Characteristic function of maximum of Levy process

Nothing to add to the title, I'm looking for the characteristic function of the maximum of a Levy process, can someone help me out? Thanks

• What is your motivation? Do you really expect that the characteristic function can be calculated explicitly.....? – saz Oct 19 '17 at 18:38

The literature might contain some examples of the characteristic function of the maximum process for some simple processes.

The general way for getting what you need is to take the inverse Laplace transform of the wiener Hopf Factor of the maxima process.

Let $X_t$ be a Levy process, and $M_t = \sup_{t\in[0,\infty)}X_t$ be the supremum process. You require the characteristic function of the supremum Levy process, i.e. $\mathbf{E}[e^{i\xi M_t}]$. However, it is often easier to find the characteristic function of the randomly stopped maxima. Let $\tau$ be an exponentially distributed stopping time with parameter $r$, then the characteristic function of the randomly-stopped maximum process is the maxima Wiener-Hopf factor, i.e. $$\phi_r^+(\xi)=\mathbf{E}[e^{i\xi M_{\tau}}]=r\left[\int_0^\infty e^{-rt}\mathbf{E}[e^{i\xi M_t}]dt\right].$$

Notice that the exponential density measure is $re^{-rt}$, and the Wiener-Hopf factor for the randomly-stopped maximum happens to be the Laplace transform of the characteristic function of the process maximum. This means $$\mathbf{E}[e^{i\xi M_t}] = r^{-1}\mathcal{L}^{-1}_{r\rightarrow t}\phi_r^+(\xi) = (2r\pi i)^{-1}\int_{\gamma-i\infty}^{\gamma+i \infty}e^{rt}\phi_r^+(\xi)dr$$.

The Wiener-Hopf factors are available exactly for some Levy processes such as the Geometric Brownian Motion, Spectrally-Negative and Specrally-Positive processes, jump-diffusions with exponentially-distributed jump sizes such as the hyper-exponential JD process, phase-type processes, meromorphic Levy processes (although some work is needed to find the zeros of the complex-valued characteristic polynomial). Generally, the Wiener-Hopf factors for any Levy process can be found by computing the Baxter-Donsker convolution.

The same arguments apply for finding the characteristic function of the process minimum, but then we work with the other Wiener-Hopf factor $\phi_r^-(\xi)$, which is the characteristic function of the randomly stopped minimum....

Hope this helps...

Tamim