Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution

A stochastic SACR epidemic model for HBV transmission

“…We derive sufficient conditions for disease extinction and persistence. We illustrate numerical simulation to support our theoretical findings by using the stochastic Milstein scheme method, see [23,24].…”

“…Since the system (1) is driven by one noise dB t , especially the first and the second equation, where the third and the fourth equations are noise-free. The double stochastic integrals required by the Milstein scheme is approximated using the Fourier series, for more details see [23]. By discretizing the time interval into 1000 equidistant time steps, we simulate the SACR system under the conditions of our theoretical results above.…”

Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series

“…As it turned out [12], [13], [27] (also see [5]- [11], [14]- [24], [29], [33]- [35], [37]- [40], [42])- [44] it is more convenient to work with Legendre polynomials for constructing of approximations of the iterated Ito stochastic integrals (2). Approximations based on the Legendre polynomials essentially simpler than their analogues based on the trigonometric functions (see [2]- [4], [45]- [55]). Another advantages of the application of Legendre polynomials in the framework of the mentioned problem are considered in [12] (Sect.…”

“…It should be noted that unlike the method based on Theorem 1, existing approaches to the meansquare approximation of iterated stochastic integrals (see, for example, [2]- [4], [45]- [55]) do not allow choosing different numbers p for approximations of different iterated stochastic integrals. Moreover, the noted approaches [2]- [4], [45]- [55] exclude the possibility for obtaining of approximate and exact expressions for the mean-square approximation error similar to the formulas (32), (34).…”

The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

“…There are various approaches to solving the problem of the mean-square approximation of iterated stochastic integrals. Among them, we note the approach based on the Karhunen-Loeve expansion of the Brownian bridge process [1]- [4], [13], [18], [21], approach based on the expansion of the Wiener process using various basis systems of functions [6], [10], [30], [31], approach based on the conditional joint characteristic function of a stochastic integral of multiplicity 2 [11], [12] as well as an approach based on multiple integral sums [1], [19].…”