This paper uses Lie symmetry group methods to study PDEs of the form u t =xu xx +f (x)u x . We show that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find a fundamental solution.

We obtain fundamental solutions for PDEs of the form u t = σ x γ u xx + f (x)u x − μx r u by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.

This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form Ex(e −λXt− t 0

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.

An algorithm is derived for a class of continuous-time minimumcost network flow problems. The algorithm is based on some recent results in the theory of separated continuous linear programs and works with a sequence of discrete approximations to the continuous-time problem. We prove the convergence of the algorithm and present some numerical results which compare its performance against that of linear network flow algorithms applied to a uniform discrete approximation of the continuous-time problem. Q 7995 ~o h n Wiley & Sons, Inc.