TABLE 3 The parameter setting of
Ornstein-Uhlenbeck process in case 1.
Wylomanska, "Time-changed
Ornstein-Uhlenbeck process," Journal of Physics A: Mathematical and Theoretical, vol.
The mean-reverting process [Y.sub.t] evolves as an
Ornstein-Uhlenbeck process with a positive mean-reverting rate [alpha], an equilibrium level m, and the volatility of the volatility [beta].
For the
Ornstein-Uhlenbeck process, the infinitesimal generator is G = -[mu]x(d/dx) + ([[sigma].sup.2]/2)([d.sup.2]/d[x.sup.2]).
As we have been working with continuous processes throughout this work, we choose an
Ornstein-Uhlenbeck process to model the daily maximum.
We assume that the asset price process {[X.sub.t]; t [greater than or equal to] 0} is conditionally lognormal and the volatility process {[Y.sub.t]; t [greater than or equal to] 0} is a fractional
Ornstein-Uhlenbeck process. {[X.sub.t]; t [greater than or equal to] 0} and {[Y.sub.t]; t [greater than or equal to] 0} satisfy the following equations:
The
Ornstein-Uhlenbeck process satisfies the part of Condition I2 on the regressor's density in [21, page 136].
(a) The Case of an
Ornstein-Uhlenbeck Process (OU) Introduce the function
yielding the
Ornstein-Uhlenbeck process, where m, q, and v are non-negative constants interpretable as the long-range mean to which [r.sub.t] tends to revert, the speed of adjustment, and the volatility, respectively.
The Hull-White [3] model uses a lognormal process for the volatility, the Scott [4] and Stein-Stein [5] models use a mean-reverting
Ornstein-Uhlenbeck process, and the Heston model [6] uses the Feller (Cox-Ingersoll-Ross) process.
A large volume of empirical researches in financial market which indicates the assumption that these variables are stochastic volatile and follow a certain stochastic process (e.g.,
Ornstein-Uhlenbeck process) is more realistic [3, 4].
The solution is often called an
Ornstein-Uhlenbeck process. In fact,