Stochastic process with ACF [$]\exp(-s^2)[$]

cfp · November 24th, 2014, 4:04 pm

By some old results of Schoenberg, for any times [$]t_1,t_2,\dots,t_n[$], the matrix with element [$]i,j[$] given by [$]\exp(-|t_i-t_j|^2)[$] is positive definite. If I am not mistaken, this is sufficient for the existence of a Gaussian stochastic process with auto-covariance function [$]\exp(-s^2)[$]. Now I know that [$]\exp(-|s|)[$] is the ACF of an Ornstein-Uhlenbeck process, but what stochastic process corresponds to the ACF [$]\exp(-s^2)[$]? Does it have a causal "MA" type representation?Thanks in advance,Tom

piterbarg · November 25th, 2014, 7:20 am

The process does not have a name but is widely used in some applications eg Oceanography. It is an example of a process whose trajectories are analytic a.s. so a full trajectory can be completely recovered by observations in an arbitrary small time interval. Here is a referenceБеляев Ю. К. Аналитические случайные процессы. Теория вероятностей и ее применения, IV, вып. 4 (1959). [c.174]I think this is the English versionhttp://epubs.siam.org/doi/abs/10.1137/1104040not sure it has much use in financehope it helpsVladimir

cfp · November 25th, 2014, 1:47 pm

Thanks, that's a useful reference.