Infinitesimal Calculus of Random Walk and Poisson Processes

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Gauge Institute Journal, H. Vic Dannon Example 2 2 Var[ X( ζ, t)] = E[ X ( ζ, t)] −( E[ X( ζ, t)]) 9 6 2 () 9 5 5 25 = − = . 32
Gauge Institute Journal, H. Vic Dannon 8. Continuity <strong>of</strong> X(,) ζ t ζ t t0 8.1 Hyper-real X(,) t is continuous at = iff for any dt , ⇔ E{[ X( ζ, t + dt) − X( ζ, t )] } = infinitesimal , ∑ X( ζ, t ), ζ∈S 0 0 0 2 2 0 ζ 0 0 [ X( ζ, t + dt) − X( , t )] dF( x, t ) = infinitesimal If dF(, x t ) = f (, x t ) dx , ⇔ ∑ X(, ζ t ), ζ∈S 0 0 0 2 0 ζ 0 0 [ X( ζ, t + dt) − X( , t )] f( x, t ) dx = infinitesimal ζ = 0 0 8.2 X(,) t is continuous at t t ⇒ EX [ ( ζ, t )] is continuous Pro<strong>of</strong>: 0 ≤ E[{[ X(, ζ t + dt) −X(, ζ t )] − E[ X(, ζ t + dt) −X(, ζ t0)]}] 0 0 0 = E{[ X( ζ, t + dt) −X( ζ, t )] } 0 0 2 − 2 E{[ X( ζ, t + dt) − X( ζ, t )] E[ X( ζ, t + dt) −X( ζ, t )]} 0 0 0 0 + { EX [ ( ζ, t + dt) −X( ζ, t )]} 0 0 2 2 2 0 0 0 0 = E{[ X(, ζ t + dt) −X(, ζ t )]} − { E[ X(, ζ t + dt) −X(, ζ t )]} Therefore, 2 33
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Infinitesimal Calculus of Random Walk and Poisson Processes

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