Space-time stochastic processes

Predicting Integrals of Stochastic Processes Using Space-Time Data

Consider a stationary spatial process Z(x) = S(x) + (x) on I R d where S(x) is the signal process and (x) represents measurement errors. This paper studies asymptotic properties of the mean squared error for predicting the stochastic integral R D v(x)S(x) dx based on space-time observations on a xed cube D I R d. The random noise process (x) is assumed to vary with time t, and the covariance st...

Modeling and estimation of space-time stochastic processes

Discrete Time Stochastic Processes

4 Martingales 35 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Doob Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Optional Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Inequalities and Convergence . . . . . . . . . . . ...

Stochastic Processes in Continuous Time

1 Basic Concepts 3 1.1 Notions of equivalence of stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Sample path properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Properties of filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Stopping times . . . . . . . . . . . . . . . . . ....

Space representation of stochastic processes with delay.

We show that a time series x(t) evolving by a nonlocal update rule x(t) =f (x(t-n),x(t-k)) with two different delays k < n can be mapped onto a local process in two dimensions with special time-delayed boundary conditions, provided that n and k are coprime. For certain stochastic update rules exhibiting a nonequilibrium phase transition, this mapping implies that the critical behavior does not ...