TD-CARMA: Painless, Accurate, and Scalable Estimates of Gravitational Lens Time Delays with Flexible CARMA Processes

CARMA processes as solutions of integral equations

A CARMA(p, q) process is defined by suitable interpretation of the formal p order differential equation a(D)Yt = b(D)DLt, where L is a two-sided Lévy process, a(z) and b(z) are polynomials of degrees p and q, respectively, with p > q, and D denotes the differentiation operator. Since derivatives of Lévy processes do not exist in the usual sense, the rigorous definition of a CARMA process is bas...

Prediction of Lévy-driven CARMA processes

The conditional expectations, E(Y (h)|Y (u),−∞ < u ≤ 0) and E(Y (h)|Y (u),−M ≤ u ≤ 0) with h > 0 and 0 < M < ∞ are determined for a continuous-time ARMA (CARMA) process (Y (t))t∈R driven by a Lévy process L with E|L(1)| < ∞. If E(L(1)2) <∞ these are the minimum mean-squared error predictors of Y (h) given (Y (t))t≤0 and (Y (t))−M≤t≤0 respectively. Conditions are also established under which the...

Gravitational lens time delays and gravitational waves.

Using Fermat’s principle, we analyze the effects of very long wavelength gravitational waves upon the images of a gravitationally lensed quasar. We show that the lens equation in the presence of gravity waves is equivalent to that of a lens with different alignment between source, deflector, and observer in the absence of gravity waves. Contrary to a recent claim, we conclude that measurements ...

Existence and Uniqueness of Stationary Lévy-driven CARMA Processes

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results gene...

CARMA Correlator CDR