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ANISE can also be used to solve stochastic equations with colored noise. Such equations are widely used in physics and also find important applications in biophysics (e.g. Hodgkin-Huxley and FitzHugh-Nagumo models of neurons) and for volatility models in finance. Here we consider a few rare models that have exact solutions which we can use to test the accuracy of ANISE.

1.   Ornstein-Uhlenbeck Inefficient Market Model

Consider an Ornstein-Uhlenbeck model

\begin{eqnarray} dS_t&=&(\mu +V_t)dt\nonumber \\ dV_t&=&-(1/\tau)V_tdt+(\sigma/\tau)dW_t\nonumber \end{eqnarray}


with colored noise volatility $V_t$ such that

\begin{eqnarray} M[V_tV_s]=(\sigma^2/2\tau)e^{-\vert t-s\vert/\tau}.\nonumber \end{eqnarray}


Here

\begin{eqnarray} V_t&=&\sigma \frac{1}{\sqrt{\tau}}\int_{-\infty}^t e^{(s-t)/\tau}dW_s\nonumber \end{eqnarray}


and so the second equation must be integrated from some large negative time to time zero before the simulation of the price $S_t$ can begin. The exact solution is

\begin{eqnarray} S_t&=&S_0e^{\mu t+\int_0^tV_sds}. \nonumber \end{eqnarray}


We have chosen $\mu=\tau=1=\sigma$. The base ten logarithm of the error in this quantity, averaged over 1000 realizations, is shown in the figure. The requested tolerance was 1 x 10-12.

2.   Non-Markovian Quantum State Diffusion

Consider the wave equation for a spin-1/2 interacting with a bath of bosons

\begin{eqnarray} d\vert\Psi(t)\rangle &=&[-i\frac{\omega}{2}\sigma_z+\lambda z_t... ...z-\lambda^2\int_0^tds \alpha(t-s)]\vert\Psi(t)\rangle dt\nonumber \end{eqnarray}


where zt represents a complex colored noise with $M[z_tz_s]=\alpha(t-s)$ where $\alpha(t)=(\gamma/2)e^{-\gamma t}$ is the environment correlation function. It is easy to show that $z_t=\gamma \int_{-\infty}^t e^{\gamma (s-t)}dW_s$ where $W_s$ is a complex Wiener process. Hence by adding an extra equation

\begin{eqnarray} dz_t & = &~-\gamma z_t dt+\gamma dW_t \nonumber \end{eqnarray}


we can treat the problem as a set of SDEs.

Choose $\omega=1$, $\lambda=\sqrt{2\omega}$, and $\gamma=\omega$. The noise is integrated from a time -10 until time 0 after which we integrate the noise and dynamics together until a time of 50. The exact solution of the wave equation is

\begin{eqnarray} \vert\Psi(t)\rangle&=&\exp\{-i\frac{\omega}{2}\sigma_zt+\int_0^... ...2\int_0^tdu \int_0^uds ~\alpha(u-s)\}\vert\Psi(0)\rangle\nonumber \end{eqnarray}


from which we may compute the expectation

\begin{eqnarray} \langle \sigma_z\rangle =\langle\Psi(t)\vert\sigma_z\vert \Psi(t)\rangle/\langle\Psi(t)\vert \Psi(t)\rangle. \nonumber \end{eqnarray}


The base ten logarithm of the error in this quantity, averaged over 1000 realizations, is shown in the figure. The requested tolerance was 1 x 10-12.

3.  System Subject to Additive and Multiplicative Noise

Consider an overdamped linear system with additive and multiplicative noise and a sinusoidal external field

\begin{eqnarray} dX_t&=&(-aX_t-\xi_tX_t+b\cos(\omega t)+\eta_t)dt.\nonumber \end{eqnarray}


Suppose first that the noises are Gaussian but uncorrelated so that the correlation functions are

\begin{eqnarray} M[\xi_t \xi_s]&=&\gamma_1e^{-\gamma_1\vert t-s\vert}\nonumber \... ...gamma_2\vert t-s\vert}\nonumber \\ M[\xi_t \eta_s]&=&0.\nonumber \end{eqnarray}


Then it follows that $\xi_t=(\gamma_1/2) \int_{-\infty}^t e^{\gamma_1 (s-t)}dW_s^1$ and $\eta_t=(\gamma_2/2) \int_{-\infty}^t e^{\gamma_2 (s-t)}dW_s^2$ where $W_t^1$ and $W_t^2$ are independent Wiener processes. Thus we obtain a set of stochastic differential equations

\begin{eqnarray} d\xi_t&=&~-\gamma_1 \xi_t dt +\gamma_1 dW_t^1\nonumber\\ d\eta_t&=&~-\gamma_2 \eta_t dt +\gamma_2 dW_t^2\nonumber \end{eqnarray}


which we can solve along with the equation for $X_t$. The exact solution for this problem is

\begin{eqnarray} X_t=X_0e^{-at-\int_0^t\xi_sds}+\int_0^tdu~(b\cos (\omega u)+\eta_u)~e^{a(u-t)+\int_t^u\xi_sds}.\nonumber \end{eqnarray}


We chose $\gamma_1=\gamma_2=1$, $\omega=1$, a = 1 and b = 2. The base ten logarithm of the numerical error ANISE makes in calculating this quantity, averaged over 1000 realizations, is shown in the figure. The requested tolerance was 1 x 10-12.

* Innovative Stochastic Algorithms *

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