We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size,

Numerical approximation methods for stochastic differential equations (SDEs) are well developed for SDEs with coefficients satisfying Lipschitz conditions. One of basic methods is the

We consider stochastic differential equations with additive noise and an endpoint singularity with respect to the time variable in the drift term. With this time singularity, the global error estimate for the Euler-Maruyama scheme can only be applied up to a fixed time before the singularity. Using fixed step sizes to generate a numerical approximation for a time closer to the singularity comes at a great cost in efficiency due to the much smaller step size required to produce the same order of global error. We seek to increase the efficiency of the algorithm by using variable step sizes adapted to the shape of the singularity. In making this adaptation, we must also ensure that the step sizes do not become so small that the sum of the steps cannot reach times arbitrarily close to the singularity.

We consider a stochastic differential equation of the form

Note that the singularity in the drift term stops us from using the standard results of numerical analysis. To overcome this problem, we use variable step sizes and stop the approximation at

In Section

Brownian bridge from

In Section

Let

The same thing can be given in integral notation assuming all the integrals exist [

Let

Consider a sequence of times

The first three terms of expansion generate the Euler-Maruyama algorithm with variable step sizes

With suitable conditions on the coefficient function

If

We return to the SDE of the form (

In this case,

We use Cauchy-Schwartz inequality and the inequality

For simplicity, we assume

To produce a better estimate and a more efficient algorithm, we consider variable step sizes defined as follows. First fix

Given that the SDE in (

If we have

Given a positive constant

Letting

With a simple example, we show that we cannot expect a much better result. Consider the SDE:

We apply the above approximation to the Brownian bridge introduced in (

For this example, we can verify the error explicitly. Using Algorithm (

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors would like to thank the referee for insightful comments and help in improving this paper.