Multidimensional Diffusion Processes Research Articles

Antithetic multilevel particle filters

Abstract In this paper we consider the filtering of partially observed multidimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate-dimensional problems is the multilevel particle filter of Jasra et al. (SIAM J. Numer. Anal.55 (2017), 3068–3096). This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of Giles and Szpruch (Ann. Appl. Prob.24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for $\epsilon>0$ given, the cost to produce a mean squared error (MSE) of $\mathcal{O}(\epsilon^2)$ in the estimation of the filter is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$ . In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE.

An improved unbiased particle filter

Abstract In this paper, we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. We assume that, for numerical reasons, one has to time-discretize the diffusion process, which typically leads to filtering that is subject to discretization bias. The approach in [A. Jasra, K. J. H. Law and F. Yu, Unbiased filtering of a class of partially observed diffusions, Adv. Appl. Probab. 54 (2022), 3, 661–687] establishes that, when only having access to the time discretized diffusion, it is possible to remove the discretization bias with an estimator of finite variance. We improve on this method by introducing a modified estimator based on the recent work [A. Jasra, M. Maama and H. Ombao, Antithetic multilevel particle filters, preprint (2023), https://arxiv.org/abs/2301.12371]. We show that this new estimator is unbiased and has finite variance. Moreover, we conjecture and verify in numerical simulations that substantial gains are obtained. That is, for a given mean square error (MSE) and a particular class of multi-dimensional diffusion, the cost to achieve the said MSE falls.

Drift estimation for a multi-dimensional diffusion process using deep neural networks

Recently, many studies have shed light on the high adaptivity of deep neural network methods in nonparametric regression models, and their superior performance has been established for various function classes. Motivated by this development, we study a deep neural network method to estimate the drift coefficient of a multi-dimensional diffusion process from discrete observations. We derive generalization error bounds for least squares estimates based on deep neural networks and show that they achieve the minimax rate of convergence up to a logarithmic factor when the drift function has a compositional structure.

A mixed singular/switching control problem with terminal cost for modulated diffusion processes

In this paper, we study the regularity of the value function associated with a stochastic control problem involving two controls that act simultaneously on a modulated multidimensional diffusion process. The first is a switching control modelling a random clock. Whenever the random clock rings, the generator matrix is replaced by another, leading to a different dynamic for the finite state Markov chain of the modulated diffusion process. The second is a singular stochastic control that is executed on the process within each regime.

Stability Remarks on Discretized Multi-dimensional Diffusion Process Models and its Application to Model Reduction

Stability Remarks on Discretized Multi-dimensional Diffusion Process Models and its Application to Model Reduction

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

General criteria for the study of quasi-stationarity

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

Approximation to portfolio liquidation value with calculation of its skewness

A boundary problem for a homogeneous multidimensional diffusion process is considered under the assumption of small perturbations. Approximations to the mean, second and third central moments of the process at the moment of the first crossing a given plane in phase space are presented as a solution of ordinary differential equations with an additional transformation (“projection onto the boundary”). The quantile of a linear combination of coordinates is estimated by the second order expansion in powers of a small parameter determining the magnitude of perturbations. In the first approximation, this expansion corresponds to the Gaussian distribution, the next term contains skewness. The result is extended to a process with multiple boundaries, upon reaching each of them the equation of the process changes. Such a model describes the liquidation of a portfolio of financial instruments in which the closing rate of each of the positions is a random process. The result is illustrated by two examples. In the first example a portfolio consists of linear instruments (such as stocks, futures), prices are correlated Geometric Brownian Motions with zero drift. The closing rates are constant, but with random noise due to daily fluctuations in trading volume. In this particular case approximations for the mean, variance, skewness, VaR and CVaR of the financial result of portfolio liquidation are given explicitly. In the second example, the liquidation of an exchange-traded option position is considered under the assumption that the closing rate depends on the ratio of the underlying price to the strike of the option. Numerical calculations demonstrate that taking into account skewness significantly increases the accuracy of estimates.

Adaptive inference for small diffusion processes based on sampled data

We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter \(\varepsilon \) from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under \(\varepsilon \rightarrow 0\), \(n\rightarrow \infty \) and the balance condition that \((\varepsilon n^\rho )^{-1} =O(1)\) for some \(\rho >0\). Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.

Multi-dimensional diffusion process of allele frequencies in population genetics.

One of the ultimate goals of population genetics is to theoretically describe the behavior of allele frequency. Diffusion theory has been commonly used for this purpose mainly in one-locus one-population models, although it is not easy to handle diffusion theory in models with multiple loci or with multiple populations. This review introduces several successful cases, where multi-dimensional diffusion equations contributed to addressing evolutionary questions, thereby demonstrating its strong potential in population genetics.

Spectral thresholding for the estimation of Markov chain transition operators

We consider nonparametric estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the singular values of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for $P$ and ${p}$, discarding most of the estimated singular triples. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in $L^2$-loss. Particularly, the effect of the dimensionality $d$ of the state space on the nonparametric rate improves from $2d$ to $d$ compared to the case without singular value decay.

Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria

This paper aims to provide some tools coming from functional inequalities to deal with quasi-stationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related to a suitable Doob transform entails exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance. A special attention is paid to multi-dimensional diffusion processes, for which the aforementioned Poincaré inequality is implied by an easier-to-check Bakry-Émery condition depending on the right eigenvector for the sub-Markovian generator, which is not always known. Under additional assumptions on the potential, it is possible to bypass this lack of knowledge showing that exponential quasi-ergodicity is entailed by the classical Bakry-Émery condition.

Параметрическое моделирование состояния объектов критической инфраструктуры в условиях деструктивного воздействия

The purpose of the study: development of a method for obtaining parameterized values of local characteristics of a diffusion Markov process. It is used to simulate the state of a critical information infrastructure object under off-normal operating conditions caused by a destructive information impact. Methods: synthesis of local characteristics of evolutionary equations which describe the state of a critical information infrastructure object, using the Markov theory for estimating multidimensional diffusion processes as well as the apparatus for studying moment functions. Results: the problems of parametric modeling of the state of critical information infrastructure objects are analyzed on the basis of diffusion Markov processes in the process of synthesis of evolutionary equations which describe the behavior of such objects. The study reveals features of the functioning of objects, which implement the management of complex technical systems in the conditions of normal operation as well as off-normal operation, caused by destructive information impact. A method has been developed for the synthesis of local characteristics of the diffusion process, which simulates the behavior of critical information infrastructure objects during off-normal operation. An example of the formation of a domain for determining the characteristics of a destructive information impact, used as a parameter of synthesized models for assessing the state of objects of critical information infrastructure, is given. The proposed method can be used as the basis for specifying technical requirements for critical information infrastructure objects that perform the functions of automated control at promising and modernized complex technical facilities.

Nonparametric Estimation for the Diffusion Coefficient of Multidimensional Time-Varying Diffusion Processes

This paper proposes a kernel estimator for the coefficient of multidimensional time-varying diffusion processes as an extension of the estimation model for one dimensional diffusion coefficient to the multidimensional case. By using “time division”, the authors overcome the problem of sample observation in time varying model. In addition, the authors prove the strong consistency and limit distribution of the estimator. Finally, the authors test the performance of the estimator through a simulation experiment and an empirical application.

Stochastic Tamed Navier\u2013Stokes Equations on {mathbb {R}}^3: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the L^4-norm of the solution from the torus to {mathbb {R}}^3, see Lemma 5.1 and thus establish the existence of an invariant measure on {mathbb {R}}^3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

Mean-variance asset–liability management with partial information and uncertain time horizon

ABSTRACT This paper studies a mean-variance asset–liability management (ALM) problem with uncertain time horizon and partial information. The ALM can invest in an incomplete financial market consisting of one riskless asset and n risky stocks. The stocks price process is described by a multidimensional diffusion process with random coefficients. The uncontrollable liability process is modelled by a general diffusion process with hedgeable risks and unhedgeable risks. The uncertainty of the time horizon is assumed to come from randomness of financial assets and the liability and some other stochastic factors. The ALM may get only partial information about the liability. We derive efficient strategies and efficient frontiers in terms of the solutions of two BSDEs under complete and partial information. The closed-form expressions of efficient strategies and efficient frontiers are obtained under some special cases.

Krylov–Safonov estimates for a degenerate diffusion process

This paper proves a Krylov–Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process and Hölder estimates for the associated partial differential equation are obtained.

Simulation of elliptic and hypo-elliptic conditional diffusions

AbstractSuppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

Recurrence of direct products of diffusion processes in random media having zero potentials

In this paper, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes. Moreover, as an application, we prove the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials.

Синтез модели объекта критической информационной инфраструктуры для безопасного функционирования технической системы в условиях деструктивного информационного воздействия

The purpose of the article: improving the security and stability of the functioning of complex technical systems managed by critical information infrastructure objects under the conditions of a destructive information impact, using parameterized evolutionary models. Methods: synthesis of parametrized evolutionary models of significant objects of critical information infrastructure based on the Markov theory of estimation of multidimensional diffusion processes. The result: within the framework of an integrated approach to ensuring the security of complex technical systems, a parametrized evolutionary model of an object of a critical information infrastructure operating under conditions of destructive information impact has been synthesized. An approach to the development of recommendations for managing a complex technical system by means of an operational assessment of the main characteristics of the functioning of a critical information infrastructure object is proposed. A sequence has been developed for developing solutions for the operation of a complex technical system, based on an assessment of the impact level and the results of an operational calculation of the main characteristics of the functioning of a critical information infrastructure object. The results obtained make it possible to reasonably formulate technical requirements for the created or modernized means of ensuring the security of significant objects of critical information infrastructure that manage complex technical systems.