A reflected moving boundary problem driven by space–time white noise

Abstract

We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and show that the solution is global when the boundary speed is bounded. We also derive the expected Hölder continuity for the process and hence for the derivative of the moving boundary. Both the case when the spatial domains are given by fixed finite distances from the shared boundary, and when the spatial domains are the semi-infinite intervals on either side of the shared boundary are considered. In the second case, our results require us to further develop the known theory for reflected SPDEs on infinite spatial domains by extending the uniqueness theory and establishing the local Hölder continuity of the solutions.The obstacle problem on the compact interval [0, 1] was originally discussed in [9], in which the proofs can be found.Definition 2.1. Let v ∈ C([0, T ]; C 0 ((0, 1))) with v(0, ·) ≤ 0. We say that the pair (z, η) satisfies the heat equation with obstacle v if:(i) z ∈ C([0, T ] × [0, 1]), z(t, 0) = z(t, 1) = 0, z(0, x) ≡ 0 and z ≥ v.(ii) η is a measure on (0, 1) × [0, T ].

Existence of Invariant Measures for Reflected Stochastic Partial Differential Equations

“…Otobe extended the existence theory to the case when the spatial domain is R in [8], proving uniqueness for the case when σ is constant. Uniqueness has also been shown by Hambly and Kalsi in [5] for the equation on an unbounded domain provided that σ satisfies a Lipschitz condition, with a Lipschitz coefficient which decays exponentially fast in the spatial variable. Some interesting properties of the solutions have been proved.…”

“…where L is a smooth function which vanishes at t = 0. To control the contributions of the first three terms, see the proof of Proposition A.1 and Proposition A.4 in [5] for details. Note that the constants will not depend on t for this case.…”

“…Setting Z T = X T + C f,α,p Y T , we have from (3.1) and (3.3) that Z T bounds the (γ/4, γ/2)-Hölder norm ofṽ. In the proof of Theorem 3.3 in [3] and Theorem 3.16 in [5], it is shown that the Hölder norm ofṽ controls the Hölder norm ofũ. More precisely, we have that |ũ(t, x) −ũ(s, y)| ≤ C γ Z T (|t − s| γ/4 + |x − y| γ/2 )…”

Stefan problems for reflected SPDEs driven by space–time white noise

“…We now derive our notion of solution, by formally multiplying our equations by test functions and integrating by parts. The following set up is as in [4], and we include the calculations here for completeness.…”

Approximation of the interface condition for stochastic Stefan-type problems

“…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.…”