Stochastic Differential Equation Software

Accurately integrate Itô stochastic differential equations (SDES)

$\begin{eqnarray} {\color{yellow} dX^j_t}&{\color{yellow}=}&{\color{yellow} a^j({\bf X}_t,t) dt+\sum_{k=1}^mb^j_k({\bf X}_t,t) dW^k_t} \nonumber \end{eqnarray}$
where j = 1,..., n and

$\begin{eqnarray} {\color{yellow} {\bf X}_t}&{\color{yellow} =}&{\color{yellow} (X^1_t,\dots,X^n_t)}\nonumber \end{eqnarray}$

dW^k_t are normally distributed differentials with mean 0 and variance dt
Integrator is ANISE (Algorithm for Numerical Integration of Stochastic Equations)
ANISE is a solver for SDES with strong solutions
User must supply derivatives

$\begin{eqnarray} {\color{yellow}\frac{\partial X^j_t}{\partial W^k_t}} &{\color{... ...,t)}{\partial X_t^i}b^i_k({\bf X}_t,t)+a^j({\bf X}_t,t)}\nonumber \end{eqnarray}$

Fortran call sequence for ANISE sde integrator is

call anise( n, m, t, x, dt, dw, tol )

C/C++ call sequence for ANISE (type void) sde integrator is

anise( n, m, t, x, dt, dw, tol );

fortran	C/C++	description
integer*4 n	int n;	number of equations (input)
integer*4 m	int m;	number of Wiener processes (input)
real*8 t	double t[1];	current time (input, t + dt on return)
real*8 x(n)	double x[n];	initial conditions (input, solutions at t + dt on return)
real*8 dt	double dt;	desired time step (input)
real*8 dw(m)	double dw[m];	stochastic differentials (input sampled N(0,dt))
real8 tol*	double tol;	desired tolerance (input)

Fortran user must supply

subroutine stod( n, m, t, v, x, y)

C/C++ user must supply (type void)

stod( n, m, t, v, x, y)

fortran	C/C++	description
real*8 v(m)	double v[m];	supplied by anise
real*8 y(m)	double y[m];	returned by stod

$\begin{eqnarray} {\color{yellow} Y(i)}&{\color{yellow} =}&{\color{yellow}\frac{\... ...um_{j=1}^m V(j) \frac{\partial X^i_t}{\partial W^j_t}} \nonumber \end{eqnarray}$

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