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

This paper studies the loss of the semimartingale property of the process $$g(Y)$$ g ( Y ) at the time a one-dimensional diffusion $$Y$$ Y hits a level, where $$g$$ g is a difference of two convex functions. We show that the process $$g(Y)$$ 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 $$g$$ g and the coefficients of $$Y$$ Y ) for $$g(Y)$$ 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 $$Y$$ Y on $$[0,\infty )$$ [ 0 , ∞ ) and a predictable finite stopping time $$\zeta $$ ζ such that $$Y$$ Y is a local semimartingale on the stochastic interval $$[0,\zeta )$$ [ 0 , ζ ) , continuous at $$\zeta $$ ζ and constant after $$\zeta $$ ζ , but is not a semimartingale on $$[0,\infty )$$ [ 0 , ∞ ) .