Consider the window Parisian feature. Associated with each time point, a moving window is defined with m) consecutive monitoring instants before and including that time point. The option is knocked out at a given time when the asset price has already stayed within the knock-out region exactly m times, m ≤ m), within the moving window. Under what condition does the window Parisian feature reduce to the consecutive Parisian feature? How can we construct the corresponding discrete grid function gwin in the forward shooting grid (FSG) algorithm? 

We define a binary string A = a₁a₂ ··· am) to represent the history of the asset price path falling inside or outside the knock-out region within the moving window. The augmented path dependence state vector has binary strings as elements (Kwok and Lau, 2001a).