= Solution
At a positive state $x$, the future is constant. At zero, the residual holding time is independent rate-one exponential and the next state is an independent uniform mark. Thus the conditional future given the natural filtration depends only on the present state: $X$ is a <Markov jump process>. For any bounded Borel $g$, its transition semigroup is
$$
P_hg(0)=e^{-h}g(0)+(1-e^{-h})\int_0^1g(y)dy,\qquad P_hg(x)=g(x)\quad(x>0).
$$
The <Lévy kernel of a Markov jump process>, written in destination-state convention, is
$$
\boxed{q(x,dy)=\mathbf1_{\{x=0\}}\mathbf1_{(0,1)}(y)dy.}
$$
In displacement convention it has the same expression $K(x,dz)=\mathbf1_{\{x=0\}}\mathbf1_{(0,1)}(z)dz$, since its only nonzero jump rate is from zero. The <Markov jump-process generator> is $Lg(x)=\mathbf1_{\{x=0\}}(\int_0^1g(y)dy-g(0))$.
We state the generator martingale theorem used here: for a nonexplosive Markov jump process and a bounded $g$ with bounded $Lg$, the process $g(X_t)-g(X_0)-\int_0^t Lg(X_{s-})ds$ is a true martingale. In this one-jump model it also follows directly from part (a), first for mark indicators and then by bounded-function approximation.
Take $g_\theta(x)=e^{\theta x}$, and put
$$
r_\theta=\int_0^1(e^{\theta y}-1)dy=\begin{cases}(e^\theta-1)/\theta-1,&\theta\ne0,\\0,&\theta=0.\end{cases}
$$
Here $Lg_\theta(x)/g_\theta(x)=r_\theta\mathbf1_{\{x=0\}}$. Therefore choose the <previsible process>
$$
\boxed{Y_t=\int_0^t r_\theta\mathbf1_{\{X_{s-}=0\}}ds=r_\theta(t\wedge T).}
$$
It is continuous adapted and thus predictable. To verify the exponential, write $dg_\theta(X_t)=dN_t+Lg_\theta(X_{t-})dt$ with the generator martingale $N$. The <semimartingale integration by parts> formula, using the continuous finite-variation factor $e^{-Y_t}$, gives
$$
d(e^{-Y_t}g_\theta(X_t))=e^{-Y_t}dN_t+e^{-Y_t}\bigl(Lg_\theta(X_{t-})-g_\theta(X_{t-})Y'_t\bigr)dt=e^{-Y_t}dN_t.
$$
Thus $Z_t=e^{\theta X_t-Y_t}$ is a local martingale. On each fixed horizon it is bounded by $e^{|\theta|+|r_\theta|t}$, so it is a true martingale with $Z_0=1$. This is the <exponential martingale from a jump generator>; for $\theta=0$ it reduces to the constant process one.
Back to article page
