A.13 Stochastic difference equations

Stochastic difference equations with random coefficients arise in studying price differences or returns of an asset. The asymptotic behavior of the solutions of these equations for large time, called the tail behavior, may exhibit the so-called power laws, and is important for that reason.

In studying stationary solutions of stochastic linear difference equations

which are of the form

are two-dimensional i.i.d. random vectors, de Haan et al. (1989) quote Kesten (1973) as providing a motivation for approximating sums by maximal terms. Kesten shows that the expression

Differentiating the characteristic function with respect to it and setting t to 0, we obtain the mean of Z n as