Runge–Kutta schemes for backward stochastic differential equations

We study the convergence of a class of Runge–Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability property. As a consequence, the overall rate of the convergence of these schemes is controlled by their local truncation error. The schemes are categorized by the number of intermediate stages implemented between consecutive partition time instances. We show that the order of the schemes matches the number $p$ of intermediate stages for $p\le3$. Moreover, we show that the so-called order barrier occurs at $p=3$, that is, that it is not possible to construct schemes of order $p$ with $p$ stages, when $p>3$. The analysis is done under sufficient regularity on the final condition and on the coefficients of the BSDE.