A Stefan-type stochastic moving boundary problem

Abstract

Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform the problem from a moving boundary problem into a stochastic evolution equation with fixed boundary conditions. Using results from interpolation theory we obtain existence and uniqueness of local strong solutions, extending results of Kim, Zheng and Sowers. In addition, we formulate conditions for existence of global solutions and provide a refined analysis of possible blow-up behavior in finite time.Comment: 34 page

Approximation of the interface condition for stochastic Stefan-type problems

“…We recall the notion of solution as introduced in [7] for the problem with n = ∞. First, to formalize the moving frame for the free boundary problem, we define for each x ∈ R the function space…”

“…We now state the main assumptions, which are the same as required for existence of the stochastic Stefan problems in [7].…”

“…Outline of the proof. For the proof and further analysis we will, as in [7], reformulate (1.5) in terms of stochastic evolution equations.…”

“…Recently, semilinear and stochastic extensions of the Stefan problem (0.1) have been studied in the context of demand and supply modeling in modern financial markets [4,7,14,17], where trading works fully electronic via so called limit order books. In this framework, x ∈ R describes a price level (e. g. in logarithmic scale or for short time also linear scale) and v(t, x) denotes the number of active buy or sell orders at time t and the price level x.…”

“…In Section 2, we extend this abstract setting by studying continuous dependence of the mild solution map on the coefficients and initial data. To overcome the issue that the solutions might explode in finite time we truncate the coefficients as in [7,8] and need to deal with convergence results on the respective explosion times, following [8]. In Section 3, we then apply the abstract setting to the stochastic moving boundary problems and finish the proof of the statements from Section 1.…”