Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
dF(2, 2) = Pr( X( ζ, 2) = 2) = ⋅ = . <br />
5 4 5<br />
9 8 18<br />
7.3 Hyper-real Probability Density <strong>of</strong> X(,)<br />
ζ t<br />
ζ<br />
0<br />
Let X(,)<br />
t be Hyper-real, <strong>and</strong> fix t = t . If there is Hyperreal<br />
f (, xt<br />
0)<br />
so that<br />
dF(, x t ) = f (, x t ) dx ,<br />
0 0<br />
Then<br />
fxt (, )<br />
0<br />
=<br />
dF(, x t )<br />
dx<br />
0<br />
is the Hyper-real Probability Density <strong>of</strong> X(, ζ t ).<br />
0<br />
7.4 Expectation <strong>of</strong> Hyper-real X(,)<br />
ζ t<br />
ζ<br />
0<br />
Let X(,)<br />
t be Hyper-real, fix t = t , <strong>and</strong> define<br />
If dF(, x t ) = f (, x t ) dx ,<br />
Example<br />
E[ X( ζ, t )] ≡ ∑ xdF( x, t ),<br />
0 0<br />
0 0<br />
x= X(, ζ t ), ζ∈S<br />
EX [ ( ζ, t )] = ∑ xf( xt , ) dx.<br />
0 0<br />
x= X(, ζ t ), ζ∈S<br />
0<br />
0<br />
30