1
$\begingroup$

I'd like to simulate a continuous Ornstein-Uhlenbeck-process.

Just recently I tried to simulate gaussian-white-noise and had the theoretical result by hand. I was told that, in order to simulate, for example, $\int_0^t\epsilon(t')\mathrm{d}t'$, where $\epsilon(t)$ is gaussian-distributed $\mathcal{N}(0,\sigma^2)$ and fullfilling the gaussian-noise condition, i had to do the step that: $\int_0^t\epsilon(t')\mathrm{d}t'\longrightarrow \Delta t\sum_i \gamma_i$, where $\gamma_i$ is distributed like $\mathcal{N}(0,\sigma^2/\Delta t)$.

In this case, i found my mistake because i could calculate the process also theoretically and found the discrepancies.

In the case of the Ornstein-Uhlenbeck-process (or possibly others) I have no clue how to compare my simulated results to 'the real ones', especially because my function-depencendence on the stochastic variables becomes more complex.

So my question is: How can i simulated the continuous Ornstein-Uhlenbeck-process in discrete time-steps.

Thanks already!

$\endgroup$
  • $\begingroup$ I tried to impose in the same manner, that $\Delta t \sum_tE\left[\gamma_t\right]&\overset{!}{=}\int_0^tE\left[\epsilon(t')\right]\mathrm{d}t'$ and same for the variance. Sadly the equation is not solvable...are there any approximations for small $\Delta t$ $\endgroup$ – Martin Aug 17 '17 at 11:46
0
$\begingroup$

For the Langevin equation \begin{align*} \dot{x} = h(x, t) + g(x,t)\Gamma(t), \end{align*} where $\Gamma(t)$ is such that \begin{align*} \langle \Gamma(t) \rangle = 0, \qquad\langle \Gamma(t) \Gamma(t') \rangle = 2\delta(t-t'), \end{align*} then the infinitesimal properties of the trajectory are given by $$ \begin{align*} D^{(1)}(x,t) &:= \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t}\langle X(t+\delta t)- X(t) \rangle_{X(t) = x(t) } = h(x,t) + \frac{\partial g(x,t)}{\partial x}g(x,t) \\ D^{(2)}(x,t) &:= \frac{1}{2}\lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \langle \left[X(t+\Delta t) - X(t) \right]^2 \rangle_{X(t)=x(t)} = g^2(x,t) \end{align*} $$ Then we can approximate a discrete sample from the stochastic differential equation by $$ \begin{align*} x_{n+1} \approx x_n + D^{(1)}(x_n,t_n)\Delta t + \left( \sqrt{D^{(2)}(x_n,t_n) \Delta t} \right)w_n, \qquad w_n \sim \mathcal{N}(0, 2) \end{align*} $$ so that as $\Delta t \rightarrow 0$ the approximated process has the same infinitesimal properties as the theoretical process. Now for an OU process let us say $$ x(t) = -\gamma x + \sigma \Gamma(t), \qquad \gamma > 0 $$ so $D^{(1)} = -\gamma x$ and $D^{(2)}=\sigma^2$ which you can now simulate as described above. However for this process you can also solve the transition probability exactly to give $$ p(x_{n+1}, t_{n+1} | x_n , t_n ) = \sqrt{\frac{\gamma}{2 \pi \sigma^2 (1 - e^{-2 \gamma(t_{n+1}-t_n)} )}} \exp\left( -\frac{\gamma(x_{n+1}-e^{-\gamma(t_{n+1}-t_n)}x_n)^2}{2\sigma^2(1-e^{-2\gamma(t_{n+1}-t_n)}) }\right) $$ whichis a Gaussian random variable with mean $e^{-\gamma (t_{n+1}-t_n)}x_{n}$ and variance $\sigma^2 (1-e^{-2\gamma(t_{n+1}-t_{n})})/\gamma$

So that is two ways to simulate your trajectories, the first an approximation and the second exact.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.