Stochastic Calculus Research Articles

Stochastic and variational approach to finite difference approximation of Hamilton-Jacobi equations

The author presented a stochastic and variational approach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic setting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. (2015)]. In the current paper, we extend these results to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equations, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks; a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition; the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and variational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs (1969)].

Solutions to the jump-diffusion linear stochastic differential equations

The jump-diffusion stochastic process is one of the most common forms in reality (such as wave propagation, noise propagation, turbulent flow, etc.), and researchers often refer to them in models of random processes such as Wiener process, Levy process, Ito-Hermite process, in research of G. D. Nunno, B. Oksendal, F. B. Hanson, etc. In our research, we have reviewed and solved three problems: (1) Jump-diffusion process (also known as the Ito-Levy process); (2) Solve the differential equation jump-diffusion random linear, in the case of one-dimensional; (3) Calculate the Wiener-Ito integral to the random Ito-Hermite process. The main method for dealing with the problems in our presentation is the Ito random-integrable mathematical operations for the continuous random process associated with the arbitrary differential jump by the Poisson random measure. This study aims to analyse the basic properties of jump-diffusion process that are solutions to the jump-diffusion linear stochastic differential equations: dX(t) = [a (t)X (t􀀀)+A(t)]dt + [b (t)X (t􀀀 ∫ )+B(t)]dW (t) + R0 [g (t; z)X (t􀀀)+G(t; z)] ¯N (dt;dz) with a set of stochastic continuous functions fa;b ;g ;A;B;Gg and assuming that the compensated Poisson process ¯N (t; z) is independent of the Wiener process W(t). Derived from the Ito-Hermite formulas for the Ito-Hermite process and for the Ito-Levy process class we presented the results for the differential and multiple stochastic integration for the Ito- Hermite process. We also provided a separation method to solve jump-diffusion linear differential equations.

The Magnus Expansion for Stochastic Differential Equations

In this paper, all the terms in the stochastic Magnus expansion are presented by rooted trees. First, stochastic Magnus methods for linear stochastic differential equations are constructed by truncating the stochastic Magnus expansion. Then, explicit stochastic Magnus methods are constructed by Picard’s iteration for nonlinear stochastic differential equations on matrix Lie group. Furthermore, general nonlinear stochastic differential equations are transformed into linear operator stochastic differential equations by using the Lie derivative. Finally, numerical methods for general nonlinear stochastic differential equations are constructed by using the theory of the stochastic Magnus expansion for the linear case. In particular, for the commutative case, it is shown that the stochastic Magnus expansion provides a novel way to construct computationally inexpensive and arbitrarily high-order numerical methods while avoiding the simulation of multiple stochastic integrals. Moreover, the proposed methods are shown to preserve the intrinsic properties of the original system well and the numerical experiments agree with the theoretical results.

Inference in heavy-tailed vector error correction models

This paper first studies the full rank least squares estimator (FLSE) of the heavy-tailed vector error correction (VEC) models. It is shown that the rate of convergence of the FLSE related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes when the tail index α∈(0,2). Furthermore, we show that the rate of convergence of the FLSE related to the short-term parameters is n1∕αL̃(n) and its limiting distribution is a functional of two stable processes when α∈(1,2), where L̃(n) is a slowly varying function. However, when α∈(0,1), we show that the rate of convergence of the FLSE related to the short-term parameters is n and its limiting distribution not only depends on the stationary component itself but also depends on the unit root component. Based on the FLSE, we then study the limiting behavior of the reduced rank LSE (RLSE). The results related to the short-term parameters of both FLSE and RLSE are significantly different from those of heavy-tailed time series in the literature, and it may provide new insights in the area for future research. Simulation study is carried out to demonstrate the performance of both estimators. A real example with application to 3-month Treasury Bill rate, 1-year Treasury Bill rate and Federal Fund rate is given.

Integrable boundedness of set-valued stochastic integrals

The paper deals with integrable boundedness of Itô set-valued stochastic integrals defined in 2003 by E.J. Jung and J.H. Kim in the paper [2]. Unfortunately integrable boundedness of integrals defined in the paper [2], has not been solved there. The problem was partially solved by M. Michta in the paper [8]. In the present paper it is proved that set-valued stochastic integrals defined in the paper [2] are integrably bounded if and only if they are single valued. Such result was obtained in the author's paper [7] by the additional assumption (see [7], Th. 8) that some Hypothesis B is satisfied.

Strategic Insider Trading in Continuous Time: A New Approach

The continuous-time version of Kyle's (1985) model of asset pricing with asymmetric information is studied, and generalized by allowing time-varying noise trading. From rather simple assumptions we are able to derive the optimal trade for an insider; the trading intensity satisfies a deterministic integral equation, given perfect inside information, which we give a closed form solution to. We use a new technique called forward integration in order to find the optimal trading strategy. This is an extension of the stochastic integral which takes account of the informational asymmetry inherent in this problem. The market makers' price response is found by the use of filtering theory. The novelty is our approach, which could be extended in scope.

A Lagrangian scheme for numerical evaluation of the noncausal stochastic integral

We are concerned with a noncausal approach to the numerical evaluation of the stochastic integral $$\int f\ dW_t$$ with respect to Brownian motion. Viewed as a special case of the numerical solution (in strong sense) of the SDE, it may be believed that the precision level of such an approximation scheme that uses only a finite number of increments $$\Delta _kW=W(t_{k+1})-W(t_k)$$ of Brownian motion, would not exceed the order $$O\big (\frac{1}{n}\big )$$ where n is the number of steps for discretization. We present in this note a simple but not trivial example showing that this belief is not correct. The discussion is developed on the basis of the noncausal theory of stochastic calculus introduced by the author.

Diffusion limits for networks of Markov-modulated infinite-server queues

This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation, meaning that the system’s parameters depend on an autonomously evolving Markov chain, called the background process. In previous papers on single-node queues with Markov modulation, two variants were distinguished. In the first variant the arrival rate and the server speed are modulated, whereas in the second variant the arrival rate and the service requirement are modulated. The setup of the present paper, however, is more general: we not only extend single-node systems to a network setting, but also allow both the server speed and the service requirement to depend on the background process. For this model we derive a Functional Central Limit Theorem. In particular, we show that, after accelerating the arrival processes and the background process, a centered and normalized version of the network population vector converges to a multidimensional Ornstein–Uhlenbeck process. The proof of this result relies on weak convergence of stochastic integrals as well as continuous-mapping arguments.

Performance Evaluation of URLLC in 5G Based on Stochastic Network Calculus

Ultra-reliable low latency communications (URLLC) is one of the most important scenarios in 5G. URLLC with strict latency and reliability requirements is widely used in delay-sensitive application such as self-driving. As the 3GPP claims, the URLLC is amenable to 99.999% transmission correctness and within 1ms delay bound. How to meet the requirements of reliability and latency is still an open issue. Few efforts have been made to applying a theoretical method to analyze the delay bound. Stochastic network calculus is an elegant way to obtain the delay bound based on traffic models and service guarantees. In this paper, we take the character of 5G architecture into account and use the stochastic network calculus to analyze the delay in URLLC. A tandem model describing the communication in the 5G network is built, and parameters which have an influence on the delay are analyzed. Numerical results are presented to verify the correctness of the delay analysis. We investigate how to optimize the parameters to reduce the delay, which would provide valuable guidelines for the design of URLLC architecture.

Understanding the Evolution of Tree Size Diversity within the Multivariate Nonsymmetrical Diffusion Process and Information Measures

This study focuses on the stochastic differential calculus of Itô, as an effective tool for the analysis of noise in forest growth and yield modeling. Idea of modeling state (tree size) variable in terms of univariate stochastic differential equation is exposed to a multivariate stochastic differential equation. The new developed multivariate probability density function and its marginal univariate, bivariate and trivariate distributions, and conditional univariate, bivariate and trivariate probability density functions can be applied for the modeling of tree size variables and various stand attributes such as the mean diameter, height, crown base height, crown width, volume, basal area, slenderness ratio, increments, and much more. This study introduces generalized multivariate interaction information measures based on the differential entropy to capture multivariate dependencies between state variables. The present study experimentally confirms the effectiveness of using multivariate interaction information measures to reconstruct multivariate relationships of state variables using measurements obtained from a real-world data set.

Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

In this paper, an averaging principle for multidimensional, time dependent, stochastic differential equations (SDEs) driven by fractional Brownian motion and standard Brownian motion was established. We combined the pathwise approach with the Itô stochastic calculus to handle both types of integrals involved and proved that the original SDEs can be approximated by averaged SDEs in the manner of mean square convergence.

Non-Lipschitz anticipated backward stochastic differential equations driven by fractional Brownian motion

In this paper, we obtain the existence and uniqueness of the solutions of anticipated backward stochastic differential equations driven by fractional Brownian motion under the non-Lipschitz condition, where Hurst index H is greater than 1∕2 and the associated stochastic integral is the Skorohod integral.

Approximations of the cumulative distribution function for infinite weighted sum of random variables

The infinite weighted sum of random variables originates from the linear process, random walk, and series representation of Brownian motion and stochastic integral in goodness-of-fit test. In general, this infinite weighted sum of random variables is approximated by the truncated one-term approximation. However, it is very time consuming to numerically obtain the cumulative distribution of the truncated one-term approximation with high accuracy. In this paper, a truncated two-term approximation is proposed, which significantly reduced the required computational time while maintaining high accuracy of approximating the cumulative distribution of the infinite weighted sum of random variables. Three examples are used to demonstrate the simplicity and accuracy of the proposed method.

Applications of supercapacitor energy storage systems in microgrid with distributed generators via passive fractional-order sliding-mode control

This paper develops a novel passive fractional-order sliding-mode control (PFOSMC) of a supercapacitor energy storage (SCES) system in microgrid with distributed generators. Firstly, a storage function is constructed and thoroughly analysed to investigate the inherent physical characteristics of SCES systems. Then, the beneficial terms are carefully retained for the sake of transient responses improvement, while the other detrimental terms are fully removed to achieve a globally control consistency. In order to further enhance the robustness of the closed-loop system, a fractional-order sliding-mode control (FOSMC) framework is synthesized as an additional input, which employs the fractional-order PDα sliding surface as well as an energy reshaping mechanism to realize a more flexible control performance. Four case studies, including (a) Active power and reactive power supply, (b) System restoration under power grid fault, (c) Power support under stochastic solar energy and wind energy integration, and (d) Robustness with system parameter uncertainties, are carried out to study the control performance of PFOSMC compared to that of PID control, interconnection and damping assignment passivity-based control (IDA-PBC), and FOSMC, Finally, a hardware-in-the-loop (HIL) experiment using dSpace platform is undertaken to validate its implementation feasibility.

Affine Volterra processes

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.

Coercivity condition for higher moment a priori estimates for nonlinear SPDEs and existence of a solution under local monotonicity

Higher moment a priori estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.

Real-time damage detection of degrading systems

In this article, a robust output-only real-time damage detection technique for multi-degree-of-freedom degrading systems using recursive canonical correlation analysis is presented. It has been observed that the impact of damage to a vibrating system gradually advances with time that sustains until the system degrades up to a considerable extent. Of significant interest is the effect of sudden damage in presence of continuous degradation in real-time, which is studied in the form of a sudden stiffness reduction in a separate floor. The proposed recursive canonical correlation analysis algorithm estimates the iterative update of eigenspace at each instant from the response data, thereby capturing the features of a time varying degrading structure in an online framework. Furthermore, recursive canonical correlation analysis algorithm is shown to reduce the computational cost by updating the eigenspace at each instant of time. This article explores newly developed recursive condition indicators: recursive Mahalanobis distance and recursive Itakura distance that elicit damage information from the eigenspace. In order to model degradation, simulations aimed at successfully capturing the behavior of the process in real-time becomes imperative. A general stochastic formulation of the coupled response-degradation problem accounting for the evolution of degradation is presented in the light of stiffness degradation problems. The evolution of time varying system responses is generated using a newly proposed Ito–Taylor expansion-based stochastic numerical integration formulation. Numerically simulated structural vibrating systems, namely, 2-degree-of-freedom base-isolated and 4-degree-of-freedom linear systems, have been used to check the performance of the recursive canonical correlation analysis method. The spatial damage detectability of the algorithm in real-time is explored through identifying crack location on a beam traversed by a vehicle. Finally, an experimental case study has been carried out to verify the robustness of the proposed algorithm. The identification results for both numerical and experimental cases demonstrate the efficacy of the proposed algorithm in identification of nonlinear and time varying behavior associated with degrading structural systems.

Approximation of stochastic processes by nonexpansive flows and coming down from infinity

This paper deals with the approximation of semimartingales in finite dimension by dynamical systems. We give trajectorial estimates uniform with respect to the initial condition for a well-chosen distance. This relies on a nonexpansivity property of the flow and allows to consider non-Lipschitz vector fields. The fluctuations of the process are controlled using the martingale technics and stochastic calculus. Our main motivation is the trajectorial description of stochastic processes starting from large initial values. We state general properties on the coming down from infinity of one-dimensional SDEs, with a focus on stochastically monotone processes. In particular, we recover and complement known results on $\Lambda $-coalescent and birth and death processes. Moreover, using Poincare’s compactification techniques for flows close to infinity, we develop this approach in two dimensions for competitive stochastic models. We thus classify the coming down from infinity of Lotka–Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.

The Stochastic-Calculus Approach to Multi-Receiver Poisson Channels

We study two-receiver Poisson channels using tools derived from stochastic calculus. We obtain a general formula for the mutual information over the Poisson channel that allows for conditioning and the use of auxiliary random variables. We then use this formula to compute necessary and sufficient conditions under which one Poisson channel is less noisy and/or more capable than another, which turn out to be distinct from the conditions under which this ordering holds for the discretized versions of the channels. We also use general formula to determine the capacity region of the more capable Poisson broadcast channel with independent message sets, the more capable Poisson wiretap channel, and the general two-decoder Poisson broadcast channel with degraded message sets.

A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations

The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. On the example of iterated Ito stochastic integrals of multiplicities 1 to 3, included in the Taylor-Ito expansion, it is shown that expansions of stochastic integrals based on Legendre polynomials are much easier and require significantly less computational costs compared to their analogues obtained using the trigonometric system of functions. The results of the article can be useful for construction of high-order strong numerical methods for Ito stochastic differential equations.