Stochastic Partial Differential Equation Models Research Articles

Optimal Incentives in a Limit Order Book: A SPDE Control Approach

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision/consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [R. Cont and M. S. Müller, 2021, A stochastic partial differential equation model for limit order book dynamics, SIAM Journal on Financial Mathematics 12(2), 744–787]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman–Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

Stochastic Integral Evolution Equations with Locally Monotone and Non-Lipschitz Coefficients

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms.

Cherenkov Radiation: A Stochastic Differential Model Driven by Brownian Motions

With the development of molecular imaging, Cherenkov optical imaging technology has been widely concerned. Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation. In this paper, time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process. Based on the original steady-state diffusion equation, we first propose a stochastic partial differential equation model. The numerical solution to the stochastic partial differential model is carried out by using the finite element method. When the time resolution is high enough, the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation, which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality. In addition, the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide in muscle tissue are also first proposed by GEANT4 Monte Carlo method. The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations, which shows that the stochastic partial differential equation can simulate the corresponding process.

Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

Abstract In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flow dynamics. 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 mean reverting. 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.

Approximate Bayesian inference for multivariate point pattern analysis in disease mapping.

We present a novel approach for analysing multivariate case-control georeferenced data in a Bayesian disease mapping context using stochastic partial differential equations (SPDEs) and the integrated nested Laplace approximation (INLA) for model fitting. In particular, we propose smooth terms based on SPDE models to estimate the underlying spatial variation as well as risk associated to pollution sources. Log-Gaussian Cox processes are used to estimate the intensity of the cases and controls, to account for risk factors and include a term to measure spatial residual variation. Each intensity is modelled on a baseline spatial effect (estimated from both controls and cases), a disease-specific spatial term and the effects of some covariates. By fitting these models, the residual spatial terms can be easily compared to detect high-risk areas not explained by the covariates. Three different types of effects to model exposure to pollution sources are considered on the distance to the source: a fixed effect, a smooth term to model non-linear effects by means of a discrete random walk of order one and a Gaussian process in one dimension with a Matérn covariance function. Spatial terms are modelled using a Gaussian process in two dimensions with a Matérn covariance function and are approximated using an approach based on solving an SPDE through INLA. Finally, this new framework is applied to a dataset of three different types of cancer and a set of controls from Alcalá de Henares (Madrid, Spain). Covariates available include the distance to several polluting industries and socioeconomic indicators. Our findings point to a possible risk increase due to the proximity to some of these industries.

Improving national level spatial mapping of malaria through alternative spatial and spatio-temporal models

The most common approach to create spatial prediction of malaria in the literature is to approximate a Gaussian process model using stochastic partial differential equation (SPDE). We compared SPDE to computationally faster alternatives, generalized additive model (GAM) and state-of-the-art machine learning method gradient boosted trees (GBM), with respect to their predictive skill for country-level malaria prevalence mapping. We also evaluated the intuition that incorporation of past data and the use of spatio-temporal models may improve predictive accuracy of present spatial distribution of malaria. Model performances varied among the countries and setting with SPDE and GAM performed well generally. The inclusion of past data is beneficial for GAM and GBM, but not for SPDE. We further investigated the weaknesses of SPDE at spatio-temporal setting and GAM at the edges of the countries. Taken together, we believe that spatial/spatio-temporal SPDE models should be evaluated alongside with the alternatives or at least GAM.

Parametric estimation for a parabolic linear SPDE model based on discrete observations

We consider parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) from high frequency data which are observed in time and space. By using thinned data obtained from the high frequency data, adaptive estimators of the coefficient parameters including the volatility parameter are proposed. Moreover, we give some examples and simulation results of the adaptive estimators of the SPDE model based on the high frequency data.

Compression of climate simulations with a nonstationary global SpatioTemporal SPDE model

Modern climate models pose an ever-increasing storage burden to computational facilities, and the upcoming generation of global simulations from the next Intergovernmental Panel on Climate Change will require a substantial share of the budget of research centers worldwide to be allocated just for this task. A statistical model can be used as a means to mitigate the storage burden by providing a stochastic approximation of the climate simulations. Indeed, if a suitably validated statistical model can be formulated to draw realizations whose spatiotemporal structure is similar to that of the original computer simulations, then the estimated parameters are effectively all the information that needs to be stored. In this work we propose a new statistical model defined via a stochastic partial differential equation (SPDE) on the sphere and in evolving time. The model is able to capture nonstationarities across latitudes, longitudes and land/ocean domains for more than 300 million data points while also overcoming the fundamental limitations of current global statistical models available for compression. Once the model is trained, surrogate runs can be instantaneously generated on a laptop by storing just 20 Megabytes of parameters as opposed to more than six Gigabytes of the original ensemble.

Deformed SPDE models with an application to spatial modeling of significant wave height

A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field becomes non-stationary. We show that if the stationary field is a Matérn field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method’s more intuitive parameterization of non-stationarity. In particular it allows for independent control over the non-stationary practical correlation range and the variance, which has not been possible with previously proposed non-stationary SPDE models.The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model parameters are estimated to data from the north Atlantic using a maximum likelihood approach. The fitted model is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicating that the model could be used for route optimization in naval logistics.

Stochastic partial differential equation models for spatially dependent predator-prey equations

Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE models are more versatile. To incorporate more qualitative features of the ratio-dependent models, the Beddington-DeAngelis functional response is also used. To analyze the systems under consideration, first existence and uniqueness of solutions of the SPDEs are obtained using the notion of mild solutions. Then sufficient conditions for permanence and extinction are derived.

Asymptotic and Numerical Analysis of a Stochastic PDE Model of Volume Transmission

Volume transmission is an important neural communication pathway in which neurons in one brain region influence the neurotransmitter concentration in the extracellular space of a distant brain region. In this paper, we apply asymptotic analysis to a stochastic partial differential equation model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model involves the diffusion equation in a three-dimensional domain with interior holes that randomly switch between being either sources or sinks. These holes model nerve varicosities that alternate between releasing and absorbing neurotransmitter according to when they fire action potentials. In the case that the holes are small, we compute analytically the first two nonzero terms in an asymptotic expansion of the average neurotransmitter concentration. The first term shows that the concentration is spatially constant to leading order and that this constant is independent of many details in the problem. Specifically, this constant first term is independent of the number and location of nerve varicosities, neural firing correlations, and the size and geometry of the extracellular space. The second term shows how these factors affect the concentration at second order. Interestingly, the second term is also spatially constant under some mild assumptions. We verify our asymptotic results by high-order numerical simulation using radial-basis-function--generated finite differences.

Boundary Static Output Feedback Control for Nonlinear Stochastic Parabolic Partial Differential Systems via Fuzzy-Model-Based Approach

This article investigates a fuzzy boundary control problem of a class of stochastic nonlinear systems modeled by the Ito-type parabolic stochastic partial differential equation (SPDE). Initially, a Takagi–Sugeno fuzzy SPDE model is proposed to accurately represent the nonlinear SPDE system. Then, on the basis of the infinite-dimensional infinitesimal operator, a fuzzy boundary static output feedback controller is developed in terms of a set of linear matrix inequalities to locally exponentially stabilize the resulting system in the mean square sense. By using an approximation argument and constructing a Lyapunov function for the mild solution, the local mean square exponential stability of the closed-loop system is proved. Meanwhile, the closed-loop well-posedness analysis is also given by virtue of the semigroup theory. Finally, a simulation study on a Belousov–Zhabotinsky reaction–diffusion system with random parameter variation is presented to illustrate the effectiveness of the proposed method.

\u201cSpatial heterogeneity of environmental risk in randomized prevention trials: consequences and modeling\u201d

BackgroundIn the context of environmentally influenced communicable diseases, proximity to environmental sources results in spatial heterogeneity of risk, which is sometimes difficult to measure in the field. Most prevention trials use randomization to achieve comparability between groups, thus failing to account for heterogeneity.This study aimed to determine under what conditions spatial heterogeneity biases the results of randomized prevention trials, and to compare different approaches to modeling this heterogeneity.MethodsUsing the example of a malaria prevention trial, simulations were performed to quantify the impact of spatial heterogeneity and to compare different models.Simulated scenarios combined variation in baseline risk, a continuous protective factor (age), a non-related factor (sex), and a binary protective factor (preventive treatment). Simulated spatial heterogeneity scenarios combined variation in breeding site density and effect, location, and population density.The performances of the following five statistical models were assessed: a non-spatial Cox Proportional Hazard (Cox-PH) model and four models accounting for spatial heterogeneity—i.e., a Data-Generating Model, a Generalized Additive Model (GAM), and two Stochastic Partial Differential Equation (SPDE) models, one modeling survival time and the other the number of events. Using a Bayesian approach, we estimated the SPDE models with an Integrated Nested Laplace Approximation algorithm.For each factor (age, sex, treatment), model performances were assessed by quantifying parameter estimation biases, mean square errors, confidence interval coverage rates (CRs), and significance rates. The four models were applied to data from a malaria transmission blocking vaccine candidate.ResultsThe level of baseline risk did not affect our estimates. However, with a high breeding site density and a strong breeding site effect, the Cox-PH and GAM models underestimated the age and treatment effects (but not the sex effect) with a low CR.When population density was low, the Cox-SPDE model slightly overestimated the effect of related factors (age, treatment). The two SPDE models corrected the impact of spatial heterogeneity, thus providing the best estimates.ConclusionOur results show that when spatial heterogeneity is important but not measured, randomization alone cannot achieve comparability between groups. In such cases, prevention trials should model spatial heterogeneity with an adapted method.Trial registrationThe dataset used for the application example was extracted from Vaccine Trial #NCT02334462 (ClinicalTrials.gov registry).

A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

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 mean-reverting. 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.

Surrogate modelling for the prediction of spatial fields based on simultaneous dimensionality reduction of high-dimensional input/output spaces.

Time-consuming numerical simulators for solving groundwater flow and dissolution models of physico-chemical processes in deep aquifers normally require some of the model inputs to be defined in high-dimensional spaces in order to return realistic results. Sometimes, the outputs of interest are spatial fields leading to high-dimensional output spaces. Although Gaussian process emulation has been satisfactorily used for computing faithful and inexpensive approximations of complex simulators, these have been mostly applied to problems defined in low-dimensional input spaces. In this paper, we propose a method for simultaneously reducing the dimensionality of very high-dimensional input and output spaces in Gaussian process emulators for stochastic partial differential equation models while retaining the qualitative features of the original models. This allows us to build a surrogate model for the prediction of spatial fields in such time-consuming simulators. We apply the methodology to a model of convection and dissolution processes occurring during carbon capture and storage.

Extremely rare collapse and build-up of turbulence in stochastic models of transitional wall flows.

This paper presents a numerical and theoretical study of multistability in two stochastic models of transitional wall flows. An algorithm dedicated to the computation of rare events is adapted on these two stochastic models. The main focus is placed on a stochastic partial differential equation model proposed by Barkley. Three types of events are computed in a systematic and reproducible manner: (i) the collapse of isolated puffs and domains initially containing their steady turbulent fraction; (ii) the puff splitting; (iii) the build-up of turbulence from the laminar base flow under a noise perturbation of vanishing variance. For build-up events, an extreme realization of the vanishing variance noise pushes the state from the laminar base flow to the most probable germ of turbulence which in turn develops into a full blown puff. For collapse events, the Reynolds number and length ranges of the two regimes of collapse of laminar-turbulent pipes, independent collapse or global collapse of puffs, is determined. The mean first passage time before each event is then systematically computed as a function of the Reynolds number r and pipe length L in the laminar-turbulent coexistence range of Reynolds number. In the case of isolated puffs, the faster-than-linear growth with Reynolds number of the logarithm of mean first passage time T before collapse is separated in two. One finds that ln(T)=A_{p}r-B_{p}, with A_{p} and B_{p} positive. Moreover, A_{p} and B_{p} are affine in the spatial integral of turbulence intensity of the puff, with the same slope. In the case of pipes initially containing the steady turbulent fraction, the length L and Reynolds number r dependence of the mean first passage time T before collapse is also separated. The author finds that T≍exp[L(Ar-B)] with A and B positive. The length and Reynolds number dependence of T are then discussed in view of the large deviations theoretical approaches of the study of mean first passage times and multistability, where ln(T) in the limit of small variance noise is studied. Two points of view, local noise of small variance and large length, can be used to discuss the exponential dependence in L of T. In particular, it is shown how a T≍exp[L(A^{'}R-B^{'})] can be derived in a conceptual two degrees of freedom model of a transitional wall flow proposed by Dauchot and Manneville. This is done by identifying a quasipotential in low variance noise, large length limit. This pinpoints the physical effects controlling collapse and build-up trajectories and corresponding passage times with an emphasis on the saddle points between laminar and turbulent states. This analytical analysis also shows that these effects lead to the asymmetric probability density function of kinetic energy of turbulence.

A stochastic partial differential equation model for the pricing of mortgage-backed securities

We develop a dynamic structural model for the wealth of individual mortgagors in a mortgage pool. We model the process of default and prepayment and, by taking a limit as the pool size goes to infinity, derive a stochastic partial differential equation (SPDE) which can be used to describe the evolution of the loss process from the pool. We prove existence and uniqueness of solutions to this SPDE and show how our model is able to capture, in a flexible way, the prices of credit risky tranches of mortgage-backed securities under different market conditions.

Using additive and coupled spatiotemporal SPDE models: a flexible illustration for predicting occurrence of Culicoides species

This paper formulates and compares a general class of spatiotemporal models for univariate space-time geostatistical data. The implementation of stochastic partial differential equation (SPDE) approach combined with integrated nested Laplace approximation into the R-INLA package makes it computationally feasible to use spatiotemporal models. However, the impact of specifying models with and without space-time interaction is unclear. We formulate an extensive class of additive and coupled spatiotemporal SPDE models and investigate the distinction between them by (1) Extending their temporal effect, allowing a random walk process in time, (2) varying the spatial correlation function and (3) running a simulation study to assess the effect of misspecifying the spatial and temporal models, and to assess the generalizability of our results to a higher number of locations. Our methods are illustrated with Culicoides data from Belgium. The Bayesian spatial predictions showed that the highest prevalence of Culicoides species was found in the Northeastern and central parts of Belgium during summer.

Numerical methods for random and stochastic partial differential equations

Mathematical models involving partial differential equations (PDE) arise in numerous applications ranging from Natural Sciences and Engineering to Economics. Random and stochastic PDE models become very powerful (and sometimes unavoidable) extensions of deterministic models when a coefficient or forcing term cannot be adequately described by a single realization, e.g. due to their natural spatial or temporal variability, or the lack of information. This advantage comes at the price that stochastic models are usually significantly more difficult to solve and their numerical simulations result often in prohibitive computational costs. Feasible and reliable computer simulations require mathematically rigorous systematic studies of stability, convergence and efficiency of numerical algorithms and approximation methods.