Forward invariance and Wong–Zakai approximation for stochastic moving boundary problems

Abstract

We discuss a class of stochastic second-order PDEs in one spacedimension with an inner boundary moving according to a possibly non-linear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong-Zakai type approximation. After a coordinate transformation the problems are reformulated and analysed in terms of stochastic evolution equations on domains of fractional powers of linear operators.

Approximation of the interface condition for stochastic Stefan-type problems

“…The following result is now a combination of the previous theorem with localization of vector-valued stochastic processes. For details on the localization we refer to [8,Section 3.3] and [10, Section 2]. Theorem 2.9.…”

“…and h r (x) = 1, for x ∈ [0, r 2 ], and h r (x) = 0, for x ∈ [(r + 1) 2 , ∞). Then define for r ∈ (0, ∞), n ∈N, Thus, [8,Proposition 3.23] (with V ∶= E α , Y n ∶= X n ), see also [10, Theorem 2.1], yields (ii) and, moreover, that for each ǫ > 0 in probability…”

“…Here, we use the convention that buy orders have a negative, and sell orders a positive sign. Demand and supply are cleared instantaneously when the price levels of orders are matching, and so there is a price level p * separating buy and sell side of v. It was shown in [8] that under reasonable assumptions on the coefficients one gets in fact that v(t, x) ≤ 0, for x < p * (t), v(t, x) ≥ 0, for x > p * (t).…”

“…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.…”