In this post, we consider one particular specification of the background driving Lévy process in the general Ornstein-Uhlenbeck stochastic clock dynamics introduced in a previous post. We show that a compound Poisson process with exponentially distributed increments yields a gamma stationary distribution for the instantaneous rate of activity. We also discuss how the problem could be approached from the other end by imposing the stationary distribution and finding the corresponding background driving Lévy process.
- General Stochastic Clocks
- Ornstein-Uhlenbeck Stochastic Clocks
- Cox-Ingersoll-Ross Stochastic Clocks
- Inverse Gaussian Ornstein-Uhlenbeck Stochastic Clocks
Construction via the Background Driving Lévy Process
where is a Poisson process with arrival rate and is a sequence of independent identically distributed (i.i.d.) exponential random variables with mean for . See Barndorff-Nielsen and Shephard (2001a) or Chapter 5.2 in Schoutens (2003) for this construction of the gamma Ornstein-Uhlenbeck process. The characteristic function of the background driving Lévy process is given by
see e.g. Proposition 3.4 in Cont and Tankov (2004). The corresponding characteristic exponent is thus
The characteristic function of the instantaneous rate of activity is
Here, we used the previously obtained result that links the expected value of an exponential stochastic integral with respect to to the exponential integral over the characteristic exponent , i.e.
Taking the limit yields
We recognize this as the characteristic function of a gamma distributed random variable as previously claimed.
Construction via the Stationary Distribution
Without assuming a particular form of the background driving Lévy process , we start from
as above. Let
be the cumulant generating function of the stationary distribution of . Then (not being fully rigorous)
This relationship between the cumulant generating function of the stationary distribution of and the cumulant generating function of the background driving Lévy process has been obtained by Barndorff-Nielsen (2001). Now, given that the stationary distribution of is with
As shown before, this is the cumulant generating function of a compound Poisson process with arrival rate and i.i.d. exponentially distributed increments with mean .
As shown in my previous post, the general solution for the characteristic function of the total activity process is given by
The integral evaluates to
Putting everything together, we get
This coincides with the expression given in Chapter 7.2 of Schoutens (2003).
Barndorff-Nielsen, Ole E. (2001) “Superposition of Ornstein-Uhlenbeck Type Processes,” Theory of Probability and its Applications, Vol. 45, No. 2, pp. 175-194
Barndorff-Nielsen, Ole E. and Neil Shephard (2001a) “Modelling by Lévy Processes for Financial Econometrics,” in Ole E. Barndorff-Nielsen, Thomas Mikosch and, and Sidney I. Resnick eds. Lévy Processes – Theory and Applications: Birkhauser, pp. 283-318
Cont, Rama and Peter Tankov (2004) Financial Modelling with Jump Processes: Chapman & Hall
Schotens, Wim (2003) Lévy Processes in Finance: John Wiley & Sons