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

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 $[0,\infty )$ and a predictable finite stopping time $\zeta$ such that $Y$ is a local semimartingale on the stochastic interval $[0,\zeta )$, continuous at $\zeta$ and constant after $\zeta$, but is not a semimartingale on $[0,\infty )$.

Appendices

Appendix 1: Bessel Process of Dimension $\delta \in (0,1)$ Is Not a Semimartingale

It is known that a Bessel process of dimension $\delta \in (0,1)$ is not a semimartingale. However, we did not find a direct reference for this. We think this can be deduced from the general Theorem 7.9 in [3], but this does not look straightforward. Therefore, we now present a direct proof.

Let $x_{0}\ge 0$. Consider a squared Bessel process $Y$ of dimension $\delta \in (0,1)$ starting from $x_{0}^{2}$, i.e. $Y$ satisfies

$$\begin{aligned} Y_{t}=x_{0}^{2}+\delta t+\int \limits _{0}^{t}2\sqrt{Y_{s}}\,{\hbox {d}}W_{s}, \quad t\ge 0, \end{aligned}$$

(8.1)

where $W$ is a Brownian motion. It is well known that SDE (8.1) has a pathwise unique strong solution, which is nonnegative, and it holds

$$\begin{aligned} \int \limits _{0}^{\infty }I(Y_{s}=0)\,{\hbox {d}}s=0\quad \text {a.s.} \end{aligned}$$

(8.2)

(see [14, Ch. XI, § 1]). A Bessel process of dimension $\delta \in (0,1)$ starting from $x_{0}$ is by definition

$$\begin{aligned} \rho _{t}:=\sqrt{Y_{t}},\quad t\ge 0. \end{aligned}$$

Assume $\rho =x_{0}+M+A$ for a continuous local martingale $M$ and a continuous finite variation process $A$ with $M_{0}=A_{0}=0$. In particular, $\rho$ has a continuous in $t$ and càdlàg in $a$ version $(L_{t}^{a}(\rho );t\ge 0,a\in {\mathbb {R}})$ of local time. The process $\int _{0}^{.}I(\rho _{s}=0)\,{\hbox {d}}M_{s}$ is a continuous local martingale starting from $0$ with the quadratic variation

$$\begin{aligned} \int \limits _{0}^{t}I(\rho _{s}=0)\,{\hbox {d}}{{{\langle }} M,M{{\rangle }}}_{s}&= \int \limits _{0}^{t}I(\rho _{s}=0)\,{\hbox {d}}{{{\langle }} \rho ,\rho {{\rangle }}}_{s}\\&= \int \limits _{{\mathbb {R}}}I_{\{0\}}(a)L_{t}^{a}(\rho )\, {\hbox {d}}a=0 \quad \text {a.s.,}\quad t\ge 0, \end{aligned}$$

where the second equality follows from the occupation times formula (see [14, Ch. VI, Cor. 1.6]), i.e.

$$\begin{aligned} \int \limits _{0}^{t}I(\rho _{s}=0)\,{\hbox {d}}M_{s}=0 \quad \text {a.s.,}\quad t\ge 0. \end{aligned}$$

(8.3)

Since $Y=\rho ^{2}$, we have

$$\begin{aligned} Y_{t}=x_{0}^{2}+\int \limits _{0}^{t}2\rho _{s}\,{\hbox {d}}M_{s} +\int \limits _{0}^{t}(2\rho _{s}\,{\hbox {d}}A_{s}+{\hbox {d}}{{{\langle }}\rho ,\rho {{\rangle }}}_{s}), \quad t\ge 0. \end{aligned}$$

(8.4)

Comparing decompositions (8.1) and (8.4) and using (8.3) and (8.2), we obtain

$$\begin{aligned} M_{t}=\int \limits _{0}^{t}I(\rho _{s}\ne 0)\,{\hbox {d}}M_{s} =\int \limits _{0}^{t}I(\rho _{s}\ne 0)\,{\hbox {d}}W_{s}=W_{t} \quad \text {a.s.,}\quad t\ge 0. \end{aligned}$$

Then ${{{\langle }}\rho ,\rho {{\rangle }}}_{t}={{{\langle }} M,M{{\rangle }}}_{t}=t$, hence, by (8.1) and (8.4),

$$\begin{aligned} \int \limits _{0}^{t}2\rho _{s}\,{\hbox {d}}A_{s} =(\delta -1)t,\quad t\ge 0, \end{aligned}$$

which yields

$$\begin{aligned} A_{t}=\int \limits _{0}^{t}I(\rho _{s}=0)\,{\hbox {d}}A_{s} +\int \limits _{0}^{t}I(\rho _{s}\ne 0) \frac{\delta -1}{2\rho _{s}}\,{\hbox {d}}s \quad \text {a.s.,}\quad t\ge 0. \end{aligned}$$

(8.5)

By the occupation times formula, for the term $\int _{0}^{t}I(\rho _{s}\ne 0) \frac{\delta -1}{2\rho _{s}}\,{\hbox {d}}s$ to be finite, we necessarily have $L_{t}^{0}(\rho )=0$ a.s., $t\ge 0$. Furthermore, $L_{t}^{0-}(\rho )=0$ a.s., $t\ge 0$, because $\rho$ is nonnegative. By [14, Ch. VI, Th. 1.7],

$$\begin{aligned} \int \limits _{0}^{t}I(\rho _{s}=0)\,{\hbox {d}}A_{s} =\frac{1}{2}(L_{t}^{0}(\rho )-L_{t}^{0-}(\rho ))=0 \quad \text {a.s.,}\quad t\ge 0. \end{aligned}$$

Thus, using (8.5), we get that $\rho$ is a nonnegative global (i.e. on $[0,\infty )$) solution of the SDE

$$\begin{aligned} {\hbox {d}}\rho _{t} =I(\rho _{t}\ne 0)\frac{\delta -1}{2\rho _{t}}\,{\hbox {d}}t +{\hbox {d}}W_{t}. \end{aligned}$$

(8.6)

But, by [2, Th. 2.13], the latter SDE does not have a nonnegative global solution. Here is a description of what happens: the singular point $0$ of SDE (8.6) has right type 1, which is one of non-entrance types, in the terminology of [2], that is, after $\rho$ reaches $0$, which happens at a finite time with probability $1$, it cannot be continued in the positive direction (also see [2, Sec. 2.4]). The obtained contradiction completes the proof.

Appendix 2: Behaviour of One-Dimensional Diffusions

Now we state some well-known results about the behaviour of a one-dimensional diffusion $Y$ of (2.1) with the coefficients satisfying (2.2) and (2.3). These results follow from the construction of solutions of (2.1) (see e.g. [5] or [9, Ch. 5.5] or [2, Ch. 2 and Ch. 4]) or can be deduced from the results in [4, Sec. 1.5].

Proposition 9.1

For any $a\in J$, with

$$\begin{aligned} \tau ^{Y}_{a}:=\inf \{t\ge 0:Y_{t}=a\} \qquad (\inf \emptyset :=\infty ), \end{aligned}$$

we have ${\mathsf {P}}(\tau ^Y_a<\infty )>0$.

Let us consider the sets

$$\begin{aligned} A&=\left\{ \zeta =\infty ,\;\limsup _{t\rightarrow \infty }Y_t=r,\;\liminf _{t\rightarrow \infty }Y_t=l\right\} ,\\ B_r&=\left\{ \zeta =\infty ,\;\lim _{t\rightarrow \infty }Y_t=r\right\} ,\\ C_r&=\left\{ \zeta <\infty ,\;\lim _{t\uparrow \zeta }Y_t=r\right\} ,\\ B_l&=\left\{ \zeta =\infty ,\;\lim _{t\rightarrow \infty }Y_t=l\right\} ,\\ C_l&=\left\{ \zeta <\infty ,\;\lim _{t\uparrow \zeta }Y_t=l\right\} . \end{aligned}$$

Proposition 9.2

Either ${\mathsf {P}}(A)=1$ or ${\mathsf {P}}(B_r\cup B_l\cup C_r\cup C_l)=1$.

Proposition 9.3

• (i) ${\mathsf {P}}(B_r\cup C_r)=0$ holds if and only if $s(r)=\infty$.

• (ii) ${\mathsf {P}}(B_l\cup C_l)=0$ holds if and only if $s(l)=-\infty$.

In particular, we get that ${\mathsf {P}}(A)=1$ holds if and only if $s(r)=\infty$, $s(l)=-\infty$.

Proposition 9.4

Assume that $s(r)<\infty$. Then either ${\mathsf {P}}(B_r)>0,\, {\mathsf {P}}(C_r)=0$ or ${\mathsf {P}}(B_r)=0$, ${\mathsf {P}}(C_r)>0$. Furthermore, we have

$$\begin{aligned} {\mathsf {P}}\left( \lim _{t\uparrow \zeta }Y_t=r,\;\;Y_t>a\;\forall t\in [0,\zeta )\right) >0 \end{aligned}$$

for any $a<x_0$.

Proposition 9.5

(Feller’s test for explosions). We have ${\mathsf {P}}(B_r)=0$, ${\mathsf {P}}(C_r)>0$ if and only if

$$\begin{aligned} s(r)<\infty \quad \text {and}\quad \frac{s(r)-s}{\rho \sigma ^2}\in L^1_{\mathrm{loc}}(r-). \end{aligned}$$

Clearly, Propositions 9.4 and 9.5, which contain statements about the behaviour of one-dimensional diffusions at the endpoint $r$, have their analogues for the behaviour at $l$. Feller’s test for explosions in this form is taken from [2, Sec. 4.1]. For a different (but equivalent) form see e.g. [9, Ch. 5, Th. 5.29].

Let us finally emphasise that the results stated in “Appendix 2” do not in general hold beyond (2.2) and (2.3).