Some Results on Strong Solutions of SDEs with Applications to Interest Rate Models

In this work, we investigate SDEs whose coefficients may depend on the entire past of the solution process. We introduce different Lipschitztype conditions on the coefficients. It turns out that for existence and uniqueness of a strong solution it suffices to have Lipschitz continuity in mean, in a sense to be made precise. We then investigate when it suffices to have local Lipschitz conditions. Furthermore we consider the case of drift coefficients which are locally Lipschitz in mean. Finally we show how these results can be applied to prove existence and uniqueness of solutions in interest rate term structure models.