Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase $[s^{-}\left(t\right),s^{+}\left(t\right)]$ defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread $s^{+}\left(t\right)-s^{-}\left(t\right)$ stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green’s function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density.