A stochastic Stefan-type problem under first-order boundary conditions

Müller, Marvin

doi:10.1214/17-aap1359

Cited by 12 publications

(30 citation statements)

References 41 publications

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“…The order book itself at a given time is simply the record of unexecuted, uncancelled limit orders at that time. There has been much interest in trying to model As in [8] and [6], we can think of our equations (1.1) as being a model for the limit order book. In this context, we would think of the spatial variable as representing the price or the logprice.…”

Section: An Application: Limit Order Booksmentioning

confidence: 99%

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

Hambly

Kalsi

2019

Stoch PDE: Anal Comp

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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 ].

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Section: An Application: Limit Order Booksmentioning

confidence: 99%

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

Hambly

Kalsi

2019

Stoch PDE: Anal Comp

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“…where we extend u t to R by setting u t (y) = 0 for y ∈ R \ [−L, L]. Using a (generalized) Ito-Wentzell formula, (see Appendix A), we can show that v may be described as the solution of a stochastic moving boundary problem (Mueller, 2018) (…”

Section: A Stochastic Pde Model For Limit Order Book Dynamicsmentioning

confidence: 99%

“…The dynamics of v may then be described, via an application of the Ito-Wentzell formula, as the solution of a stochastic moving boundary problem (Mueller, 2018):…”

Section: Eigenfunctions Ofmentioning

confidence: 99%

A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

Cont

Mueller²

2019

SSRN Journal

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We propose an analytically tractable class of models for the dynamics of a limit order book, described as the solution of a stochastic partial differential equation (SPDE) with multiplicative noise. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient methods for estimation and computation. We study two examples of parsimonious models in this class: a two-factor model and a model in which the order book depth is meanreverting. For each model we perform a detailed analysis of the role of different parameters, study the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

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“…as the moving interface is approached. More recent work on such problems include the models in [7] and in [5]. In these papers, as in [6], the two sides satisfy SPDEs driven by spatially coloured noise, as this makes it easier to establish the existence of the spatial derivative at the interface.…”

Section: Introductionmentioning

confidence: 99%

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

Hambly

Kalsi

2020

Stochastic Processes and their Applications

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We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space-time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary.

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A stochastic Stefan-type problem under first-order boundary conditions

Cited by 12 publications

References 41 publications

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

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

A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

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

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