In the previous post it was shown how the existence and uniqueness of solutions to stochastic differential equations with Lipschitz continuous coefficients follows from the basic properties of stochastic integration. However, in many applications, it is necessary to weaken this condition a bit. For example, consider the following SDE for a process X
where Z is a given semimartingale and are fixed real numbers. The function has derivative which, for , is bounded on bounded subsets of the reals. It follows that f is Lipschitz continuous on such bounded sets. However, the derivative of f diverges to infinity as x goes to infinity, so f is not globally Lipschitz continuous. Similarly, if then f is Lipschitz continuous on compact subsets of , but not globally Lipschitz. To be more widely applicable, the results of the previous post need to be extended to include such locally Lipschitz continuous coefficients.
In fact, uniqueness of solutions to SDEs with locally Lipschitz continuous coefficients follows from the global Lipschitz case. However, solutions need only exist up to a possible explosion time. This is demonstrated by the following simple non-stochastic differential equation
For initial value , this has the solution , which explodes at time . Continue reading “SDEs with Locally Lipschitz Coefficients”