On the Loss of the Semimartingale Property at the Hitting Time of a Level

Abstract

Abstract. This paper studies the loss of the semimartingale property of the process g(Y )at the time a one-dimensional diffusion Y hits a level, where g is a difference of two convex functions. We show that the process g(Y ) can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the first and second kind. We give a deterministic if and only if condition (in terms of g and the coefficients of Y ) for g(Y ) to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion Y on [0, ∞) and a predictable finite stopping time ζ, such that Y is a local semimartingale on the stochastic interval [0, ζ), continuous at ζ and constant after ζ, but is not a semimartingale on [0, ∞).

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“…For results in a similar vein, see C ¸inlar et al (1980), especially Theorems 5.8 and 5.9, and Mijatović and Urusov (2015).…”

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“…Therefore the density of the first time that the 5-Bessel bridge X hits level b (with 0 < b < a) is given by Q(τ ∈ dt) = h(t, b 3 ) h(0, a 3 ) P(τ ∈ dt). (29) With h as in (24). Some numerical simulations to observe the form of the density of Q yield Figure 1.…”

“…It is known that Y is a semimartingale for δ ≥ 1. And for δ ∈ (0, 1), as pointed out in [24], process Y is a semimartingale up to the time it hits zero. This allows us to apply Itô's formula to process Z, which gives rise to the SDE…”

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“…It is known that Y is a semimartingale for δ ≥ 1. And for δ ∈ (0, 1), as pointed out in [18], process Y is a semimartingale up to the time it hits zero. This allows us to apply Itô's formula to process Z, which gives rise to the SDE…”