STOCHASTIC

APPENDIX A
EXAMPLE
Geometric Brownian motion {X(t)} is defined as the Ito process which solves
dX = aX dt + bX dz for constants a and b.
To solve for X(t), let Y= logX and use Eq. (iv). Thus
dY= (a-$b 2 ) dt + b dz.
We have logX(t)/X(0) = Y(t)-Y(0) = (a-$b 2 )t+bz(t); thus logX(t)/X(0)
is normally distributed, with mean (a — \b 2 )t and variance bt. We may therefore
also refer to {X(t)} as a (stationary) log-normal process.
II. Poisson Differential Equations
Consider a stochastic process {q(t)}, t ^ 0, such that the value of q(t)
jumps by an amount y (a random variable) at times T which are arrival times
of a Poisson process. Thus X5 + o(