Distribution Of Stochastic Differential Equations Research Articles

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.

A note on sufficient conditions of asymptotic stability in distribution of stochastic differential equations with [formula omitted]-Brownian motion

This paper investigates a sufficient condition of asymptotic stability in distribution of stochastic differential equations driven by G-Brownian motion (G-SDEs). We define the concept of asymptotic stability in distribution under sublinear expectations. Sufficient criteria of the asymptotic stability in distribution based on sublinear expectations are given. Finally, an illustrative example is provided.

The stochastic \u03b8 method for stationary distribution of stochastic differential equations with Markovian switching

In this paper, stationary distribution of stochastic differential equations (SDEs) with Markovian switching is approximated by numerical solutions generated by the stochastic θ method. We prove the existence and uniqueness of stationary distribution of the numerical solutions firstly. Then, the convergence of numerical stationary distribution to the underlying one is discussed. Numerical simulations are conducted to support the theoretical results.

The explicit expression of non-trivial stationary distribution of SDEs under regime switching

This paper mainly focuses on the stationary probability distribution of stochastic differential equations (SDEs) under regime switching. We give the explicit expression of stationary distribution according to the limit probability distribution of SDEs under some conditions. As a special case, we obtain the stationary distribution of SDEs with Markovian switching as the weighted mean of the stationary distribution of its subsystems. In the end, an example and some conclusions are given to illustrate our theoretical results of this paper.

Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond.

Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to construct the landscape for multi-stable complex systems. We aim to summarize and elucidate the relationships among these theories from a mathematical point of view. We systematically investigate and compare three different but closely related realizations in the recent literature: the Wang's potential landscape theory from steady state distribution of stochastic differential equations (SDEs), the Freidlin-Wentzell quasi-potential from the large deviation theory, and the construction through SDE decomposition and A-type integral. We revisit that the quasi-potential is the zero noise limit of the potential landscape, and the potential function in the third proposal coincides with the quasi-potential. We compare the difference between local and global quasi-potential through the viewpoint of exchange of limit order for time and noise amplitude. We argue that local quasi-potentials are responsible for getting transition rates between neighboring stable states, while the global quasi-potential mainly characterizes the residence time of the states as the system reaches stationarity. The difference between these two is prominent when the transitivity property is broken. The most probable transition path by minimizing the Onsager-Machlup or Freidlin-Wentzell action functional is also discussed. As a consequence of the established connections among different proposals, we arrive at the novel result which guarantees the existence of SDE decomposition while denies its uniqueness in general cases. It is, therefore, clarified that the A-type integral is more appropriate to be applied to the decomposed SDEs rather than its primitive form as believed by previous researchers. Our results contribute to a deeper understanding of landscape theories for biological systems.

A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching

The aim of this paper is to improve the results on stability in distribution of nonlinear stochastic differential equations in Yuan and Mao (2003). Both conditions for the stability in distribution given in that paper are weakened.

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.

The estimation of probability distribution of SDE by only one sample trajectory

This paper focuses on the method of the simulation of a stochastic system and the main method of our paper is the Monte Carlo computation simulation method. Taking the stochastic Logistic equation as an example, we present the simulation of the sample trajectory by Euler scheme and the invariant probability distribution of stochastic differential equations with the Monte Carlo method. We also compare the simulation result with the analytical result for the autonomous stochastic Logistic model. Moreover, the stochastic Logistic equation with Markovian switching which is described by a Markov chain taking values in a finite state space is considered.

A note on stability in distribution of Markov-modulated stochastic differential equations with reflection

We extend the stability criterion in distribution as in Yuan and Mao [C. Yuan and X. Mao. Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 103 (2003) 277–291] to multi-dimensional reflected Markov-modulated stochastic differential equations on a closed positive orthant.

Numerical method for stationary distribution of stochastic differential equations with Markovian switching

In principle, once the existence of the stationary distribution of a stochastic differential equation with Markovian switching is assured, we may compute it by solving the associated system of the coupled Kolmogorov–Fokker–Planck equations. However, this is nontrivial in practice. As a viable alternative, we use the Euler–Maruyama scheme to obtain the stationary distribution in this paper.

Numerical method for stationary distribution of stochastic differential equations with Markovian switching

In principle, once the existence of the stationary distribution of a stochastic differential equation with Markovian switching is assured, we may compute it by solving the associated system of the coupled Kolmogorov–Fokker–Planck equations. However, this is nontrivial in practice. As a viable alternative, we use the Euler–Maruyama scheme to obtain the stationary distribution in this paper.

Asymptotic stability in distribution of stochastic differential equations with Markovian switching

Stability of stochastic differential equations with Markovian switching has recently been discussed by many authors, for example, Basak et al. (J. Math. Anal. Appl. 202 (1996) 604), Ji and Chizeck (IEEE Trans. Automat. Control 35 (1990) 777), Mariton (Jump Linear System in Automatic Control, Marcel Dekker, New York), Mao (Stochastic Process. Appl. 79 (1999) 45), Mao et al. (Bernoulli 6 (2000) 73) and Shaikhet (Theory Stochastic Process. 2 (1996) 180), to name a few. The aim of this paper is to study the asymptotic stability in distribution of nonlinear stochastic differential equations with Markovian switching.