# Measure changes for gamma process

GENERAL THEORY

In his book Ken-Iti Sato ("Lévy Processes and Infinitely Divisible Distributions") provides the theory for measure change for Lévy processes in Theorems 33.1 and 33.2. It can be summarised as:

PROPOSITION (Sato Theorems 33.1 and 33.2)

Let $$X_t$$ be a Lévy process under the probability measure $$\mathbb{P}$$ with the characteristic triplet $$(A, \nu, \gamma)$$, and a Levy process under the probability measure $$\mathbb{Q}$$ with the triplet $$(A^Q, \nu^Q, \gamma^Q)$$.

Then the measures $$\mathbb{P}|_{\mathcal{F}_t}$$ and $$\mathbb{Q}|_{\mathcal{F}_t}$$ are equivalent for all $$t$$ if and only if the three conditions are satisfied:

1. $$A=A^{Q}$$

2. The \levy{} measures are equivalent with $$\begin{equation} \int_{-\infty}^{\infty} (e^{\phi(x)/2}-1)^2 \nu(dx) < \infty \end{equation}$$ where $$\phi(x) = \ln(\frac{d \nu^Q}{d \nu})$$

3. If $$A=0$$ then in addition we must have: $$\begin{equation} \gamma^Q = \gamma+\int_{|x|\leq 1} x(\nu^Q-\nu)(dx) \end{equation}$$

When $$\mathbb{P}$$ and $$\mathbb{Q}$$ are equivalent, then the Radon-Nikodym derivative is given by the following exponential martingale: $$\begin{equation} \left. \frac{d \mathbb{Q}}{d \mathbb{P}} \right|_{\mathcal{F}_t} = e^{U_t}, \end{equation}$$ where $$U_t$$ is a Lévy process with characteristic triplet $$(A_U, \nu_U, \gamma_U)$$ given by: $$\begin{equation} A_U = \langle\eta, A\eta\rangle \end{equation}$$ $$\begin{equation} \nu_U = \nu\phi^{-1} \end{equation}$$ $$\begin{equation} \gamma_U = -0.5\langle\eta, A\eta\rangle - \int_{-\infty}^\infty (e^y -1 -y \mathbf{1}(0<|y|<1))\nu_U(dx) \end{equation}$$ where $$\eta=0$$ if $$A=0$$, otherwise it solves the equation: $$\begin{equation} \gamma^Q - \gamma+\int_{|x|\leq 1} x(\nu^Q-\nu)(dx) = A\eta \end{equation}$$

Note, that $$\gamma_U$$ is determined in the equation above by the condition that $$e^{U_t}$$ is a martingale.

GAMMA PROCESS

By a gamma process $$\{\gamma_t\}$$ on a given probability space with shape $$m$$ and scale $$\kappa$$ we mean a process with independent increments, such that $$\gamma_0 = 0$$ and the random variable $$\gamma_t$$ has a gamma distribution with mean $$\kappa mt$$ and variance $$\kappa^2 mt$$. It has the density function: $$\begin{equation}\label{eq:gamma_density} g(x) = \frac{\kappa^{-mt}x^{mt-1}e^{-x/\kappa}}{\Gamma[mt]}\quad \text{ for }x\geq 0, \end{equation}$$ where $$\Gamma[a]$$ denotes the standard gamma function. The characteristic function of the gamma process is given by: $$\begin{equation} \mathbb{E}\left[e^{i\lambda\gamma_t}\right]=\frac{1}{(1-i\kappa\lambda)^{mt}} = \exp(t m \log (1 + \kappa u)) \end{equation}$$ By doing some algebraic transformation (cf. Protter, 2003, p. 33) one can show that the associated Lévy measure is given by: $$\begin{equation} \nu(x) = mx^{-1}e^{-x/\kappa} \end{equation}$$ Let $$\gamma_t$$ be a standard gamma process with shape parameter $$m$$ and scale parameter $$\kappa=1$$ under the probability measure $$\mathbb{P}$$. For a given $$\kappa>0$$ we can define a measure change to an equivalent probability measure $$\mathbb{Q}$$ as: $$\begin{equation} \left. \frac{d \mathbb{Q}}{d \mathbb{P}} \right|_{\mathcal{F}_t} = \kappa^{-mt}e^{\frac{\kappa-1}{\kappa}\gamma_t}. \end{equation}$$ It is straight forward to check that the Radon-Nikodym process is a martingale. Under the $$\mathbb{Q}$$ measure the process $$\gamma_t$$ is a gamma process with the same shape $$m$$, but with scale given by $$\kappa$$. To show this, lets calculate the characteristic function: $$\begin{equation} \mathbb{E}^Q[e^{ia\gamma_t} = \mathbb{E} \left[e^{ia\gamma_t} \kappa^{-mt} e^{\frac{\kappa-1}{\kappa}\gamma_t} \right] \end{equation}$$ $$\begin{equation} =\kappa^{-mt}\mathbb{E} \left[ \exp\left(\frac{ia\kappa + \kappa -1}{\kappa}\gamma_t\right) \right] \end{equation}$$ $$\begin{equation} =\kappa^{-mt}\frac{1}{\left[\frac{1-ia\kappa}{\kappa}\right]^{mt}} \end{equation}$$ $$\begin{equation} =\frac{1}{\left[1-ia\kappa\right]^{mt}} \end{equation}$$ which indeed shows that $$\gamma_t$$ has the scale parameter $$\kappa$$ under $$\mathbb{Q}$$.

We can calculate the measure change above using the proposition above. Because $$\frac{d \nu^Q}{d \nu} = e^{x/\kappa - x}$$, we have: $$\begin{equation} \phi^{-1}(x) = \frac{\kappa y}{1-\kappa}. \end{equation}$$ The Lévy density of $$U_t$$ is thus given by: $$\begin{equation} \nu_U = \nu(\phi^{-1}(x)) = m\left( \frac{1-\kappa}{\kappa}\right) \exp\left(\frac{\kappa x}{\kappa -1} \right) \end{equation}$$ which is also a gamma process with the same scale as in the previous method given by $$\frac{1-\kappa}{\kappa}$$. However, the shape parameter is different! To my best knowledge $$U_t$$ should be unique up to null sets, so what am I doing wrong here?

• \begin{align} & A_U = <\eta,A\eta> \\ {} \\ & A_U = \langle\eta, A\eta\rangle \end{align} The second item above is the right way to do it, and I edited accordingly. Apr 7, 2022 at 1:48