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A Stochastic Stefan-Type Problem Under First-Order Boundary Conditions

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A Stochastic Stefan-Type Problem Under First-Order Boundary Conditions Submitted to the Annals of Applied Probability 2018 Vol. 28, No. 4, 2335–2369 DOI: 10.1214/17-AAP1359 A STOCHASTIC STEFAN-TYPE PROBLEM UNDER FIRST-ORDER BOUNDARY CONDITIONS By Marvin S. Muller¨ ∗,† ETH Z¨urich Abstract Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in eco- nomics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which in- troduces additional challenges in the analysis. Working on the dis- tribution space, It¯o-Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochas- tic evolution equations and stochastic maximal Lp-regularity we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary inter- action even when the coefficients of the original problem have linear growths. Multi-phase systems of partial differential equations have a long history in applications to various fields in natural science and quantitative finance. Recent developments in modeling of demand, supply and price formation in financial markets with high trading frequencies ask for a mathematically rigorous framework for moving boundary problems with stochastic forcing terms. Motivated by this application, we consider a class of semilinear two- phase systems in one space dimension with first order boundary conditions at the inner interface. While the deterministic problems have been extensively studied in the second half of the past century - see e. g. [30] and references therein - the stochastic equations are much less understood. In the past decade, several arXiv:1601.03968v3 [math.PR] 29 Oct 2018 authors have started to study stochastic extensions of the classical Stefan ∗Supported by the Swiss National Science Foundation through grant SNF 205121 163425. †Most of this work was carried out within the scope of the author’s dissertation [34] at TU Dresden with funding from the German Research Foundation (DFG) under grant ZUK 64. MSC 2010 subject classifications: Primary 60H15; secondary 91B70, 91G80 Keywords and phrases: Stochastic partial differential equations, Stefan problem, mov- ing boundary problem, limit order book 2335 2336 M. S. MULLER¨ problem. In 1888, Josef Stefan introduced this problem as a model for heat diffusion in the polar sea [36]. However, to the best of the author’s knowl- edge so far the stochastic perturbations have been limited to the systems behaviour inside the respective phases or, in the context of Cahn-Hilliard equations [2], on the boundary. As a step towards more realistic models, we also extend the Stefan-type dynamics of the free interface by Brownian noise which introduces additional challenges in the analysis. Barbu and da Prato [4] used the so called enthalpy function in the set- ting of the classical Stefan problem with additive noise to transform the free boundary problem into a stochastic evolution equation of porous media type. In a series of papers, Kim, C. Mueller, Sowers and Zheng [21, 22, 40] studied a class of linear stochastic moving boundary problems in one space dimension. After a coordinate transformation, the resulting SPDEs have been solved directly using heat kernel estimates. Extending these results, Keller-Ressel and M. S. M¨uller [20] used classical estimates from interpola- tion theory to established a notion of strong solutions for stochastic moving boundary problems. The framework for existence, regularity and further analysis of the solution is based on the theory of mild and strong solutions of stochastic evolution equations in the sense of [13]. The change of coordi- nates was made rigorous by imposing a stochastic chain rule. Unfortunately, this way is no longer accessible when the path of the moving interface has unbounded variation or when the solution itself has a discontinuity at the in- ner boundary. Instead, we will switch into the space of generalized functions and use It¯o-Wentzell formula for SPDEs to perform the transformation. It turns out that the terms describing the evolution of the density close to the boundary are distribution-valued, which gives the need for an extension of the concepts of solutions. The setting for analysis of the centered problems will still be based on semigroup theory for stochastic evolution equations, but can be located at the borderline case for existence. Recently, various systems based on both stochastic and deterministic parabolic partial differential equations have been applied in finance as dy- namical models for demand, supply and price formation, see e. g. [5, 14, 26, 32, 33] which is just a short list and far away from being complete. For mod- ern financial markets with high trading frequencies, we introduce a class of continuous models for the limit order book density with infinitesimal tick size, where the evolution of buy and sell side is described by a semilinear second-order SPDE and the mid price process defines a free boundary sepa- rating buy and sell side. Based on empirical observations [10, 28], we assume that average price changes can are determined by the bid-ask imbalance. Ex- tending the models presented in [40, 20] we allow the price process to have A STOCHASTIC STEFAN-TYPE PROBLEM 2337 unbounded variation. The paper is structured in the following way. In the first section, we in- troduce the moving boundary problem and the centered SPDE, define the notions of solutions and present the results on existence and regularity of local solutions and characterize its explosion times. Equations of this type arise, for instance, in macroscopical descriptions of demand and supply evo- lution in nowadays financial markets. In Section 2, we set up a dynamic model for the density of the so called limit order book. We then switch back to the analysis and solve the centered equation using existence theory for stochastic evolution equations in the framework of stochastic maximal Lp- regularity in Section 3. This theory was studied in detail on a general class of Banach spaces by Weis, Veraar and van Neerven [38, 39]. The required results, adapted to the Hilbert space setting, are sketched in the appendix. In Section 4, we switch to Krylov’s framework of solutions of SPDEs in the sense of distributions, see [24, 25], and translate the existence and reg- ularity results for the equations on the moving frames. In Section 5, we present heuristically a toy example which illustrates our notion of solutions for stochastic moving boundary problems. Without further mentioning, we work on (Ω, , ( ), P), a filtered proba- F Ft bility space with the usual conditions. For a stopping time τ we denote the closed stochastic interval by J0,τK := (t,ω) [0, ) Ω t τ(ω) . Re- { ∈ ∞ × | ≤ } spectively, we define J0,τJ, K0,τJ and K0,τK. For stochastic processes X and Y we say X(t) = Y (t) on J0,τJ, if equality holds for almost all ω Ω and ∈ all t 0 such that (t,ω) J0,τJ. Given Hilbert spaces E and H, we write ≥ ∈ E ֒ H when E is continuously and densely embedded into H. As usual, we → denote by Lq the Lebesgue space, q 1, and with Hs, s > 0, the Sobolev ≥ spaces of order s > 0. Moreover, ℓ2(E) is the space of E-valued square summable sequences and HS(U; E), for separable Hilbert spaces U and E, is the space of Hilbert-Schmidt operators from U into E. The scalar prod- uct on E will be denoted by ., . When working on the distribution space h iE D, we denote the dualization by ., . We will work only with real separa- h i ble Hilbert spaces and implicitely use their complexification when necessary to apply results from the literature. We typically denote positive constants by K which might change from line to line and might have subindices to indicate dependencies. 2338 M. S. MULLER¨ 1. A Stochastic Moving Boundary Problem. We consider the stochas- tic moving boundary problem in one space dimension (1.1) ∂2 ∂ dv(t,x)= η+ 2 v + µ+ x p (t),p (t), v, v dt ∂x − ∗ ∗ ∂x h i + σ+ (x p (t),p (t), v) dξt(x), x>p (t), − ∗ ∗ ∗ ∂2 ∂ dv(t,x)= η 2 v + µ x p (t),p (t), v, v dt − ∂x − − ∗ ∗ ∂x h i + σ (x p (t),p (t), v) dξt(x), x<p (t), − − ∗ ∗ ∗ dp (t)= ̺ v(t,p (t)+), v(t,p (t) ) dt + σ dBt, ∗ ∗ ∗ − ∗ with Robin boundary conditions ∂ ∂p v(t,p (t)+) = κ+v(t,p (t)+), (1.2) ∗ ∗ ∂ v(t,p (t) )= κ v(t,p (t) ), ∂p ∗ − − − ∗ − 4 3 for t> 0, µ : R R, σ : R R, η > 0, σ 0, and κ+, κ [0, ). ± → ± → ± ∗ ≥ − ∈ ∞ On (Ω, , ( ), P), B is a real Brownian motion and ξ the spatially colored F Ft noise, (1.3) ξt(x) := Tζ Wt(x), Tζw(x) := ζ(x,y)w(y) dy, x R, t 0, R ∈ ≥ Z where W is a cylindrical Wiener process W on U := L2(R) with covariance operator Id, independent of B, and ζ : R2 R an integral kernel. It was → shown in [20], that for σ = 0, and κ = κ+ = , one can shift the equation ∗ − ∞ onto the fixed domain R˙ := R 0 , and there exists a strong solution of the \ { } integral equation corresponding to (1.1). Following this procedure at least informally, we get for u(t,x) := v(t,x + p (t)), x = 0, t> 0, ∗ 6 (1.4) 1 2 ∂2 ∂ du(t,x)= (η+ + 2 σ ) ∂x2 u + µ+ x,p (t), u, ∂x u ∗ ∗ h +̺(u(t, 0+), u(t, 0 )) ∂ u(t,x ) dt − ∂x ∂ i + σ+ (x,p (t), u)) dξt(x + p (t)) + σ u(t,x) dBt, x> 0, ∗ ∗ ∗ ∂x 1 2 ∂2 ∂ du(t,x)= (η + 2 σ ) ∂x2 u + µ x,p (t), u, ∂x u − ∗ − ∗ h + ̺(u(t, 0+), u(t, 0 )) ∂ u(t,x) dt − ∂x ∂ i + σ (x,p (t), u) dξt(x + p (t)) + σ u(t,x) dBt, x< 0, − ∗ ∗ ∗ ∂x dp (t)= ̺ u(t, 0+), u(t, 0 ) dt + σ dBt, ∗ − ∗ A STOCHASTIC STEFAN-TYPE PROBLEM 2339 with boundary conditions ∂ ∂ (1.5) u(t, 0+) = κ+u(t, 0+), u(t, 0 )= κ u(t, 0 ).
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