Infinitesimal Calculus of Random Walk and Poisson Processes

Gauge Institute Journal,
H. Vic Dannon
( x − μ)
dx is at most infinitesimal (it vanishes at x = μ ),
− μ 2
d )
2
1
( x ) ( x is at most infinitesimal,
2
e
1
2 2
− ( x−μ
) ( dx ) 2 1 2 2
1 ( x μ) ( )
2
infinitesimal
≈ − − dx ≈ 1,
f ( x) ≈ ( dx) = infinitesimal.
1
2π
x = infinite hyper-real Then,
x
1
= α , where α is finite hyper-real,
dx
1 2 2 1 1
2 2 dx
( ) 2
2
)
( x − μ) ( dx) = α −μ
( dx ,
e
1
2π
1 2 1 2
α αμ dx μ
2 2
= − ( ) + ( dx)
2 ,
1
α 2
2
≈ ,
1 2 2
( x μ) ( dx)
1
2 2
− − −
−
1
2
α
2
≈ e
fx ( ) ≈ ( dxe ) = infinitesimal.
α
2
,
6.2 Infinitesimal Variance
σ = dx ⇒ f ( x ) = Delta Function
Proof:
We’ll show that
fx ( )
=
1 e
2πdx
−
1
( x−μ
) 2
2 dx
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