In this pose, we consider an Ornstein-Uhlenbeck stochastic clock whose instantaneous rate of activity process has an inverse Gaussian stationary distribution. We use the previously obtained relationship between the cumulant generation function of the stationary distribution and that of the background driving Lévy process and show that the latter can be represented as the sum of an inverse Gaussian processes and a compound Poisson process with chi-squared distributed increments.

**Related Posts**

- General Stochastic Clocks
- Ornstein-Uhlenbeck Stochastic Clocks
- Gamma Ornstein-Uhlenbeck Stochastic Clocks
- Cox-Ingersoll-Ross Stochastic Clocks

**Construction via the Background Driving Lévy Process**

We assume that the stationary distribution of is inverse Gaussian with characteristic exponent

This is the distribution of the first hitting time of a drifted Brownian motion to the level , where ; see e.g. Schoutens (2003). As shown in a previous post, the following relationship holds

Thus,

As shown by Barndorff-Nielsen (1998) we can decompose the background driving Lévy process as where and

Here is a Poisson process with arrival rate and is a sequence of independent identically distributed (i.i.d.) chi-squared random variables with degree of freedom one; see also Schoutens (2003). We have

Next,

and thus

Here, we used the general form of the characteristic function for compound Poisson processes. Summing the cumulant generating functions yields

as claimed.

**Integrated Time-Change**

We again start from the previously obtained general solution for the characteristic function of the total activity process ,

We evaluate the two integrals corresponding to and separately. The first one is

Here, we defined

The second one is

Combining these two expressions and grouping terms yields

This expression coincides with the one given in Schoutens (2003), who defines

instead.

**References**

Barndorff-Nielsen, Ole E. (1998) “Processes of Normal Inverse Gaussian Type”, *Finance and Stochastics*, Vol. 2, No. 1, pp. 41-68

Schotens, Wim (2003) *Lévy Processes in Finance*: John Wiley & Sons