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Covariance Function for Processes

The CovarianceFunction at times s and t is given by Expectation[(x[s]-μ[s])(x[t]-μ[s]),x], where x[t] is the process state and μ[t] is the mean function.

    
A discrete-time and discrete-state random process:
Wolfram Language code: CovarianceFunction[BinomialProcess[1 / 3], s, t]
Wolfram Language code: DiscretePlot3D[%, {s, 0, 10}, {t, 0, 10}, ExtentSize -> 1 / 2, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]
A discrete-time and continuous-state random process:
Wolfram Language code: CovarianceFunction[ARProcess[{2 / 3}, 1], s, t]
Wolfram Language code: DiscretePlot3D[%, {s, 0, 10}, {t, 0, 10}, ExtentSize -> 1 / 2, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]
A continuous-time and discrete-state random process:
Wolfram Language code: CovarianceFunction[PoissonProcess[1], s, t]
Wolfram Language code: Plot3D[%, {s, 0, 10}, {t, 0, 10}, Mesh -> None, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]
A continuous-time and continuous-state random process:
Wolfram Language code: CovarianceFunction[OrnsteinUhlenbeckProcess[0, 1, 1 / 5], s, t]
Wolfram Language code: Plot3D[%, {s, 0, 10}, {t, 0, 10}, Mesh -> None, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]