Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Abstract

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of ϑ -methods for Volterra integral equations.

User-Friendly Expressions of the Coefficients of Some Exponentially Fitted Methods

“…Further developments of this research will be oriented to the reformulation, through C and S functions, of existing methods for ordinary differential equations [ 2 , 17 , 20 , 25 , 26 , 28 , 37 – 39 , 41 , 42 , 44 , 48 , 51 , 53 , 56 , 77 , 80 ], integral equations [ 5 – 8 , 10 , 11 , 24 , 29 , 32 , 34 , 55 , 71 ], stochastic problems [ 9 , 12 , 13 , 18 , 19 , 29 , 47 ], fractional equations [ 12 , 13 , 21 , 22 , 36 ], partial differential equations [ 57 – 59 , 66 , 74 ].…”

An integrative model of cancer cell differentiation with immunotherapy*

“…One way to characterize this phenomenon is with the addition of multiplicative noise to the system (under the previously mentioned arguments), turning it into a system of stochastic differential equations, as indicated in the literature [59,60]. The multiplicative noise added in each equation of the system 2.2, is by the expression √ DX i ξ(t), where D is the intensity of the multiplicative noise, X i is the population cell of the type i, and ξ(t) is an independent Gaussian variable, this characterization can be seen more deeply in appendix B.…”

“…The generation of these stochastic fluctuations, previously discussed in the body of this work, can be given by intrinsic factors (for example demographic) and extrinsic (access to nutrients, to mention just one). One way to incorporated this fluctuations is by adding multiplicative noise √ DX i ξ i (t) to the system equations as a source of stochasticity [59,60]:…”

Advanced Numerical Methods in Applied Sciences

“…Stochastic differential equations are considered in [8], where the authors review stability issues related to stochastic ordinary and Volterra integral equations. Two-step methods are then considered for the numerical solution in the ordinary case, and the θ method in the case of Volterra equations.…”