If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the **joint** characteristic function of this process and its quadratic variation process?


If not possible in general, are there specific examples of compound Poisson processes for which this function is known in closed form?



Characteristic function of quadratic variation of compound Poisson process
Suppose that $N=\{N(t), t\in\mathbb{R}\_+\}$ is a Poisson process with rate $\lambda $, and $\{D\_j,j\ge 1\} $ are i.i.d. random variables, with distrbution function $F$, which are also independent of $N$. Let
\begin{equation\*}
 X(t) = \sum\_{j=1}^{N(t)}D\_j, \quad t\in\mathbb{R}\_+, \tag{1}
 \end{equation\*}
then the continuous-time stochastic process $X=\{X(t), t\in\mathbb{R}\_+\}$ is a
compound Poisson process. $X$ is Lévy process and also step process(all its trajectories are cádlág step funtion having at most a finite number of jumps in every finite interval). The quadratic variation process of $X$ is
\begin{equation\*}
 Y(t) \stackrel{\triangle}{=} [X,X]\_t =\sum\_{s\le t}\Delta X^2\_s 
 = \sum\_{j=1}^{N(t)}D\_j^2, \tag{2}
\end{equation\*}
hence $Y=\{Y(t), t\in\mathbb{R}\_+\}$ is also a compound Poisson process. Now the characteristic function of $(X,Y)$ is
\begin{align\*}
 &\varphi\_{(X(t),Y(t))}(u,v)\\
 &\quad=\mathsf{E}[\exp\{\mathrm{i}(uX(t)+vY(t))\}]\\
 &\quad=\mathsf{E}[\mathsf{E}[\exp\{\mathrm{i}(uX(t)+vY(t))\}|N(t)]]\\
 &\quad=\sum\_{k=0}^{\infty}\Big[\mathsf{E}\Big[\exp\Big\{\sum\_{j=1}^{N(t)} \mathrm{i}(uD\_j+vD\_j^2)\Big\}\Bigm|N(t)=k\Big]\mathsf{P}(N(t)=k)\Big]\\
 &\quad=\sum\_{k=0}^{\infty}\Big[\int\_{\mathbb{R}}e^{\mathrm{i}(ux+vx^2)}\, \mathrm{d}F(x) \Big]^k \frac{(\lambda t)^k}{k!} \mathrm{e}^{-\lambda t} \\ 
 &\quad=\exp\Big[\lambda t\int\_{\mathbb{R}}(e^{\mathrm{i}(ux+vx^2)}-1)\, \mathrm{d}F(x)\Big]. 
\end{align\*}


