An Integration by Parts Type Formula for Stopping Times and its Application

In this article, we shall prove an integration by parts (IBP) type formula for stopping times. In order to obtain the formula, we will first construct a process which works as if it is an “alarm clock” telling us whether the stopping times are already achieved or not. Then, we shall use the Girsanov theorem. Applications of the formula to the numerical computation of the risk called the delta for options depending on the stopping times will be also considered and show the gain of efficiency compared with a classical method.