Stability Analysis of Numerical Schemes for Stochastic Differential Equations

Stability in Distribution of Numerical Solutions for Stochastic Differential Equations

“…In fact, in this case, the second moment of the true solution as well as that of the EM approximate solution (when ∆ small enough) will converge to zero exponentially (see e.g. [10,20,24]) so Π = π ∆ = δ 0 (·), a delta distribution on {0}. However, to the best knowledge of the authors, this property has not been proved in the general case.…”

“…There are many papers that study the numerical stability of SDEs, we here only mention, Artemiev [1], Hernandez [9], Higham [10], Kloeden and Platen [14], Petersen [23], Saito and Mitsui [24] and Talay [26]. Most of these papers are concerned with the numerical stability of SDEs with respect to sample paths or moments.…”

The partially truncated Euler–Maruyama method and its stability and boundedness

“…However, a nice numerical method for an SDE should also preserve some asymptotic properties of the underlying SDE, for example, stability and boundedness (see, e.g., [4,8,9,15,19,23]). …”

“…Although the stability of numerical methods for SDEs has been studied intensively (see, e.g., [4,8,9,19,23] asymptotic boundedness of numerical methods (see, e.g., [15]). …”

Explicit stabilized integration of stiff determinisitic or stochastic problems

“…We focus here on mean-square stability. To gain insight on the stability behavior of a numerical method, a widely used problem is the linear scalar test problems (d = m = 1) [13] dX = λ Xdt + μXdW (t), X(0) = 1, with fixed complex scalar parameters λ , μ. The exact solution of the test problen, given by X(t) = exp((λ + 1 2 μ 2 )t + μW (t)), is mean-square stable if and only if…”