Some Results Related to Brownian Motion
We record in this appendix for convenience a few results on Brownian motion which are used in the text. First we look at first-passage-time distributions, and the value of receiving a payment at the time of first passage if this occurs before a finite maturity date. Then we look at some expressions for expected values of time-independent functions of Brownian motions which are used in models with perpetual claims.
Throughout this appendix, the process W is a standard Brownian motion on a probability space (Ω, F, P), i.e. W0 = 0, W has independent increments and Wt – Ws is normally distributed with mean 0 and variance t – s for 0 ≤ s < t. If W is a standard Brownian motion, then the process X defined by Xt = σWt + µt is a Brownian motion with drift µ and volatility σ. For short, we will refer to X as a (µ, σ)-Brownian motion.
The results we need for the distribution of first-passage times and for conditional distributions given that a Brownian motion with drift has not hit an upper boundary can be found, for example, in the concise treatment of Harrison (1990). In this section we simply restate these results for a lower boundary, which is the case we typically need for default modeling.
Consider a (µ, σ)-Brownian motion X starting at some positive level X0 = x0, and think of the lower barrier as being at 0. Hence x0 represents the distance to a lower barrier. Let mt be the minimum of the process up to time t:
Then, using the same techniques as Harrison (1990), we have the joint distribution of the process and its running minimum, given as