Stochastic
Calculus
! "# $ % & ' ( " ) *+ " ! " " , ( "
- !" , ". , ,
!
!
" , /, #
! #
"0 " "#
1 (
2 " ! " , #
$"
,
"0 " 3
- ", "
,! "
4 # )
. ,
" 5
"" " " ), 3 " "
6
7 -"" 78 - 9 : $ ( ; ' ;8 -" ;7 $"<0 ! = > - =8 - "
8 5? 7 9 >,
" "# H
""
$ @IA " "" < " I ) " /J I/ @AJB
I @A JB
"" @IA @ A " " @A @
A 6 " K " / I
% "" @IA " @ A @
A
" B
4
@
A
"" @A
@A " I <) @A
6 <"
B )
"
J
@
A
@
A
"
/J@
/A @
A
" @
A
" B @
A
" ) 25
@
A
! " # ! B$
@A 25
% J @
A
@IA$
J @ A
@IA &
#
! ! $B
$
$ @
½
¾
AJ
@
½
¾
25 %
@IA &
! ! ' !
@ I/
@AJ
@AAJ B 25
%
@IA &
( ! ' ! @ I/
@AJ
6 " / I MBA / @A
% J @
A
" ) "
@A/J
@A
/J /
$ 716 669
% (( $
&!+( .#-0
)* ( )*
( ! ( )* @
AJ@
A ! ( )#* @
A J
! ( 1! $ !' ( )#* ! ( ) ' *
$ 724( :9 % J
B !'( ($
/J
&
)* 2" 1.' 3 B$ @
A M
N )*
" .'3$
H "
. ) "
JM@
AN
$ 7 , 9
% J
B ( )* 1!
@A/J
@A /-)* 1! # !' (
@A J
@A
& !
@
A$ " (
J $
J@
A 4
/J
)* 1!#
J
$ !
@
A$ " (
J$
J@
A 25
%
!
- %, ((
6 " ) "#) <
H )
"
@A
Q " ) $ ( "
" ! . "0 '"0
4 "
25 J
B ( ! ' ! " . !
(
( '
( ! & ! (
$ ! (
& @
A
" " , B
@A $ 7; ;9
5#' ( ( /!
$
" . ) . )
@A
@AC @ " A B
@
AJ@
A 1
@ A @
AJ
J
! @
AJ
@
AJ
J
2
6
& !+( .# )* J
B ( )* J
B ( ! B$
/ !'( /!
@A @A 1
J @
* /
,
, $
" 1 ( " 1
@ , AD -> # 7
@AJ
J
H
J
@8A
,
8
"
" 2
@8A
,
8
" " " " "
< "
!
!
!
) @8AP@A @ !
A . @8A
") 7
H
H @
A "
@
A
@ "" " 1( A
""
B "" 6
B
@
A J
H " "
@8A
P"
"
6 " "
B 1
J
B
@A @A . @
A J @
A " "
@
A J@
A "
@
AJ@
$ / MBA " 6$6% @ A $ RJ
BJ
J! ) "
Æ@RAJ ,
) # " #
/J
½
/BJ
J 6 ) "
#
/J
" " . @ 6$6% A @ A#
#
" #
#
6$6%
25
# # ! # # ) ! ! #' #' $ #
@A
1 "# S " " @ IA
$
@AJ
@A
@A
! " @ + A ) " # MBN $"<0
) .
S @ " + ) 4 58 58A 1 "
) D
) +
6
/!
$# " # H)
.
)# ,
/J
B/#
" Æ " "
D,B BJ
J" MBN
JB
J
½
@
AJ
½
#
½
" "
@ )
A , + +"" 2 B
!
" H )
25
&# ) *- -
I( ' @
A
$
$ +! @BA &$ <
<) " @ TT < A
@
A
25 @
A
# $ ! ( @A
@A + # -
"# $
M $"<0 ) ! "N
$ " 1 @BA
@@NA@AJ
@A
@A
$ " " @ ) "A $"<0 ) "
@A
@AJ
@A
B @A
6
@A
@A
) ", ) < " ) O J
J@# A8@ # A ) ) "
@A
@A/J@
A
@A@
A
@A @#A
! " $"<0 , @A
@ " ) A $ 7; ;9
1! -(# ( @
A
-( 1!' $ @
AJ@
A !
(
/! 6
.
D
@
AJ
6
B D @
AJ@
A
@
AJ
J
2 @
A
% @
AJ@
A$
/J /
@
A
/J
J /
@
AB @A J @
A
@
A @
A %
/
@
2
@
A
# ,
@
A@
A %
" " +! @
A +
J@
' # , ) # (@") A (! " " ) " !", "
.
? " . ) H # S S
"# O 4 "3 % &
6
@A
@AJ&@A
@A
& "
" "
< , # $ "/
" .
@AJ
&
JP
J P Æ &
' #
' # & "
" $ " "
&
@AJM& ' N# 1) ' )
" . "#
# # JI#
! &
%
6 '@
A ,B
"
" " &
J
@A
@A
&
@A
" , @ A , . B
&
"
" "
J
@A
@AP
&
·½
6 " " "
"
' ( ¾ Ä ¾ Å D
" " ) " S @ A
J@ A
/J
&
·½
6 "
1! !$
($ ( & )* @A
J@
A J@
A )* @A
)* M@A
NJ
M&
@
·½
AN
@
A /!
79
6
J
&
·½
&
" " " 1 )<
@ A
J
&
·½
@A
J@ A
@A
J
&
·½
" @
A
J
@
A " 2 @A
J@ AJ@
79
) " , "
J&
" & "
" " $ " " '
@@A
AJ@@A
A
@ A
H
@A
J&@
A &
" " @ A
J@@&@
AA
AJ@&@@
A
AA JB
? @A @ A
J@
A @ A
J@
A
JB @ A
" 1 , @ A
79
'@A
" ' "
J&
" " & "
" "
@A
J
&
@
A
J
&
@
A
&
,
" " C
@A
J
&
@
A
J
@&
6
" & "
@ A
8&@A
@
A
. 2"
@
A! " 2" 6@
A 1 "
.
@
A
@
A
9
M@AN
B 2 @
A D @""A .
J8
P
H #
@
A
@
A )
@A )
@A @
A @
A )@A /J
)
@A 2 $
)
" $ 2 +!
J 8
C @
" A" , .
C . )
+ " ) .
+
C
C " " .
C + ) " +
" " " # ) !
) - "
% ($( ! 6 &
J8
P2
/!
. . <
J
8@
A
" "
8
@A @
A @A
J8
P(
$ "
& . # 2
! ( . ( $ ( !6
( /!
' . , ) .
"
. , <
", ) SU SU" . S U SU ! ) " SU' # ) 3
$
J /
) . H ) (
/J
B # )
½ J
½
.
( ) C . )
+ +
. " S ! " + ) ! " + H ) $ ! ,
' +!
D) . ) <
" '
/J
P 9
) . " ) #
$ "
3 +(+'
". #(!6 (& #!
)
/
25 "
3 + ( $ " . # $ ( & # !
" ) " < "
P J P J
J ' &, (
@A
') @ A @ A
2
@A
J
&
½
&
( " 2
@
A
J@
AJ@(
(
A "
@ A
J
&
(
(
½
J
@
( A
)
J
@
( A
J
(
@A ) " " )
2 # " " 2" "
@A
JB + ) . #
JB ' )
@A/J. "
2"
@ " 2" " "
A
" @ 17 'A"
@A
@
A "
$ /
@A
@
A $ / @A
&" " "0 "
2"
@
A . " "
@A
@
A""
2
2" "
@AJ@
A
' , ) @ A
@A & #
,
@A . !0 $" 2
&
& $"
#
S *
@V+A / @A *
@A " , @A C ( *
* *
" * < V "
@A '*J*
*
"*
*
< " *
S
$
% $ -' (( -( ) @$A J * 1! % ! ! ! I !'(
)* B
$ +
% ! / I $ % $
% " <" /I
<" " "
,4 =5 $6
@,/ # +A *$$ "!
& -( ( ' ! # -( -# -( -( + + #' /!
Æ "
" <" " < $
" " J
,
,
"
' ( % @BA @BAI % % " " @BA
>$4
$ . , & J
@A
@A 1 % J & < ' & " " !
& J
@A ,
@A
@
A $ # D "
@A "
@A ) .
' $@A $@
A .
H
6
B @@
AAB "
@
A
@
A
'
' , ) .
" , "
' # , .
" D " " 2 " ' , .
) "
D @ " A
.
#
@A
J@
A
@ < A " " ) @A
H )
.
" @
A ) , 6 J
# @
A
½
J@A
½
J@
½
½
A ) @A
MBA @A
J@
A
'# ) + .
) @ A /
@
A
J@
AJ@
A
) @A
.
' -"
' , "
. " ! " . " #
) .
. . ) # "0 < ' ) . "
!
$ "#
% ' # $ ( % ' ( &0
)* @A
J@
AJ@
A )* @A ( )* J
2 )#* @ AJ@A /!
@A @A / @ A
J@
AJ@
A " "
@A
" C Æ @A @A . $- " @ A
79
@A
J
A
J
!
(
" J
(
2 @A
@
( A @ A
" . . " J
( 7,9
" @ A , " "
!
@@ AA
" J
J
@
( A
J
@A
(
J
!
@@AA
" 6 "
A
(
"
H , ) 25 #
(! %,+
J
P P
+ ($ 6 6$ + ! #
H .,
) Æ "
25 #
( @
A
+ -2' J
P P +
- $
J
JB$
(
! #
$ #
& -2'! (. /!
%
J
P
P
+
( !
# & $ + J
J
' .'$ 8 ! + ($ ! (9 ' .' # 4 ' )* 6 4
J
J
' 5' .
J
P P U /J(
' ) " # " .
P
P P ) )
/J
' , ) ")
@A/J@AP@A
! "
D , . " )
.
. /+
' "" <
#
25 % .
#( !' ') # * ! ! B$
B ""
6 ) " . ) !
, "" " . Æ " . # $
J
6$6%
O @
A
+ "" B . Æ " . +
+ ""
6$6% +"< " "
@AJ
8
@ @@A
AA
6$6% " " D
0 # ""D ) / /
6. "
@A@A
44-6 .
"" %B @
%A B
,
6 . "
B
""
""
$ % 1!
#(
! B$
#( ' /!
6 " . @AJ
%B @
%A%
@
AB
$ % 1!
'$ " .
$ %
1!
$
'
. 0, ! - "
D (".
$ %
% ($ ' # %,( !6 &
J
/!
$
J
) . ,
/J
@A .
'
J
P
P
)
" +
$"<0 $
" .
4 6
" ) .
" " Æ . "
"
@
A
@A
1 "
.
@@
A@ AA
9M@
A@AN
" 9
9 @
A
( 6
B "
" 2 " $"<0
@
A
(
B 2
M@
A@ AN
B
! " M@
A@ AN
B "" 2
M@
A@ ANB
.
.
" . " ,
$ %
& . # :
. !+( ' :
J
"" /!
@=8A" , . @""A
J
8@
A
H
8
J
J
O .
J
P
# ""2 " "
&$ <
" " W < O
U
J
D " ) 0 O " "0
( ! ,
< ) ! ) * ' "
$ 78=?4 -9 %
' # & @ A
. !
@
A ! #'
+ #0 M@A
NJM@ A
N 2# + #
@A J /!
J&
&
"" "
M@A
NJ M&@
A
N J M&@
AN J M
@ A
N
H " " , " @ A ' ) , " " " . @ ,A
@A , ".
@A
@
A
@ A
J
@
A
#
M@
AAN
#
@
A
#
M@
A@AN
#
@
A 1
@A
B 1
!
@@
AA
" J
@
A
@A
2
" " , + @
A
@A
@@
A
AB 2 "
) @A@A P@ A@A J 8 1
J
P 9
J
@PA
@A 9
2
8@ AJ
@PA
@A
(PH (<H
H @A J @
( A " @=@AA
8@ AJ
@ PA@ PA @PA@ A @A@ PA P@A@ A
)
@PA@ PA @ PA@ A J
J@PA P@PA @PA @PA J9@PA@PA "
J9@ P P P A
@A@ PA @ A@ A J
6
8@ AJ@ A@A P@A@A -
8 J P
Æ @ A J @ A .
@A @A
@ ) A 1 Æ " #
@@ A
AJ@@A
A " " )
@@A
AJ@@A
A Æ "
@A
J @A
@A
J @A
-
% ( ' # -$
@A@A J@ A /!
H
@A@A J@@A A J@@ AA A J@ A
' "
2 , #
0 ( 12 3" $
" 1 ! , " /
J
8
8
, # < " " S,
@AJ
' " @
A@
AJ
-@
A
P 8
@
A
JB 1 (
J8
P 8
8
J
8
8
D
" " " )
. 2 ,
! "
&,
>, + ) /
&
J,
8
U
&
@6 $$
O .
&
J&
&
JP
&
" " &
" ! @
¼ A
A 2
¼
B
@,@,@0
AA
AJ,
8
@A0
2 0 " @BA! " $W
" " @B@AA2 % ( " " @B(
A
6
" -
% @A ! ! $ '
/J
@A
#
B
@A
/!
D )
" * <' " . "
(
J
@A
" " $W & )
&
J,
,0
P 8 0
(
$W " "
@&
AJ@&
A
@,@,0
AAJ
,
8 0
@A
%
+ ( "
' ! " " >-?='! ) .#
25
&+ .#! '$ ! ' #
@AB(@AB
1+ '+ ' +
$
1! .# $
- !+(
É
@
AJ È
É @
AJ
È
@
A +
J È
2
& <H# ) "
1
/!
)" & <H#
" . " " $ "
" " É
@A'
%
@
AJ
È M
N @ A
) ,
É
È M
N
J
È @
È M
NA J
È M
NJ
% M
N " J
25 @ '
( ! $!
J@A
A@A
A
@
A ( !
@
A
>
" " 8
@
A )
&
/J8@A
A8@BA
A
-P
8@A
A
&
A
> ) "
&
"
6
$ I " "
" -I /
)
9P8JB I 9J -I
' O J
6
@A
A
P 8
7
@A
A
" > >
6 , -" $ . " "
)
9J B : 9 J -: $
3 '
@-IA " . 9 '
@IA #( ! 2# " ! (/I@BA !
'
@-IA (# '
J@9@A
AP8@A
AAP
@A
A -9
@A
A
J
@A
A -9
@A
A
" "
2
" MBN
8@A B " ! "
9@@AJ
@
AJ
@
AJ
M9@A
AP@ !AN $
@A # 9 " -> , !
'
" 1 "8
9
P8
J Ü
9@@AP Ü
8@@A
@A " "
" "
J@A
AP
8@A
A 2
J
9@@AJ
@
AJ
@
AJ
& @A
AP
8@A
A '
+" '/
'
-9
-J9 I
9@B@AJ@@A @ I
9@@AJ@@A B @-I 6 " .
-9 J
)
9 I " ) +
9@B@AJ@@A I " ), +
9@B@AJB -I
I 0 " 9@AJ
@@
AA
1 ( $ 7- 3@9 3
'
@
A". 9 '
@
A9 # ' / ! 2# " ! ) =*
/@BA
@BA !
'
@
A$ '/ ! (# ' 9@@AJ
Ê
@@-A $
% 9 '
@
A ! ' / ! & /J B/A
I ( &
9@@AJ
M@A
AN /!
D, @BA
/J9@@A!A
A
! B
J 9@@A!A
A
-I "
J@A
A
JB D B B ! " "
J -9
-@A
AP
-9
@A
A P
8
9
@A
A
J
-9
-P9
@A
AP
-9
@A
A
@A
A
' " B " " 9 -"
-9
-P9
JB
1
"
9@@AJ
@
AJ
@
AJ
@@A
AA
3#78 -
D " -> , " 1 ( S "*E
' " -9
-J9 I
9@B@AJ@@A @ I
9@@AJ@@A B @-I >
J 6
@A
A
P 8
7
@A
A
H " -" /
P;J 6
@A
A
P 8
7
@A
A
P;@A
A D ,
J 8
)
P;@A
A , "
-9
-J 8
)
9@A
AP;@A
A9@A
A
D <*E & ,
-> , 1 ( # I J
" 1 ( "
$ 7)-$=8B &6 9 % 9
'
@
A ! ' / + ( ! # !! $
;+ 2
( @ & 9@@AJ
@
A,
;@
A /!
D, @BA
J9@A
A,
;@ A!
D
J,
;@ A!
D B
J
-9
@A
A
P
-9
-P9P;9
@A
A
J
-9
@A
A
2
! "
) " " " " > <
$
B " ( ) " O .
J@
AP@
A#
# 1 Æ < @A@AP@A@A
@A
P@A
@P
A . .
9!
" ) O .
J*@
AP5
JB 5
( 1 * )
*@A
@P
A 6 . B
@PM
NA2
MBN " < " 6 "
&
*@
A
'
% . <" " )
*
/J@
/BA " I
* /J
*
3
" "
" > " " "
@ " 5A 6 , " %- "0 " " " %- "0 " H @ >A " %-@ A ) " ) " )
@8A/J@8@
A*
A MBN
" ) " # "0 .
+ ( "
) "% "# " 1 $
&
/J,
*
@
A5
8
*@
A
6
&
B ( /!
&
)
&
J
&
*
@
A5
&
>
&
J&
*
@
A5
&
J % , &
<
$
" . &
. ) &
1
D
&
J
&
&
J
. *, )
&
*
@
A @P%&
A
5
# , @""A "
&
P%&
J
P%
%&
*@
A @P%&
A
! + %B 1 %&
*@
A @P%&
A
J
%& @P%&
A
&
*@
A
P%& @P%&
A
&
*@
A &
P
*@
A
@P
A &
,
&
2
&
@P
A " 2 " %B
%&
*@
A @P%&
A
B @&
A
&
P%&
@&
A %B
2
@&
AJ &
*$$ 70 /( :-9
@A ! + B@A=@AP>
@A! B + > B =/MBN ( &
@A=@AP>
=@A2 *
/!
D %
J#
P
*@
A 1 (
L
H # @A MBN " @#A MBN .
& <H# @ , " A&
" "
È
M@A&
NJ È
M@#AN
2
È
M@ANJ È
M@#AN L
"
6
% ' (
# &
È
M'*
N J È
M'*N /!
)
*
/J@ +
/9BA * J*
*
C L
<" *
1 ( 2
È
M'*
NJ È
M'*
*
NJ È
M'*N
/ + 6 " )
L &
J&
L
L &
J L &
*
@
A
@"A
L &
L
2
L &
J,
*
@
A
8
*@
A
% 7"# &<"
G#. "" >)" < " @
@8A . ) " "
@8A & " "
@
@8AJ L !
8@
A L &
*
" "
@8A/JM8@
A*
N 6 . * <"
@8AJ @
@8A @
@A
H G# . * < .
$ 78 = ,9 & + !
B
D-# .
@8AJ
@8AP
@
8AP
@*
8A
@*
A
@8A
@