Numerical solution of quadratic SDE with measurable drift

Abstract

In this paper we are interested to solve numerically quadratic SDEs with
non-necessary continuous drift of the from
\begin{equation*}
X_{t}=x+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}f(X_{s})\sigma
^{2}(X_{s})ds+\int_{0}^{t}\sigma (X_{s})dW_{s},
\end{equation*}%
where, $x$ is the initial data $b$ and $\sigma $ are given coefficients that
are assume to be Lipshitz and bounded and $f$ is a measurable bounded and
integrable function on the whole space $\mathbb{R}$.

Numerical simulations for this class of SDE of quadratic growth and
measurable drift, induced by the singular term $f(x)\sigma ^{2}(x)$, is
implemented and illustrated by some examples. The main idea is to use a
phase space transformation to transform our initial SDEs to a standard SDE
without the discontinuous and quadratic term. The Euler--Maruyama scheme
will be used to discritize the new equation, thus the numerical
approximation of the original equation is given by taking the inverse of the
space transformation. The rate of convergence are shown to be of order $%
\frac{1}{2} $.