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Sm ( )S(m )
mB( ) as m
• 0 = t0n< t1
n< ... < tn
n= t
n
B(ti+1n ) B(ti
n )tin ,ti+1
n n
as n 0
• 0 = t0n< t1
n< ... < tn
n= t
n
(B(ti+1n
tin ,ti+1
n n
) B(tin ))2 t as n 0
B(ti+1n ) B(ti
n )tin ,ti+1
n n
as n 0
•
•
0 = t0n< t1
n< ... < tn
n= tn
(B(ti+1n
tin ,ti+1
n n
) B(tin ))2 t as n 0
B(ti+1n ) B(ti
n )tin ,ti+1
n n
as n 0
•
•
0 = t0n< t1
n< ... < tn
n= tn
(B(ti+1n
tin ,ti+1
n n
) B(tin ))2 t as n 0
B(ti+1n ) B(ti
n )tin ,ti+1
n n
as n 0
Formally: "(dB)2= dt"
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B(a )=d
a1/2B( ), a > 0
Let f :R R be twice continuously differentiable
f (B(t)) f (B(0)) = ( f (B(ti+1n
ti+1n ,ti
n n
) f (B(tin ))
Let f :R R be twice continuously differentiable
f (B(t)) f (B(0)) = ( f (B(ti+1n
ti+1n ,ti
n n
) f (B(tin ))
= { f '(B(tin
ti+1n ,ti
n n
)(B(ti+1n ) B(ti
n ))
+ 12 f ''(B(ti
n ))(B(ti+1n ) B(ti
n ))2}
Let f :R R be twice continuously differentiable
f (B(t)) f (B(0)) = ( f (B(ti+1n
ti+1n ,ti
n n
) f (B(tin ))
= { f '(B(tin
ti+1n ,ti
n n
)(B(ti+1n ) B(ti
n ))
+ 12 f ''(B(ti
n ))(B(ti+1n ) B(ti
n ))2}
f '(B(s))dB(s) + 12
0
t
f ''(B(s))ds as n
0
t
0
For f :R R twice continuously differentiable
f (B(t)) f (B(0)) = f '(B(s))dB(s) + 12 f ''(B(s))ds
0
t
0
t
• B = (B1,...,Bd )
B1,...,Bd
• B = (B1,...,Bd )
B1,...,Bd
B1
B2
For f C 2 (Rd ),
f (B(t)) f (B(0)) = f (B(s)) dB(s)0
t
+12 f (B(s))ds
0
t
where f =
f
x1
f
xd
and f =2 f
xi2
i=1
d
For f C 2 (Rd ),
f (B(t)) f (B(0)) = f (B(s)) dB(s)0
t
+12 f (B(s))ds
0
t
If f is bounded, then for all t 0,
Ex f (B(s)) dB(s)0
t
= 0
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Rd
g a continuous function on the boundary D f
f = 0 in D
f = g on D
f = 0 f = g
D
g = 4sin(5 )
g = 0
= inf{t > 0 :B(t) D}
B( )D
= inf{t > 0 :B(t) D}
B( )
f (x) = Ex[g(B( ))]
D
f (x) = Ex[g(B( ))] for all x D
•
f (B(t)) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
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•
•
•
f (B(t)) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
f (B(t )) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
•
•
•
f (B(t)) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
f (B(t )) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
0
f (B(t)) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
•
•
•
•
f (B(t )) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
0
Ex[ f (B(t ))] = Ex[ f (B(0))] = f (x)
f (B(t)) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds
•
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•
•
•
f (B(t )) = f (B(0)) + f0
t
(B(s)) dB(s) + 12 f0
t
(B(s))ds0
Ex[ f (B(t ))] = Ex[ f (B(0))] = f (x)
t ,Ex[ f (B( ))] = f (x)
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