COLLEGIO CARLO ALBERTO
STOCHASTIC PROCESSES 2014
2.E WIENER PROCESS
GIOVANNI PISTONE
1. Exercise. The random variables
(1)
Wt1 , pWt2 ´ Wt1 q, . . . , pWtn ´ Wtn´1 q
are independent if 0 ď t1 ď ¨ ¨ ¨ ď tn .
Proof. Note that we can assume 0 ă t1 ă cdots ă tn . Proceed by induction on n:
˘
`
(2) E f1 pWt1 qf2 pWt2 ´ Wt1 q ¨ ¨ ¨ fn pWtn ´ Wtn´1 q “
ˇ
`
`
˘˘
E f1 pWt1 qf2 pWt2 ´ Wt1 q ¨ ¨ ¨ E fn pWtn ´ Wtn´1 q ˇFtn1 “
`
˘
E pf1 pWt1 qf2 pWt2 ´ Wt1 q ¨ ¨ ¨ q E fn pWtn ´ Wtn´1 q .
2. Exercise.
˘The random vector pWt1 , . . . , Wtn q, 0 ă t1 ă ¨ ¨ ¨ ă tn is Gaussian with covariance
`
Cov Wti , Wtj “ minpti , tj q. The density exists.
Proof. The increments (1) are independent and Np0, tj ´ tj´1 q, j “ 1, . . . , n, t0 “ 0, with joint
density
ˆ
˙
n
ź
1
´1{2
´1{2
2
ppx1 , . . . , xn q “
p2πq
ptj ´ tj´1 q
exp ´
x
2ptj ´ tj´1 q j
j“1
˜
¸´1{2
˜
¸
n
n
ź
x2j
1ÿ
´n{2
“ p2πq
ptj ´ tj´1 q
exp ´
(3)
2 j“1 tj ´ tj´1
j“1
The transformation from the
»
1
—1
—
(4)
y “ —1
–
..
.
increments x to the values y, A :
fi
»
1
0
0 0 ¨¨¨
—´1 1
1 0 ¨ ¨ ¨ffi
—
ffi
x
x “ — 0 ´1
1 1 ¨ ¨ ¨ffi
–
fl
..
.
x ÞÑ y is
fi
0 ¨¨¨
0 ¨ ¨ ¨ffi
ffi
y
1 ¨ ¨ ¨ffi
fl
The determinant of A is 1, hence
˜
¸´1{2
˜
¸
n
n
2
ź
ÿ
py
´
y
q
1
j
j´1
ppy1 , . . . , yn q “ p2πq´n{2
ptj ´ tj´1 q
exp ´
, t0 “ 0
2 j“1 tj ´ tj´1
j“1
˜
¸´1{2
˜
¸
n
n
2
ź
´ 2yj´1 yj
1 ÿ yj2 ` yj´1
´n{2
(5)
.
“ p2πq
ptj ´ tj´1 q
exp ´
2 j“1
tj ´ tj´1
j“1
Date: June 9, 2014.
1
The quadratic form
n
2
ÿ
yj2 ` yj´1
´ 2yj´1 yj
“
(6) y ÞÑ
tj ´ tj´1
j“1
˙
n´1 ˆ
n
ÿ
1 2 ÿ
1
1
1
1
y1 `
`
yj2 `
yn2 ´ 2
yj´1 yj
t1
tj ´ tj´1 tj`1 ´ tj
tn ´ tn´1
t ´ tj´1
j“2
j“1 j
is better described in terms of the covariance matrix Γ “ CovpWt1 , . . . , Wtn q
(7)
´
¯
´
¯
`
˘
Γij “ Cov Wti , Wtj “ Cov Wminpti ,tj q , Wmaxpti ,tj q “ Cov Wminpti ,tj q , Wminpti ,tj q “ minpti , tj q,
and
(8) Γ “ A diag
`a
˘
`a
˘
tj ´ tj´1 : j “ 1, . . . , n pA diag
tj ´ tj´1 : j “ 1, . . . , n qT “
A diag pptj ´ tj´1 q : j “ 1, . . . , nq AT
and
ˆ
(9)
´1
Γ
“ pA
´1 T
q diag
˙
1
: j “ 1, . . . , n pA´1 q
tj ´ tj´1
3. Exercise. The distribution of Wt given W1 “ 0, t ă 1 is equal to the distribution of the
Brownian bridge Wt ´ tW1 .
ˆ „
˙
„
t t
t t
Proof. The joint distribution of pWt , W1 q is N2 0,
, with det
“ t ´ t2 and
t 1
t 1
„
´1
„
t t
1 ´t
1
“ t´t
. The joint density is
2
t 1
´t t
ˆ
˙
` 2
˘
1
2
´1
2 ´1{2
(10)
pt,1 py1 , y2 q “ p2πq pt ´ t q
exp ´
y ´ 2ty1 y2 ` ty2
2pt ´ t2 q 1
and the conditional density is
´
˘¯
` 2
1
2
p2πq´1 pt ´ t2 q´1{2 exp ´ 2pt´t
2 q y1 ´ 2ty1 y2 ` ty2
´ 2¯
(11) pWt |W1 py1 |y2 q “
“
y
p2πq´1{2 exp ´ 22
ˆ
˙
` 2
˘
1
´1{2
2 ´1{2
2
2 2
p2πq
pt ´ t q
exp ´
y ´ 2ty1 y2 ` ty2 ´ pt ´ t qy2
“
2pt ´ t2 q 1
ˆ
˙
` 2
˘
1
´1{2
2 ´1{2
2 2
p2πq
pt ´ t q
exp ´
y ´ 2ty1 y2 ` t y2 .
2pt ´ t2 q 1
In particular, because of the continuity, we can define
(12)
´1{2
pWt |W1 py1 |0q “ p2πq
2 ´1{2
pt ´ t q
ˆ
exp ´
˙
` 2˘
1
y
.
2pt ´ t2 q 1
We have Var pWt ´ tW1 q “ Var pWt q ` t2 Var pW1 q ´ 2t Cov pWt , W1 q “ t ` t2 ´ 2t2 “ t ´ t2 .
4. Exercise. The Wiener process is a Markov process an a martingale.
2
Proof. For all φ such that φ ˝ Wt is integrable and s ă t
(13)
˙
ˆ
x2
´1{2
´1{2 ´ 2pt´sq
E pφpWt q |Fs q “ E pφpWt ´ Ws ` Ws q |Fs q “ φpx`Ws q p2πq
dx “
pt ´ sq
e
˙
ˆ
ż
ż
py´Ws q2
´1{2
´1{2 ´ 2pt´sq
dy “ φpyqkpWs , yq dy,
φpyq p2πq
pt ´ sq
e
ż
that is the conditional distribution of Wt given Fs is NpWs , t ´ sq. In particular, if φpyq “ y,
then E pWt |Fs q “ Ws .
5. Exercise. Define δf pxq “ xf pxq ´ f 1 pxq and δ n 1 “ Hn pxq. For Z „ Np0, 1q, the random
variables Hn pZq are orthogonal.
Proof. As each Hn is a monic polynomial of degree n, and
ż
ż
¯
` 1
˘
d ´
2
´1{2
1
´z 2 {2
´1{2
gpzqe´z {2 dz “
(14) E f pZqgpZq “ p2πq
f pzqgpzqe
dz “ ´p2πq
f pzq
dz
ˆ
˙
ż
d
2
2
´ p2πq´1{2 f pzq
gpzqe´z {2 ´ zgpzqe´z {2 dz “ E pf pZqδgpZqq ,
dz
for m ă n
(15)
E pHm pZqHn pZqq “ E pHm pZqδ n 1q “ E pdn Hm pZqq “ 0.
6. Exercise. Let φ : R ˆ R` Ñ R and φ P C 2,1 . Under integrability conditions, if t P R` and
B
1 B2
2 By 2 φpy, tq ` Bt φpy, tq “ 0, then M “ φpW, ¨q is a martingale.
?
2
1
Proof. As kpx, y; s, tq “ ?t´s
f ppy ´ xq{ t ´ sq with f pzq “ p2πq´1{2 e´z {2 , in the limit t Ó s
#
0
if y ‰ x
2
2
(16)
lim kpx, y; s, tq “ lim ue´u py´xq {2 “
“ kpx, y, s, sq.
uÑ8
tÓs
`8 if y “ x
We have f 1 pzq “ ´zf pzq, f 2 pzq “ pz 2 ´ 1qf pzq and
d
´1{2
dt pt ´ sq
“ ´ 12 pt ´ sq´3{2 . It follows that
(17)
B2
B2
kpx,
y;
s,
tq
“
pt ´ sq´1{2 f ppt ´ sq´1{2 py ´ xqq “ pt ´ sq´3{2 f 2 ppt ´ sq´1{2 py ´ xqq “
By 2
By 2
pt ´ sq´3{2 ppt ´ sq´1 py ´ xq2 ´ 1qf ppt ´ sq´1{2 py ´ xqq
and
(18)
B
B
kpx, y; s, tq “ pt ´ sq´1{2 f ppt ´ sq´1{2 py ´ xqq “
Bt
Bt
ˆ
˙
1
1
´ pt ´ sq´3{2 f ppt ´ sq´1{2 py ´ xqq ` pt ´ sq´1{2 f 1 ppt ´ sq´1{2 py ´ xqq ´ pt ´ sq´3{2 py ´ xq “
2
2
1
1
´ pt ´ sq´3{2 f ppt ´ sq´1{2 py ´ xqq ´ pt ´ sq´2 py ´ xqf 1 ppt ´ sq´1{2 py ´ xqq “
2
2
1
1
´3{2
´1{2
´ pt ´ sq
f ppt ´ sq
py ´ xqq ` pt ´ sq´5{2 py ´ xq2 f ppt ´ sq´1{2 py ´ xqq “
2
2
`
˘
1
pt ´ sq´3{2 pt ´ sq´1 py ´ xq2 ´ 1 f ppt ´ sq´1{2 py ´ xqq,
2
hence
B
1 B2
(19)
kpx, y; s, tq “ kpx, y; s, tq.
2 By 2
Bt
3
ş
We want to show that E pφpWşt , tq |Fs q “ φpWs , sq, that is φpy, tqkpWs , y; s, tq dy “ φpWs , sq,
which,
in turn, is implied by φpy, tqkpx, y; s, tq dy “ φpx, sq, x P R. The function t ÞÑ
ş
φpy, tqkpx, y; s, tq dy is defined for t ą s and with derivative equal to
ż
ż
B
B
φpy, tqkpx, y; s, tq dy “
(20)
φpy, tqkpx, y; s, tq dy “
Bt
Bt
˙
ż ˆ
B
B
φpy, tqkpx, y; s, tq ` φpy, tq kpx, y; s, tq dy “
Bt
Bt
˙
ż ˆ
B
1
B2
φpy, tqkpx, y; s, tq ` φpy, tq 2 kpx, y; s, tq dy “
Bt
2
By
ż
ż
1
B2
B
φpy, tqkpx, y; s, tq dy `
φpy, tq 2 kpx, y; s, tq dy “
Bt
2
By
ż
ż 2
B
1
B
φpy, tqkpx, y; s, tq dy “
φpy, tqkpx, y; s, tq dy `
Bt
2 By 2
ż
ż
B
φpy, tqkpx, y; s, tq dy ´ φpy, tqkpx, y; s, tq dy “ 0.
Bt
For example,
ˆ
˙
1 B2
B
(21)
`
py 2 ´ tq “ 0
2 By 2 Bt
ˆ
˙
1 B2
B
2
(22)
`
eay´a t{2 “ 0
2
2 By
Bt
Other proofs are possible, namely
ż
ż
2
(23)
t ÞÑ φpy, tqkpx, y; s, tq dy “ phipx ` pt ´ sq1{2 z, tqp2πq´1{2 e´z {2 ,
has zero derivative for t ą s.
7. Exercise. The series of continuous function on r0, 1s
n´1 ˆż
˙
8 2ÿ
8
t
ÿ
ÿ
pnq
(24)
t ÞÑ tZo `
hj,n psq ds Zj,n “
Wt ,
n“1 j“1
0
n“0
with Z0 , Zj,n , n P N, j “ 1 . . . 2n´1 , are IID Np0, 1q, and hj.n is the j-th Haar function of the
n-th order, is a Wiener process on r0, 1s for its filtration. See [1, §2.3]
Proof.
(1) The Haar function are orthogonal in L2 r0, 1s because different functions of the same
order have disjoint supports and functions of different orders are such that the one with
lower order is constant on the support of the other. The proof of the completeness use
a monotone class argument taking as a π-class the binary intervals.
p0q
p0q
(2) Wt “ tZ0 has the correct distribution at t “ 1, i.e. W1 „ Np0, 1q. If W is a Wiener
p0q
p0q
process, then Wt „ tW1 , in general pt ÞÑ Wt q „ pt ÞÑ tW1 q.
n´1
(3) If n “ 1, 2
“ 1, and the Haar function is h1,1 psq “ p0 ď s ă 1{2q ´ p1{2 ď s ă 1q and
şt
the Shauder function is S1,1 ptq “ 0 h1,1 psq ds “ tp0 ď t ă 1{2q ` p1{2 ´ tqp1{2 ď t ă 1q.
p1q
(25)
The approximation of order 1 is Wt “ tZ0 ` S1,1 ptqZ1,1 . At the points t “ 0, 1{2, 1
the Gaussian vector is
» p1q fi »
fi »
fi »
fi
W0
0
0 0 „
0Z0 ` S1,1 p0q
`
˘
— p1q ffi – 1
1 fl
1
1
1
1 fl Z0
–
fl
–
“ 2 Z0 ` 2 Z1,1 “ 2 2
–W1{2 fl “ 2 Z0 ` S1,1 2
Z1,1
p1q
1Z
`
S
p1q
Z0
1 0
0
1,1
W
1
4
with covariance
fi
fi »
fiT »
0 0
0
0
0
0
0
p1q
p1q
p1q
CovpW0 , W1{2 , W1 q “ –1{2 1{2fl –1{2 1{2fl “ –0 1{2 1{2fl ,
1
0
1
0
1
0 21
»
(26)
p1q
p1q
p1q
so that pW0 , W1{2 , W1 q „ pW0 , W1{2 , W1 q.
?
(4) If n “ 2, then 2n´1 “ 2 and 2pn´1q{2 “ 2. The Haar function of order 2 are
?
?
h1,2 psq “ 2p0 ď s ă 1{4q ´ 2p1{4 ď s ă 1{2q,
(27)
?
?
(28)
h2,2 psq “ 2p1{2 ď s ă 3{4q ´ 2p3{4 ď s ă 1q,
(29)
(30)
and the Shauder functions are
?
?
S1,2 ptq “ 2tp0 ď t ă 1{4q ` 2p1{2 ´ tqp1{4 ď t ă 1{2q,
?
?
S2,2 ptq “ 2pt ´ 1{2qp1{2 ď t ă 3{4q ` 2p1 ´ tqp3{4 ď t ă 1q.
The approximation of order 2 is
p2q
(31)
Wt
“ tZ0 ` S1,1 ptqZ1,1 ` S1,2 ptqZ1,2 ` S2,2 ptqZ2,2 .
At the binary points t “ 0, 1{4, 1{2, 3{4, 1 the Gaussian vector is
»
(32)
fi
p2q
fi
»
W0
0
— p2q ffi
?
—W ffi —
— 1{4 ffi —p1{4qZ0 ` p1{4qZ1,1 ` p 2{4qZ1,2 ffi
ffi
— p2q ffi —
ffi “
p1{2qZ0 ` p1{2qZ?1,1
—W1{2 ffi “ —
ffi
— p2q ffi –
p3{4qZ0 ` p1{4qZ1,1 ` p 2{4qZ2,2 fl
—W ffi
– 3{4 fl
Z0
p2q
W1
»
fi
»
fi
0
0 ?0
0
Z0
—1{4 1{4
ffi
2{4
0 ffi —
—
ffi
—1{2 1{2
ffi —Z1,1 ffi
0
—
? 0 ffi –Z1,2 fl
–3{4 1{4
2{4fl
0
Z2,2
1
with covariance
(33)
p2q
p2q
p2q
p2q
p2q
CovpW0 , W1{4 , W1{2 , W3{4 , W1 q “
»
fi »
0
0 ?0
0
0
ffi
—
—1{4 1{4
2{4
0 ffi —1{4
—
—1{2 1{2
ffi —
0
0
—
? ffi —1{2
–3{4 1{4
0
2{4fl –3{4
1
0
0
0
1
»
0 0
0
0
—0 1{4 1{4 1{4
—
—0 1{4 1{2 1{2
—
–0 1{4 1{2 3{4
0 1{4 1{2 3{4
fiT
0 ?0
0
1{4
2{4
0 ffi
ffi
ffi
1{2
0
? 0 ffi “
1{4
0
2{4fl
0
0
0
fi
0
1 ffi
ffi
1{2ffi
ffi “ CovpW0 , W1{4 , W1{2 , W3{4 , W1 q.
3{4fl
1
(5) To proceed, we want a better organization of the computations. At order 0 we have a
zero at t “ 0; at order 1 we have zeros at t “ 0, 1; at order 2 we have zeros at t “ 0, 1{2, 1.
p2q
p1q
At the points t “ 0, 1{2, 1 we have Wt “ Wt . As the new point 1{4, 3{4 the values of
p1q
Wt are interpolated values, to which are added the vertex values of the new Shauder
functions. The increments on binary points are:
5
»
p2q
p2q
fi
W1{4 ´ W0
(34)
— p2q
ffi
—W ´ W p2q ffi
— 1{2
1{4 ffi
— p2q
ffi “
—W ´ W p2q ffi
1{2 fl
– 3{4
p2q
W1
p2q
´ W3{4
fi
p1{4 ´ 0qZ0 ` pS1,1 p1{4q ´ S1,1 p0qqZ1,1 ` pS1,2 p1{4q ´ S1,2 p0qqZ1,2 ` pS2,2 p1{4q ´ S2,2 p0qqZ2,2
—p1{2 ´ 1{4qZ0 ` pS1,1 p1{2q ´ S1,1 p1{4qqZ1,1 ` pS1,2 p1{2q ´ S1,2 p1{4qqZ1,2 ` pS2,2 p1{2q ´ S2,2 p1{4qqZ2,2 ffi
ffi
—
–p3{4 ´ 1{2qZ0 ` pS1,1 p3{4q ´ S1,1 p1{2qqZ1,1 ` pS1,2 p3{4q ´ S1,2 p1{2qqZ1,2 ` pS2,2 p3{4q ´ S2,2 p1{2qqZ2,2 fl “
p1 ´ 3{4qZ0 ` pS1,1 p1q ´ S1,1 p3{4qqZ1,1 ` pS1,2 p1q ´ S1,2 p3{4qqZ1,2 ` pS2,2 p1q ´ S2,2 p3{4qqZ2,2
»
fi
p1{4qZ0 ` S1,1 p1{4qZ1,1 ` S1,2 p1{4qZ1,2
—p1{4qZ0 ` pS1,1 p1{2q ´ S1,1 p1{4qqZ1,1 ´ S1,2 p1{4qZ1,2 ffi
ffi
—
–p1{4qZ0 ` pS1,1 p3{4q ´ S1,1 p1{2qqZ1,1 ` S2,2 p3{4qZ2,2 fl “
p1{4qZ0 ´ S1,1 p3{4qZ1,1 ´ S2,2 p3{4qqZ2,2
»
fi
p1{4qZ0 ` p1{2qS1,1 p1{2qZ1,1 ` S1,2 p1{4qZ1,2
— p1{4qZ0 ` p1{2qS1,1 p1{2qZ1,1 ´ S1,2 p1{4qZ1,2 ffi
—
ffi
– p1{4qZ0 ´ p1{2qS1,1 p1{2qZ1,1 ` S2,2 p3{4qZ2,2 fl “
p1{4qZ0 ´ p1{2qS1,1 p1{2qZ1,1 ´ S2,2 p3{4qqZ2,2
»
fi »
fi
1 1
1
0
p1{4qZ0
—1 1 ´1 0 ffi —p1{2qS1,1 p1{2qZ1,1 ffi
—
ffi —
ffi “
–1 ´1 0
1 fl – S1,2 p1{4qZ1,2 fl
1 ´1 0 ´1
S2,2 p3{4qZ2,2
fi
»
fi »
p1{4qZ0
1 1
1
0
—1 1 ´1 0 ffi — p1{4qZ1,1 ffi
ffi “
—
ffi — ?
–1 ´1 0
1 fl –p?2{4qZ1,2 fl
1 ´1 0 ´1
p 2{4qZ2,2
fi »
»
fi
fi »
1{4 0
0
0
Z0
1 1
1
0
—
—1 1 ´1 0 ffi — 0 1{4
ffi
0 ffi
?0
ffi —Z1,1 ffi
—
ffi —
fl
–
–1 ´1 0
fl
–
2{4 ? 0
Z1,2 fl
1
0
0
Z2,2
1 ´1 0 ´1
2{4
0
0
0
»
and the covariance is
(35)
p2q
p2q
CovpW1{4 ´ W0
»
1 1
—1 1
—
–1 ´1
1 ´1
p2q
p2q
p2q
p2q
p2q
p2q
, W1{2 ´ W1{4 , W3{4 ´ W1{2 , W1 ´ W3{4 q “
fiT
fi »
fi »
1 1
1
0
1
0
1{16
0
0
0
—
ffi
—
1{16 0
0 ffi
´1 0 ffi
ffi —1 1 ´1 0 ffi “
ffi — 0
–
–
fl
fl
1 ´1 0
1 fl
0
0
1{8 0
0
1
1 ´1 0 ´1
0
0
0 1{8
0 ´1
»
fi »
fiT
1{2 1{2
1
0
1 1
1
0
ffi —
ffi
1—
—1{2 1{2 ´1 0 ffi —1 1 ´1 0 ffi
fl
–
–
1 ´1 0
1 fl
1
8 1{2 ´1{2 0
1 ´1 0 ´1
1{2 ´1{2 0 ´1
(6) Let us prove the convergence. This proof is due to [2] and uses the Borel-Cantelli lemma
[3, §2.7]. First note that Z „ Np0, 1q implies the following estimate of the queues:
(36)
c ż
c ż
c
ż
2
2 8 ´z 2 {2
2 8 z ´z 2 {2
2 e´x {2
´1{2
´z 2 {2
P p|Z| ą xq “ p2πq
e
dz “
e
dz ď
e
dz “
.
π x
π x x
π x
|z|ąx
6
It follows that
(37)
ˆ
˙
P max |Zj,n | ą Apnq
j
“ P pYj t|Zj,n | ą Apnquq ď
ÿ
P p|Zj,n | ą nq “ 2n´1 P p|Z| ą Apnqq
j
c
ď2
n´1
2
2 e´Apnq {2
.
π Apnq
References
1. Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts
in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940 (92h:60127)
2. Henry P. McKean, Stochastic integrals, AMS Chelsea Publishing, Providence, RI, 2005, Reprint of the 1969
edition, with errata. MR 2169626 (2006d:60003)
3. David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University
Press, Cambridge, 1991.
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