note on explicit bounds for a stopped Feynman–Kac functional

Let Q t = ( x t , y t ) be a two-dimensional geometric Brownian motion which is possibly correlated starting at ( x , y ) in the positive quadrant, and let τ be an F t Q -stopping time generated by the process Q t . Under certain conditions, we prove that E x , y [ ( x τ − y τ ) e − ∫ 0 τ Φ ( x s , y s ) d s ] ≤ C x n y 1 − n g ∗ , x μ y where Φ is a bounded Borel function, C > 0 , μ > 1 , n > 1 are constants and g ∗ is an explicit bound for a solution of a certain second order ordinary differential equation. The present result extends and supplements the explicit upper bound in Hu and Oksendal (1998) .