Stochastic Differential Equations

Abstract

Starting with coefficients a(t, x) = ((a^ij(t, x)))_{1≤i, j≤d} and b(t, x) = (bⁱ(t,x))_1≤i≤d, we saw in Chapter 3 how the associated parabolic equation

$$ \frac{{\partial u}} {{\partial t}} + L_t u = 0 $$

((0.1))

can be a source of a transition probability function on which to base a continuous Markov process. Although this is a perfectly legitimate way to pass from given coefficients to a diffusion process, it has the unfortunate feature that it involves an intermediate state (namely, the construction of the fundamental solution to (0.1)) in which the connection between the coefficients and the resulting process is somewhat obscured. Aside from the loss of intuition caused by this rather circuitous route, as probabilists our principal objection to the method given in Chapter 3 is that everything it can say about the Markov process from a knowledge of its coefficients must be learned by studying the associated parabolic equation. The result is that probabilistic methods take a back seat to analytic ones, and the probabilist ends up doing little more than translating the hard work of analysts into his own jargon.