linear_sde_statistics

probnum.filtsmooth.linear_sde_statistics(rv, start, stop, step, driftfun, jacobfun, dispmatfun)

Computes mean and covariance of SDE solution.

For a linear(ised) SDE

\[d x_t = [G(t) x_t + v(t)] d t + L(t) x_t d w_t.\]

mean and covariance of the solution are computed by solving

\[\frac{dm}{dt}(t) = G(t) m(t) + v(t), \frac{dC}{dt}(t) = G(t) C(t) + C(t) G(t)^\top + L(t) L(t)^\top,\]

which is done here with a few steps of the RK4 method. This function is also called by the continuous-time extended Kalman filter, which is why the drift can be any function.

Parameters

• rv – Normal random variable. Distribution of mean and covariance at the starting point.

• start – Start of the time-interval

• stop – End of the time-interval

• step – Step-size used in RK4.

• driftfun – Drift of the (non)linear SDE

• jacobfun – Jacobian of the drift function

• dispmatfun – Dispersion matrix function

Returns

• Normal random variable – Mean and covariance are the solution of the differential equation

• dict – Empty dict, may be extended in the future to contain information about the solution process, e.g. number of function evaluations.