Infinitesimal Calculus of Random Walk and Poisson Processes

Gauge Institute Journal,
H. Vic Dannon
(2)
1
EB [ ] = EB [ ] 0
dt i
= .
(3) Var[ B] = E[ B
] −( E
[ B])
0
2 2
0
=
=
1
2
( dt)
2
( dx) 1
dt
dt
2D
2
i
EB [ ]
,
2
( dx )
= (2 D) δ( t ).
0
11.3 EB( [ ζ , t)] has unbounded Variation in [ ab ,]
Proof:
2 Db− ( a) = (2 Ddt ) + (2 Ddt ) + ... + (2 Ddt )
2 2
( dx) ( dx) ( dx) 2
⎡ 2⎤ ⎡
2
= E ⎢{ B(,) ζ b −B(, ζ b − dt) } ⎥ + .. + E ⎢{
B(, ζ a + dt) −B(, ζ a)
} ⎤ ⎥
⎣ ⎦ ⎣
⎦
≤ max B(, ζ t + dt) −B(,) ζ t E ⎡ B(,) ζ b −B(, ζ b −dt) ⎤ ..
a≤≤
t b
⎣
⎦
+
infinitesimal
... + max B(, ζ t + dt) − B(,) ζ t E ⎡ B(, ζ a + dt) − B(, ζ a)
⎤
⎣
⎦
=
a≤≤
t b
= infinitesimal{ EB( ζ, b) −B( ζ, b− dt) + ... + EB( ζ, a+ dt) − B( ζ, a) },
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