Linear Stochastic Differential Equations Research Articles

Bayesian inference and optimisation of stochastic dynamical networks

Network inference has been extensively studied in several fields, such as systems biology and social sciences. Learning network topology and internal dynamics is essential to understand mechanisms of complex systems. In particular, sparse topologies and stable dynamics are fundamental features of many real-world continuous-time (CT) networks. Given that usually only a partial set of nodes are able to observe, we consider linear CT systems to depict networks since they can model unmeasured nodes via transfer functions. Additionally, measurements tend to be noisy and with low and varying sampling frequencies. This paper applies dynamical structure functions (DSFs) derived from linear stochastic differential equations (SDEs) to describe networks of measured nodes. A numerical sampling method, preconditioned Crank–Nicolson (pCN), is used to refine coarse-grained trajectories to improve inference accuracy. The proposed method can handle sparsely sampled data and unmeasurable nodes. Monte Carlo simulations indicate that the proposed method outperforms state-of-the-art methods with various network topologies. The developed method can be applied under a wide range of contexts, such as gene regulatory networks, social networks and communication systems.

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

This paper is concerned with robust performance analysis of linear stochastic systems acting as an integral operator on a standard Wiener process at the input and producing a stationary Gaussian random process at the output. We propose a performance criterion which uses the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the “cost-shaping” analytic function, the covariance-analytic performance criterion depends on higher-order Hardy-Schatten norms of the system transfer function. We discuss the links of these norms with the asymptotic behaviour of cumulants of finite-horizon quadratic functionals of the system output and their variational properties pertaining to system robustness to statistically uncertain inputs which differ from the standard Wiener process. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of mixed products of noncommuting annihilation and creation operators in quantum mechanics. This computational procedure, which is one of the main results of the paper, involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H2-norm of an auxiliary cascaded system. These results are also compared with an alternative approach, which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations, and illustrated by a numerical example.

Precision in disease dynamics: Finite difference solutions for stochastic epidemics with treatment cure and partial immunity

The complex and ever-changing characteristics of epidemic modelling, particularly when considering random elements, provide a substantial obstacle in creating precise and practical numerical methods for solving differential equations. This study contributes to this effort by introducing an innovative finite difference method for linear and non-linear stochastic and deterministic differential equations. This scheme expands explicitly upon the Euler Maruyama method, improving its precision for the deterministic aspect while ensuring coherence in dealing with stochastic terms. This contribution provides a numerical scheme that can be used to find solutions to linear and non-linear stochastic and deterministic differential equations. The scheme can be considered as the extension of the Euler Maruyama scheme for solving stochastic differential equations. The Euler Maruyama scheme offers a first-order accuracy of the deterministic model. Still, this scheme provides second-order accuracy for the deterministic part, whereas the integration of stochastic terms is the same in both schemes. The scheme is employed in a stochastic diffusive epidemic model with the effect of treatment, cure, and partial immunity. The comparison of the proposed scheme with the existing nonstandard finite difference method is made, and it is shown that the proposed scheme performs better than the nonstandard finite difference method in accuracy for the deterministic differential equations. It is also demonstrated that susceptible people rise whereas infected and recovered people decline by enhancing treatment cure rate. How does the cure rate of the treatment influence the number of the three populations, i.e., S(t), I(t), and R(t)? The results from the numerical simulation have provided useful insights into the dynamics of the epidemic model under various settings. This is particularly useful for influencing any public health plan and intervention. Thus, this work contributes numerical approaches and is an essential tool for epidemiological studies.

Numerical Mean-square and Asymptotic Stability Analysis for the Weak Simpson Method

A weak Simpson method has order of weak convergence one in general and has order of weak convergence three under certain additional assumptions. The proposed method has the potential to overcome some of the numerical instabilities that are often experienced when using explicit Euler methods. This work aims to determine the mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and step-sizes for the numerical method where errors propagation are under control in a well-defined sense are given. The main results are illustrated with numerical examples.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

On near-martingales and a class of anticipating linear stochastic differential equations

The goals of this paper are to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. For a class of anticipating linear stochastic differential equations, we prove the existence and uniqueness of solutions using two approaches: (1) Ayed–Kuo differential formula using an ansatz, and (2) a braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin–Wentzell type large deviations result for the solution of such equations. In addition, we prove large deviation results for small noise where the initial conditions are random.

On the mean‐square solution to the Legendre differential equation with random input data

In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conjecture posed in a recent paper, concerning the power‐series solution in a Lebesgue sense. By using the finiteness of the moment‐generating function of the equation's coefficient, mean‐square convergence is derived.

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

First-order linear Marcus SPDEs

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)].

Trajectory fitting estimation for reflected stochastic linear differential equations of a large signal

Abstract In this paper we study the drift parameter estimation for reflected stochastic linear differential equations of a large signal. We discuss the consistency and asymptotic distributions of trajectory fitting estimator (TFE).

Nash Equilibrium of Stochastic Partial Differential Game with Partial Information via Malliavin Calculus

In this article, we consider the Nash equilibrium of stochastic differential game where the state process is governed by a controlled stochastic partial differential equation and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functionals are assumed to be random. We find an explicit strong solution of the linear stochastic partial differential equation with a generalized probabilistic representation for this solution with the benefit of Kunita’s stochastic flow theory. We use Malliavin calculus to derive a stochastic maximum principle for the optimal control and obtain the Nash equilibrium of this type of stochastic differential game problem.

Applied Lyapunov Stability for Some Nonlinear Stochastic Differential Equations

In this paper, we applied and explain the stability to some linear and non-linear stochastic differential equations by using the Lyapunov direct second method, after finding the stochastic differential equation which obtained by applying the (Ito-integrated formula) and the quadratic Lyapunov function be taken, we use the Lyapunov theorems to find and explain if the trivial (zero) solution are stochastically stabile (p-stable, mean square stable and stochastically asymptotically stable in the large ), then we explain the methods by some examples.

Applications of the Linear Quadratic Regulator Optimal Control With Completely and Partially Observed Markovian Switching

Abstract We study optimal control problems where the dynamic system evolves according to a linear stochastic differential equation with (a) multiplicative and additive white noise, (b) a mixture of white noise, and (c) white noise and colored noise. The drift rate and the diffusion matrix of the linear dynamic system depend on a continuous-time Markov chain with finite state space that is (I) partially observed and (II) completely observed. Using the results of Wonham filter theory, we reduce the partially observed problems to one with complete observation. We solve the control optimal problems explicitly with the help of dynamic programing technique, and three applications are presented to illustrate our theoretical results.

On the calculation of functionals from the solution of the linear Skorohod SDE with first-order chaos in the coefficients

This article is devoted to the precise and approximate calculation of the mathematical expectation of non-linear functionals from the solution of the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In [1–4], approximate methods for calculating the mathematical expectation of functionals from solutions of the linear Skorohod stochastic differential equation with a random initial condition and deterministic coefficient functions were proposed and investigated. This paper considers the calculation of the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In this case, the solution is obtained in an analytical form [5]; however, it contains an unknown random parameter, determined as the solution of an auxiliary integral stochastic equation. In this paper we investigate the cases when the solution of this integral equation is found in an explicit form and then evaluate the moments and the mathematical expectations of some types of functional from the solution of the initial Skorohod equation. The construction of approximate formulas for calculating more general nonlinear functionals from the solution is considered. Numerical examples are given to illustrate the accuracy of the obtained formulas.

Constrained stochastic differential games with Markovian switchings and additive structure: The total expected payoff

The main objective of this work is to give conditions for the existence of Nash equilibria for a nonzero-sum constrained stochastic differential game with additive structure and Markovian switchings. In this type of game, each player is interested only in maximizing their finite-horizon total payoff when an additional cost function of the same type is required to be dominated above by another function (in particular, by a constant). The dynamic system for this game is controlled by two players and evolves according to a Markov-modulated diffusion (also known as switching diffusions or piecewise diffusion or diffusion with Markovian switchings). Given that, each player has to solve an optimization problem with constraints. The existence of a Nash equilibrium is thus proved using the Lagrange multipliers approach combined with standard dynamic programming arguments. The Lagrange approach allows the transformation of a constrained game into an unconstrained game. Therefore, this work gives conditions under which a Nash equilibrium for the unconstrained stochastic differential game is also a Nash equilibrium for the corresponding nonzero-sum constrained stochastic differential game. The theory developed here is illustrated by a pollution accumulation problem with two players. Therein, the evolution is governed by a linear stochastic differential equation with Markovian switching, and the decay pollution rate depends on a Markov chain.

Two-point boundary value problem for linear stochastic differential equations

We present the extension of the two-point boundary problem in the case where the problem doesn’t always have a unique solution or the condition on boundary coefficients described in [Linear stochastic differential equations with boundary conditions, Probab. Th. Rel. Fields, 82, 489–526 (1989)] is not satisfied.

Partially observed mean-field Stackelberg stochastic differential game with two followers

This paper studies a partially observed stochastic Stackelberg game with two followers, where state satisfies a linear stochastic differential equation of mean-field type, and cost functionals are quadratic. Using decomposition technique and backward separation approach, we derive the followers' optimal strategy. The leader focuses on an optimal control problem driven by a fully coupled mean-field forward–backward stochastic differential equation with conditional expectation. By layer-by-layer decomposition technique, some inequalities and Riccati equations, we not only get the existence and uniqueness of solution to the corresponding Hamiltonian system but also give a feedback form of Stackelberg solution. Finally, we tackle a government debt problem by the above theoretical results.

Application of stochastic filter to three-phase nonuniform transmission lines

ABSTRACT This paper implements the stochastic filters for state and parameter estimation of nonuniform transmission lines (NTL). In general, transmission line (TL) problem is a continuous time and space problem. By taking the line loading noise into account, the TL equations become a stochastic partial differential equation (PDE) rather than a simple set of coupled finite stochastic differential equations (SDE). By transforming the spatial variables into the Fourier domain, the stochastic PDE can be transformed into an infinite sequence of SDE. After truncation to a finite set of Fourier series coefficients, it becomes a finite set of coupled linear SDE, which is the required domain in which extended Kalman filter (EKF) and unscented Kalman filter (UKF) can be applied. For state space equation of EKF and UKF, the voltage and current of periodic NTL are expanded into an infinite set of spatial harmonics. In this way, the voltage and current measurement of NTL become an eigenvalue problem. The NTL is considered as cascade of small infinite NTL and the four distributed primary parameters of the periodic NTL are expressed using Fourier series expansion. Finally, the Kalman filter (KF)-based state estimation and the EKF- and UKF-based parameter estimation have been compared with recursive least squares (RLS) method. The simulation results present the superiority of the method.

Finite-Time Stability of Linear Conformable Stochastic Differential Equation with Finite Delay

Finite-Time Stability of Linear Conformable Stochastic Differential Equation with Finite Delay

Solving nonlinear filtering problems with correlated noise based on Hermite–Galerkin spectral method

Nonlinear filtering problem has important applications in various fields. One of the core issues in nonlinear filtering is to numerically solve the Duncan–Mortensen–Zakai (DMZ) equation, which is an evolution equation satisfied by the unnormalized conditional density of state process under noisy observations, in a real-time and memoryless manner. When the noise in observations is correlated to the state process, the DMZ equation we need to deal with is a second-order stochastic partial differential equation. In this paper, we will propose an algorithm to solve the DMZ equation in this case, based on Hermite–Galerkin spectral method. According to this method, the DMZ equation is converted into a system of linear stochastic differential equations generated by the observation process. The effects of different discretization schemes on this stochastic differential system will also be discussed. Moreover, rigorous convergence analysis of the algorithm is given under mild conditions. Numerical results show that the method proposed in this paper can provide an instantaneous and accurate estimation to the state process of the system.