The stochastic solution to a Cauchy problem for degenerate parabolic equations

Abstract

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ ∈ (0, 1], the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement δ ≥ 1/2. Uniqueness results, including a FeynmanKac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

Uniqueness for stochastic scalar conservation laws on Riemannian manifolds revisited

“…• the kinetic function h = sign + (u − ξ) adapted with respect to the filtration {F t } satisfies (31) with the initial conditions h(0, x, ξ) = sign + (u 0 (x) − ξ) in the sense of weak traces and h satisfies (36) with the initial conditions h(0, x, ξ) = 1 − sign + (u 0 (x) − ξ) in the sense of weak traces.…”

“…Moreover, such equations have rich mathematical structure and therefore, they are very interesting and challenging from the mathematical point of view. We have numerous results in different directions beginning with the stochastic conservation laws [5,6,12,13,18,19,34], then velocity averaging results for stochastic transport equations [7,25], stochastic degenerate parabolic equations [15,36]. We remark that latter list of references is far from complete.…”

Well-posedness for stochastic scalar conservation laws on Riemannian manifolds

“…Moreover, such equations have rich mathematical structure and therefore, they are very interesting and challenging from the mathematical point of view. We have numerous results in different directions beginning with the stochastic conservation laws [5,6,12,13,16,17,28], then velocity averaging results for stochastic transport equations [7,21], stochastic degenerate parabolic equations [14,30]. We remark that latter list of references is far from complete.…”