Endogenous Downward Jump Diffusion and Blow Up Phenomena Before Crash

Abstract

Abstract. We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the processes are interpreted as crashes and we assume that the jump intensity is a nondecreasing function of the current process say λ(X) (X = X(t): process).For the case of λ(X) = X α , α > 0, we show that the process X should explode in finite time, say te, conditional on no crash.For the case of λ(X) = (ln X) α , we show that α = 1 is the borderline of two different classes of processes. We generalize the model by adding a Brownian noise and examine the blow up properties of the sample paths.