Consider the integral process {Z (t) : t ≥ 0}defined by Z (t) = ∫0^t X (s) ds for X (t) equal to some random variable Aj in each time increment j Δ ≤ t

Question

Consider the integral process [latex] \{Z (t) : t \geq 0\}[/latex] defined by

[latex] Z (t) =\int_{0}^{t}{X(s)ds} [/latex]

for [latex] X (t)[/latex] equal to some random variable [latex] A_{j}[/latex] in each time increment [latex] j \Delta \leq t \leq (j + 1) \Delta [/latex], and with the [latex] A_{j}[/latex] being identically distributed, mean-zero, and independent. Find the mean value function of [latex] \left\{Z (t)\right\} [/latex] , the cross-correlation function of [latex] \left\{X (t)\right\} [/latex] and [latex] \left\{Z (t)\right\}[/latex] , and the autocorrelation function for [latex] \left\{Z (t)\right\}[/latex].

Accepted Answer

Because [latex] \mu _{X}(t)= \mu _{A}=0 [/latex], we have [latex] \mu _{Z} (t) =\mu _{A} t=0 [/latex]. In Example 4.9 we found that the autocorrelation function for [latex] \left\{X (t) ight\}[/latex] is equal to [latex]E(A^{2})[/latex] in squares of dimension [latex] \Delta [/latex] along the [latex] 45^{\circ}[/latex] line of the plane and zero elsewhere. To simplify the presentation of the results, we define a new integer time function as the number of full time increments included in [latex][0,t][/latex]:

[latex] k(t)= Int (t/\Delta )=j [/latex] if [latex] j\Delta \leq t\lt (j+1)\Delta [/latex]

This allows us to classify more simply any point into one of three sets: points within the squares of dimension [latex] /\Delta [/latex] along the [latex] 45^{\circ} [/latex] line of the [latex] (t_{1},t_{2})[/latex] plane are described by [latex] k(t_{1}) =k(t_{2})[/latex] , by [latex] k(t_{1})\Delta \leq t_{2} \lt [k(t_{1})+1] \Delta [/latex]; or by [latex] k(t_{2})\Delta \leq t_{1} \lt [k(t_{2})+1] \Delta [/latex]

points below the squares are described by [latex] k(t_{1}) \gt k(t_{2})[/latex], by [latex] t_{2} \lt k(t_{1}) \Delta [/latex], or by [latex] t_{1} \geq [k(t_{2})+1] \Delta [/latex]; and points above the squares are described by [latex] k(t_{1}) \lt [k(t_{2})[/latex], by [latex] t_{2} \geq [k(t_{1})+1] \Delta [/latex], or by [latex] t_{1}\lt k(t_{2}) \Delta [/latex].

Then using

[latex] \phi _{XZ} (t_{1},t_{2})=\int_{0}^{t_{2}}{\phi _{XX}(t_{1},s_{1})ds_{2}} [/latex]

gives

[latex] \phi _{XZ} (t_{1},t_{2})=0 [/latex] if [latex]k(t_{1}) \gt k(t_{2}) [/latex]

[latex] \phi _{XZ} (t_{1},t_{2})=E(A^{2}) [t_{2} –k(t_{1}) \Delta] [/latex] if [latex]k(t_{1}) = k(t_{2}) [/latex]

[latex] \phi _{XZ} (t_{1},t_{2})=E(A^{2}) \Delta [/latex] if [latex]k(t_{1}) \lt k(t_{2}) [/latex]

The function [latex] \phi _{XZ} (t_{1},t_{2})[/latex] versus [latex] t_{2}[/latex] grows linearly from zero to [latex] E(A^{2}) \Delta [/latex] as [latex] t_{2} [/latex] varies from [latex] k(t_{1}) \Delta [/latex] to [latex] [k(t_{1})+ 1] \Delta [/latex]. The same function versus [latex] t_{1}[/latex] is a stairstep that is constant at the level [latex] E(A^{2}) \Delta [/latex] for [latex] t_{1} \lt k(t_{2})\Delta [/latex], at the level [latex] E(A^{2}) [t_{2} -k(t_{1})\Delta ][/latex] for [latex] k(t_{2})\Delta \leq t_{1} \lt [k(t_{2})+1] \Delta [/latex], and at zero for [latex] t_{1} \geq [k(t_{2})+1] \Delta [/latex].

Now

[latex] \phi _{ZZ} (t_{1},t_{2})=\int_{0}^{t_{1}}{\phi _{XZ}(s_{1},t_{2})ds_{1}}[/latex]

gives

[latex] \phi _{ZZ} (t_{1},t_{2})=E(A^{2})t_{1}\Delta [/latex] if [latex] k(t_{1})\lt k(t_{2})[/latex]

[latex] \phi _{ZZ} (t_{1},t_{2})=E(A^{2})\left(k(t_{1})\Delta^{2}+[t_{2}-k(t_{1})\Delta ][t_{1}-k(t_{2})\Delta ] ight) [/latex] if [latex] k(t_{1}) = k(t_{2})[/latex]

[latex] \phi _{ZZ} (t_{1},t_{2})=E(A^{2})t_{2}\Delta [/latex] if [latex] k(t_{1})\lt k(t_{2})[/latex]

Note that if [latex] \Delta [/latex] is small compared with [latex] t_{1}[/latex] and [latex] t_{2}[/latex] then one can say that [latex] \phi _{ZZ} (t_{1},t_{2}) \approx E(A^{2})\Delta \min (t_{1},t_{2})[/latex]. This expression is exact for [latex]k(t_{1}) eq k(t_{2})[/latex] and has an error of order [latex] \Delta ^{2} [/latex] for [latex]k(t_{1}) = k(t_{2})[/latex].

Question 4.22: Consider the integral process {Z (t) : t ≥ 0}defined by Z (t...

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