2023/02/20 by D. C. Antonopoulou, Antonopoulou, D. C., D. Farazakis +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K40 #35K55 #60H15 #60H30 #91B70 #91G80 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2302.09841
openalex publication_date 2023/02/20 · openalex created_date 2023/02/22 · openalex updated_date 2026/07/28
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 non-smooth noise in the sense of Walsh, 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-(t),s+(t)] 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+(t)-s-(t) stays a.s. non-negative and bounded. For this, we use a Picard approximation scheme and some of the estimates of \citeBH 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. Additionally, we apply some formal asymptotics when the noise depends only on time to derive that the spread is given by the integral of the solution of a linear diffusion stochastic equation.