Processes with jumps

Suppose $a(t)$ is a deterministic right-continuous nondecreasing function of $t$ with $a(0)=0$. Prove the following formulas:

Hint: First do the case where $a$ has only finitely many discontinuities.

If $A_{t}$ is an increasing process and $\widetilde{A}_{t}$ is its compensator, show that $\widetilde{A}$ jumps only when $A$ does.

Let $P_{t}^{j}, j \in \mathbb {Z}$, be independent Poisson processes with parameter $\lambda_{j}$. Suppose $\lambda_{j}=\lambda_{-j}$ for each $j \neq 0$. Suppose $\lambda_{j}$ decreases as $j$ increases for $j \geq 1$. Let

Determine reasonable conditions on the sequence $\lambda_{j}$ so that $X$ is a semimartingale. A local martingale. A martingale. A locally square integrable martingale.

Show that if $f(t)$ is a purely discontinuous function, then $e^{f(t)}$ is also.

Suppose $M$ is a non-negative right-continuous martingale and $T=\inf \left\{ t>0: M_{t}=0\right\}$. Show that $M_{t}=0$ on $(t>T)$.

Suppose $\mathbb {P}$ and $\mathbb {Q}$ are two equivalent probability measures, $M_{\infty }$ is the Radon-Nikodym derivative of $\mathbb {Q}$ with respect to $\mathbb {P}$, and $M_{t}=\mathbb {E}\left[M_{\infty } \mid \mathcal{F}_{t}\right]$. Show that $Y_{t}$ is a local martingale with respect to $\mathbb {Q}$ if and only if $Y_{t} M_{t}$ is a local martingale with respect to $\mathbb {P}$.

Suppose $X_{t}$ is an increasing process with paths that are right continuous with left limits, $X_{0}=0$, a.s., $X$ is purely discontinuous, and all jumps are of size +1 only. Suppose $X_{t}-t$ is a martingale. Prove that $X$ is a Poisson process.

Hint: Imitate the proof of Theorem 12.1. When using Itô's formula, it is important to use the fact that $\Delta X_{t}$ is always 0 or 1 .

Suppose $X_{t}$ is an increasing process with paths that are right continuous with left limits, $X_{0}=0$, a.s., $X$ is purely discontinuous, and all jumps are of size +1 only. Suppose $\lim_{t \rightarrow \infty } X_{t}=\infty$, a.s. Prove that $X$ is a time change of a Poisson process.

Suppose $P_{t}$ is a Poisson process with parameter $\lambda ,\left\{ \mathcal{F}_{t}\right\}$ is the minimal augmented filtration for $P$, and $M_{t}=P_{t}-\lambda t$. Suppose $Y$ is a $\mathcal{F}_{1}$ measurable random variable with finite mean and variance. Prove that there exists a predictable process $H$ such that

Let $P_{1}$ and $P^{2}$ be two independent Poisson processes with the same parameter. Let $X_{t}=P_{t}^{1}-P_{t}^{2}$ and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration for $X$. Find a bounded mean zero random variable $Y$ that is $\mathcal{F}_{1}$ measurable which does not satisfy

for any predictable process $H$.