Stochastic B–series and order conditions for exponential integrators

Alemayehu Adugna Arara (Hawassa University, Hawassa, Ethiopia), Kristian Debrabant, Anne Kværnø

We will discuss B–series for the solution of a stochastic differential equation of the form \[\mathrm{d}X(t)=\bigg(AX(t) + g_0\big(X(t)\big)\bigg)\mathrm{d}t+\sum_{m=1}^{M}g_{l}(X(t))\star\mathrm{d}W_m(t),\quad X(0)=x_{0},\] for which the exact solution can be written as \[X(t) = e^{tA}x_0 + \int_{0}^t e^{(t-s)A}g_0(X(s))\mathrm{d}s + \sum_{m=1}^M \int_{0}^t e^{(t-s)A}g_m(X(s))\star\mathrm{d}W_m(s).\] Based on this, we will derive an order theory for exponential integrators for such problems. The integral w.r.t. the Wiener process has to be interpreted e.g. as an Itô or a Stratonovich integral.

References

[1] https://arxiv.org/pdf/1801.02051.pdf