Regularity of solutions to the parabolic fractional obstacle problem

In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jumpdiffusion processes where incorporated by Merton [14] into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock’s prices, in contrast with the classical lognormal diffusion model of Black and Scholes [2]. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes. Let us recall that an American option gives its holder the right to buy a stock at a given price prior (but not later) than a given time T > 0. If v(τ, x) represents the rational price of an American option with a payoff ψ at time T > 0, then v will solve (in the viscosity sense) the following obstacle problem: { min{Lv, v − ψ} = 0, v(T ) = ψ.