Weak Equations

* A Brief Introduction to Stochastic Differential Equations *

Weak Equations

Unfortunately, there are sdes that do not have strong solutions. Consider for example the case of a scalar equation with one Wiener process. If the derivatives ${\frac{{\partial X_t}}{{\partial W^k_t}}}$ and ${\frac{{\partial X_t}}{{\partial t}}}$ are constructed by the recipe given above and

$\begin{eqnarray} \frac{ \partial^2 X_t}{\partial t \partial W_t}\neq \frac{ \partial^2 X_t}{\partial W_t \partial t}\nonumber \end{eqnarray}$

or if these derivatives do not exist then the sde is not really a true differential but rather a Pfaffian differential and its solutions will depend on the path of integration. This situation should be familiar from thermodynamics where both heat and work have this property. ANISE may not work for these equations. The free program weak4.f available from our website may work in this case.

* Innovative Stochastic Algorithms *