This section shows runnable TME examples in Python and Matlab. In particular, we use the following two example models.
Model 1,
d X(t) = \mathrm{tanh}(X(t)) dt + dW(t).
Model 2,
d X_1(t) &= X_2(t), \\ d X_2(t) &= (X_1(t)\, (\kappa - (X_1(t))^2)) dt + X_1(t) dW(t).
We want to compute their mean, covariance/variance, or more generally \mathbb{E}[\phi(X(t + \Delta t)) \mid X(t)] for any function of interest \phi.
See, Jupyter Notebook (SymPy) for Model 1.
See, Jupyter Notebook (JaX) for Model 1.
See, Jupyter Notebook (JaX) for Model 2.
See, Jupyter Notebook (JaX) for a stochastic Lorenz model.