We construct closed-form asymptotic formulas for the Green's function of parabolic equations (e.g., Fokker-Planck equations) with variable coefficients in one space dimension. More precisely, let u(t, x) = Gt(x, y)f(y)dy be the solution of δtu-(au''+bu'+cu) = 0 for t > 0, u(0, x) = f(x). Then we find computable approximations G[n] t of Gt. The approximate kernels are derived by applying the Dyson-Taylor commutator method that we have recently developed for short-time expansions of heat kernels on arbitrary dimension Euclidean spaces. We then utilize these kernels to obtain closed-form pricing formulas for European call options. The validity of such approximations to large time is extended using a bootstrap scheme. We prove explicit error estimates in weighted Sobolev spaces, which we test numerically and compare to other methods.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Applied Mathematics