ENDOGENOUS DOWNWARD JUMP DIFFUSION AND BLOW UP PHENOMENA BEFORE CRASH
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We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the pro-cesses 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) = Xα; 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) = (lnX)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.
Abstract
We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the pro-cesses 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) = Xα; 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) = (lnX)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.
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