Existence of the Laplace Exponent for a Jump-Diffusion

Summary

The document defines a log asset-value process combining deterministic drift, Brownian motion, and compound Poisson jumps. It asks when the process has a Laplace exponent: a function that expresses the exponential moment of the process at time t as an exponential in t. The setup specifies a constant jump intensity and independent, identically distributed jump sizes, but does not state their distribution or its moment properties.

For this jump-diffusion, the Brownian and drift components have finite exponential moments for real arguments, while the jump contribution requires the jump-size distribution to have a finite moment-generating function at the argument being considered. Thus the exponent is defined on the domain where that jump exponential moment exists; it need not exist for every real argument without further assumptions. The document poses the question but supplies no answer or conditions, so the domain of existence cannot be made more specific from the information given.

Key ideas

• The process combines drift, Brownian motion, and compound Poisson jumps.
• A Laplace exponent describes exponential moments that scale exponentially with elapsed time.
• The jump-size distribution determines which real arguments have finite exponential moments.
• The exponent is defined only on the domain where the jump moment-generating function exists.
• The document does not specify a jump distribution, so it cannot establish the full domain of existence.

Tags

# Laplace Exponent of a Jump-Diffusion Process


# Laplace Exponent of a Jump-Diffusion Process












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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