Some results on strong solutions of SDEs with applications to interest rate models

Abstract

In this work, we investigate SDEs whose coefficients may depend on the entire past of the solution process. We introduce different Lipschitz-type 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.The well-known existence and uniqueness result for strong solutions of SDEs with Lipschitztype coefficients can be obtained in several settings of varying generality. The setting chosen in this work is motivated by and in fact tailor-made for applications to term structure models arising in mathematical finance, which are typically of the following form. Let I ⊂ [0, ∞) be an interval, and X an infinite-dimensional process (t, ω 1 ) → X (t, T, ω 1 ) T ∈I (describing a collection of market observables) on a space Ω 1 , which satisfies an SDE of the form X (0, T ) = X 0 (T ), dX (t, T ) = α(t, T, X )dt + σ (t, T, X )dW 1 t (0.1) * Tel.

Numerical Integration of Heath-Jarrow-Morton Model of Interest Rates

“…The imposed conditions are sufficient to ensure that the SDE (2.4)-(2.5) has a unique strong solution f (t, T ), which is sufficiently smooth in the last argument (see [13,24] and also [16,8] for differentiating SDE solutions with respect to a parameter). Further, it is not difficult to show that they imply boundedness of exponential moments of f (t, T ), i.e., for a c ∈ R there is a constant C > 0 such that…”

“…Remark 2.1 As it was shown in [24], for the SDE (2.4)-(2.5) to have the unique strong solution it suffices to require a weaker assumption than Assumption 2.1:…”

Arbitrage-free market models for option prices: the multi-strike case

“…5, we provide explicit and fairly general examples of arbitrage-free dynamic models for the price level and the local implied volatilities. To prove the existence and uniqueness of a solution to the corresponding infinite system of SDEs, we adapt results from [44] to our setting. Section 6 concludes and points out a number of open questions.…”

“…and we have recently used in [41] new techniques from [44] for infinite-dimensional SDE systems to prove existence results for this class of models. The main contribution in [41] is to show how one can handle the complicated SDE systems that arise via the drift restrictions coming from absence of dynamic arbitrage.…”

“…The key feature of these new parameters is that they have a simple state space and yet capture precisely all the static arbitrage restrictions. Once they have been constructed, dynamic arbitrage conditions and existence results for the corresponding dynamic option models still need to be dealt with, but this can be achieved by using our techniques developed in [44] and [41].…”