### 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} $.

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