Functionals of Wiener Processes

Chapter 2 discusses scalar-and multidimensional processes which are based on the Wiener process, and consequently we apply them in the context of the benchmark approach introduced in Chap. 1. The first part of the chapter collects results from the literature on important functionals of scalar versions of the Wiener process, geometric Brownian motion, the Ornstein-Uhlenbeck process, and the geometric Ornstein-Uhlenbeck process. In the second part of the chapter we present the multidimensional versions of the results. The third part of the chapter demonstrates how to apply these results to compute important functionals under the benchmark approach introduced in Chap. 1. For purposes of illustration, we choose the Black-Scholes model, and show how to derive the formulas well-known from the classical risk-neutral literature under the benchmark approach. This illustrates that the benchmark approach can recover all the results known from the classical risk neutral literature, however, as we demonstrate in the next chapter, it can go beyond the classical risk-neutral paradigm.