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I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value:

\begin{equation} V_t = V_0e^{X_t},\quad\text{where } X_t = \bigg(r - \delta - \frac{\sigma^2}{2} - \lambda \xi \Bigg)t + \sigma W_t + \sum_{k=1}^{N_t}Y_k. \end{equation}

The parameters $r$, $\delta$, $\sigma$, $\lambda$ and $\xi$ are constants. $W$ is standard Brownian motion, $N$ is a Poisson process with constant intensity $\lambda>0$ and $Y$ is a sequence of i.i.d. random variables.

The author then goes on to derive the Laplace exponent of the process, which is the function $G(\cdot)$ defined by:

\begin{equation} \mathbb{E}\big[\text{exp}(\beta X_t)\big] = \text{exp}\big[G(\beta)t\big],\quad\beta \in \mathbb{R}. \end{equation}

I am able to derive the value of $G(\cdot)$. However, in the literature I have read, I have seen the Laplace exponent being used only when $X$ is a spectrally negative Levy process or a X is a subordinator. What guarantees the existence of the Laplace exponent for the process described above?

Many thanks

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