A particular Poisson Compound process with drift

Question

• $c_1,c_2 \geq 0$
• $\lambda_1, \lambda_2 >0$
• $\alpha_1, \alpha_2 <0$
• $\nu$ is a measure with density on $\mathbb{R}^{*}$ with respect to Lebesgue measure
• $\nu(x)= \dfrac{ c_1 }{ |x|^{1+ \alpha_1} } e^{ - \lambda_1 |x|} \mathbb{1}_{x<0} + \dfrac{ c_2 }{ x^{1+ \alpha_2} } e^{ - \lambda_2 |x|} \mathbb{1}_{x>0}$
• $X$ is the Levy process with characteristic $(0, \nu , \gamma)$
• $\gamma$ real ? not defined in the exercise .

1. Show that $X$ is a Poisson compound process with drift
2. What is the intensity of the Poisson process and the distribution of the jumps
3. Under which conditions is the process increasing ?
4. What is the limit a.e. of the process.

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• $\gamma t$ is the drift
• It is a Compound process, because $\nu$ is bounded.

The jumps take values in ${\mathbb{ R}_{-}}^{*}$ and ${\mathbb{ R}_{+}}^{*}$ with respective distributions $ h_1(x)=\dfrac{ c_1 }{ |x|^{1+ \alpha_1} } e^{ - \lambda_1 |x|}$ and $h_2(x)=\dfrac{ c_2 }{ x^{1+ \alpha_2} } e^{ - \lambda_2 |x|}$

• Let $\pi$ the distribution of the jumps, we have $ \nu =K \pi$.
• $\pi$ is a density distribution.
• We compute $\int_{ \mathbb{R}\ \setminus{0} }\nu(x)dx$
• According to this computation, we obtain
• $r=c_1 \lambda_1^{\alpha_1} \Gamma( -\alpha_1) +c_2 \lambda_2^{\alpha_2} \Gamma( -\alpha_2)$
• $r$ is the intensity of the compound process.
• The jumps have density $\dfrac1r ( h_1 +h_2) $

3. The process is increasing if $c_1=0$

Answer 1

• $X(r)=X_1+\cdots+X_{N(r)}$
• $r=r_1+r_2=c_1\Gamma(-\alpha_1)\lambda_1^{\alpha_1}+c_2\Gamma(-\alpha_2)\lambda_2^{\alpha_2}$
• $\lim_{t\to \infty}\frac{X(rt)}{rt}=\lim_{t\to \infty}\frac{X(rt)}{N(rt)}\times \frac{N(rt)}{rt}=\mathbb{E}(X_1)=r_1\frac{\alpha_1}{r\lambda_1}-r_2\frac{\alpha_2}{t\lambda_2}$

Indeed $N(rt) \sim \mathcal{P}(r t)$ so $\frac{N(rt)}{rt} \to 1$

So if $\mathbb{E}(X_1)>0$ then $X(rt) \to \infty$