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Department of Mathematical Sciences

# Seminar Archives

## Statistics Seminars: On the loss of the semimartingale property at the hitting time of a level

Presented by Alex Mijatovic, Imperial

22 April 2013 14:00 in CM221

This talk describes 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 \textit{first}
and \textit{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. As an application we
construct an adapted diffusion $Y$ on $[0,\infty)$ and a \emph{predictable} finite stopping time
$\zeta$, such that $Y$ is a semimartingale on the stochastic interval $[0,\zeta)$, continuous at
$\zeta$ and constant after $\zeta$, but is \emph{not} a semimartingale on $[0,\infty)$. This
is joint work with M. Urusov.