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Here we outline a set of mathematical test problems. In all cases the equations were solved with a requested tolerance of 1 x 10-12. In the figures we show the average base ten logarithm of the absolute error plotted against time for 1000 realizations. In each case the cpu time on a 600 MHz processor was just a few seconds. The fortran codes used to generate the tests are available elsewhere on our website (examples).

Problem 1

The first test problem is an autonomous linear scalar equation

\begin{eqnarray} dX_t&=& a_0 X_t dt + b_1 X_t dW^1_t + b_2 X_t dW^2_t\nonumber \end{eqnarray}


which has an exact solution

\begin{eqnarray} X_t&=&X_0\exp\{[a_0-\frac{1}{2}(b_1^2+b_2^2)]t+b_1 W_t^1+b_2W_t^2\}.\nonumber \end{eqnarray} Accuracy vs Time


We set a0 = .36, b1 = .5 and b2 = .75.

Problem 2

Example 2 is a scalar non-autonomous equation

\begin{eqnarray} dX_t&=&[\frac{2}{1+t}X_t+\frac{1}{2}(1+t)^2]dt+\frac{1}{2}(1+t)^2 dW_t\nonumber \end{eqnarray}

with known solution

\begin{eqnarray} X_t&=&\left(1+t\right)^2X_0+\frac{1}{2}(1+t)^2(W_t+t).\nonumber \end{eqnarray} Accuracy vs Time

Problem 3

Test problem three is a nonlinear scalar equation

\begin{eqnarray} dX_t&=&[-X_t^3+(\alpha+\frac{1}{2}\sigma^2)X_t]dt+\sigma X_t dW_t\nonumber \end{eqnarray}

with solution

\begin{eqnarray} X_t&=&X_0\frac{\exp\{\alpha t+\sigma W_t\}}{\sqrt{1+2X_0^2\int_0^t\exp\{2\alpha s+2\sigma W_s\}ds}}.\nonumber \end{eqnarray} Accuracy vs Time

We chose $ \alpha$, $ \sigma$, X0 = 1.

Problem 4

Finally, we consider an example in which the exact solution is expressed in terms of a Itô stochastic integral. Consider the sde

\begin{eqnarray} dX_t&=& -[a+\frac{1}{2}b^2{\rm sech}^2 X_t]\tanh X_t dt+b{\rm sech} X_t dW_t \nonumber \end{eqnarray}

with exact solution

\begin{eqnarray} X_t&=&{\rm arcsinh} \left(e^{-at}\sinh X_0+e^{-at}\int_0^t e^{as}dW_s\right).\nonumber \end{eqnarray} Accuracy vs Time

We set a = .02, b = 1 and X0 = 1.

Problem 5

Example 5 is a set of two coupled linear autonomous sdes

\begin{eqnarray} dX_t^1&=&-\frac{3}{2} X_t^1 dt +X_t^1 dW_t^1-X_t^1dW_t^2-X_t^2d... ...rac{3}{2} X_t^2 dt +X_t^2 dW_t^1-X_t^2dW_t^2+X_t^1dW_t^3\nonumber \end{eqnarray}
Accuracy vs Time for Component 1

with three Wiener processes. Here the solutions are

\begin{eqnarray} X_t^1&=&\exp\{-2t+W_t^1-W_t^2\}\cos W_t^3\nonumber\\ X_t^2&=&\exp\{-2t+W_t^1-W_t^2\}\sin W_t^3.\nonumber \end{eqnarray}
 Accuracy vs Time for Component 2

* Innovative Stochastic Algorithms *

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