Long-memory stable {O}rnstein-{U}hlenbeck processes
The authors carry on work that has been done on stationary solutions of Langevin equations driven by noise processes that are more general than Brownian motion or fractional Brownian motion. If the driving process is Brownian motion \(W_t\), \(X_t = X_0 - \lambda \int_0^t X_s ds + W_t \), then, given \(X_0\), the solution is \[ X_t = e^{-\lambda t} [X_0 + \int_0^t e^{\lambda s} dW_s] = W_t + e^{-\lambda t} X_0 - \lambda \int_0^t e^{ \lambda s}W_s ds, \] and the initial condition \(X_0 = \int_{-\infty}^t e^{\lambda s} W_s ds \) yields the unique stationary solution \( X_t = W_t + \int_{-\infty}^t e^{- \lambda (t-s)} W_s ds \). The analogous result is shown to be true, if \(W_t\) is replaced by the linear fractional stable motion \( \Delta_{H, \alpha}(t) \) with \( 1 < \alpha < 2\), \(1/\alpha < H \). Here \( \Delta_{H, \alpha}(t) \) is a non-Gaussian self-similar process with possibly infinite variance and dependent increments. The authors investigate the dependence structure of the stationary solution and prove as main result long-range dependence.
stochastic differential equation with additive noise
scientific article; zbMATH DE number 2142115