Simulation of Rare Events in Stochastic Systems

The paper presents algorithms for simulation rare events in stochastic systems based on the theory of large deviations. Here, this approach is used in conjunction with the tools of optimal control theory to estimate the probability that some observed states in a stochastic system will exceed a given threshold by some upcoming time instant. Algorithms for obtaining controlled extremal trajectory (A-profile) of the system, along which the transition to a rare event (threshold) occurs most likely under the influence of disturbances that minimize the action functional, are presented. It is also shown how this minimization can be efficiently performed using numerical-analytical methods of optimal control for linear and nonlinear systems. These results are illustrated by an example for a precipitation-measured monsoon intraseasonal oscillation (MISO) described by a low-order nonlinear stochastic model.

Optimal Liquidation Behaviour Analysis for Stochastic Linear and Nonlinear Systems of Self-Exciting Model with Decay

When the market environment changes, we extend the self-exciting price impact model and further analysis of investors’ liquidation behaviour. It is assumed that the model is accompanied by an exponential decay factor when the temporary impact and its coefficient are linear and nonlinear. Using the optimal control method, we obtain that the optimal liquidation behaviours satisfy the second-order nonlinear ODEs with variable coefficients in the case of linear and nonlinear temporary impact. Next, we solve the ODEs and get the form of the investors’ optimal liquidation behaviour in four cases. Furthermore, we prove the decreasing properties of the optimal liquidation behaviour under the linear temporary impact. Through numerical simulation, we further explain the influence of the changed parameters ρ , a , b , x , and α on the investors’ liquidation strategy X t in twelve scenarios. Some interesting properties have been found.

Stability analysis of fractional-order systems with randomly time-varying parameters

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.

Invariance under discretization for positive systems

Positive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated.

Analytical solutions of linear and non-linear incommensurate fractional-order coupled systems

<p>In this paper, a new analytical method is developed for solving linear and non-linear fractional-order coupled systems of incommensurate orders. The system consists of two fractional-order differential equations of orders . The proposed approach is performed by decoupling the system into two fractional-order differential equations; the first one is a fractional-order differential equation (FoDE) of one variable of order , while the second one depends on the solution of the first one. The general solution of the coupled system is obtained using the adomian decomposition method (ADM). The main ideas of this work are verified via several examples of linear and nonlinear systems, and the numerical simulations are performed using Mathematica.</p>

On the optimization of n-sub-step composite time integration methods

A family of n-sub-step composite time integration methods, which employs the trapezoidal rule in the first $$n-1$$ n - 1 sub-steps and a general formula in the last one, is discussed in this paper. A universal approach to optimize the parameters is provided for any cases of $$n\ge 2$$ n ≥ 2 , and two optimal sub-families of the method are given for different purposes. From linear analysis, the first sub-family can achieve nth-order accuracy and unconditional stability with controllable algorithmic dissipation, so it is recommended for high-accuracy purposes. The second sub-family has second-order accuracy, unconditional stability with controllable algorithmic dissipation, and it is designed for heuristic energy-conserving purposes, by preserving as much low-frequency content as possible. Finally, some illustrative examples are solved to check the performance in linear and nonlinear systems.