Introductiont to Malliavin calculus with applications to economics

83 Pages·1997·0.388 MB·English

by  Oksendal B.

AN INTRODUCTION TO
MALLIAVIN CALCULUS
WITH APPLICATIONS TO ECONOMICS
Bernt ´ksendal
Dept. of Mathematics, University of Oslo,
Box 1053 Blindern, N{0316 Oslo, Norway
Institute of Finance and Management Science,
Norwegian School of Economics and Business Administration,
Helleveien 30, N{5035 Bergen-Sandviken, Norway.
Email: [email protected]
May 1997
Preface
These are unpolished lecture notes from the course BF 05 \Malliavin calculus with appli-
cations to economics", which I gave at the Norwegian School of Economics and Business
Administration (NHH), Bergen, in the Spring semester 1996. The application I had in
mind was mainly the use of the Clark-Ocone formula and its generalization to flnance,
especially portfolio analysis, option pricing and hedging. This and other applications are
described in the impressive paper by Karatzas and Ocone [KO] (see reference list in the
end of Chapter 5). To be able to understand these applications, we had to work through
the theory and methods of the underlying mathematical machinery, usually called the
Malliavin calculus. The main literature we used for this part of the course are the books
by Ustunel [U] and Nualart [N] regarding the analysis on the Wiener space, and the
forthcoming book by Holden, ´ksendal, Ub¿e and Zhang [H´UZ] regarding the related
white noise analysis (Chapter 3). The prerequisites for the course are some basic knowl-
edge of stochastic analysis, including Ito integrals, the Ito representation theorem and the
Girsanov theorem, which can be found in e.g. [´1].
The course was followed by an inspiring group of (about a dozen) students and employees
at HNN. I am indebted to them all for their active participation and useful comments. In
particular, I would like to thank Knut Aase for his help in getting the course started and
his constant encouragement. I am also grateful to Kerry Back, Darrell Du–e, Yaozhong
Hu, Monique Jeanblanc-Picque and Dan Ocone for their useful comments and to Dina
Haraldsson for her proflcient typing.
Oslo, May 1997
Bernt ´ksendal
i
Contents
1 The Wiener-Ito chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . 1.1
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8
2 The Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1
The Skorohod integral is an extension of the Ito integral . . . . . . . . . . 2.4
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6
3 White noise, the Wick product and stochastic integration . . . . . . . . . . 3.1
The Wiener-It^o chaos expansion revisited . . . . . . . . . . . . . . . . . . . 3.3
Singular (pointwise) white noise . . . . . . . . . . . . . . . . . . . . . . . . 3.6
The Wick product in terms of iterated Ito integrals . . . . . . . . . . . . . 3.9
Some properties of the Wick product . . . . . . . . . . . . . . . . . . . . . 3.9
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12
4 Difierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1
Closability of the derivative operator . . . . . . . . . . . . . . . . . . . . . 4.7
Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8
Difierentiation in terms of the chaos expansion . . . . . . . . . . . . . . . . 4.11
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13
5 The Clark-Ocone formula and its generalization. Application to flnance . . 5.1
The Clark-Ocone formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5
The generalized Clark-Ocone formula . . . . . . . . . . . . . . . . . . . . . 5.5
Application to flnance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10
The Black-Scholes option pricing formula and generalizations . . . . . . . . 5.13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15
ii
6 Solutions to the exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1
iii
1 The Wiener-Ito chaos expansion
The celebrated Wiener-Ito chaos expansion is fundamental in stochastic analysis. In
particular, it plays a crucial role in the Malliavin calculus. We therefore give a detailed
proof.
The flrst version of this theorem was proved by Wiener in 1938. Later Ito (1951) showed
that in the Wiener space setting the expansion could be expressed in terms of iterated Ito
integrals (see below).
Before we state the theorem we introduce some useful notation and give some auxiliary
results.
Let W(t) = W(t;!); t ‚ 0, ! 2 › be a 1-dimensional Wiener process (Brownian motion)
on the probability space (›;F;P) such that W(0;!) = 0 a.s. P.
For t ‚ 0 let F be the (cid:190)-algebra generated by W(s;¢); 0 • s • t. Fix T > 0 (constant).
t
A real function g : [0;T]n ! R is called symmetric if
(1:1) g(x ;:::;x ) = g(x ;:::;x )
(cid:190)1 (cid:190)n 1 n
for all permutations (cid:190) of (1;2;:::;n). If in addition
Z
(1:2) kgk2 := g2(x ;:::;x )dx ¢¢¢dx < 1
L2([0;T]n) 1 n 1 n
[0;T]n
we say that g 2 Lb2([0;T]n), the space of symmetric square integrable functions on [0;T]n.
Let
(1:3) S = f(x ;:::;x ) 2 [0;T]n; 0 • x • x • ¢¢¢ • x • Tg:
n 1 n 1 2 n
The set S occupies the fraction 1 of the whole n-dimensional box [0;T]n. Therefore, if
n n!
g 2 Lb2([0;T]n) then
Z
(1:4) kgk2 = n! g2(x ;:::;x )dx :::dx = n!kgk2
L2([0;T]n) 1 n 1 n L2(Sn)
Sn
If f is any real function deflned on [0;T]n, then the symmetrization f~of f is deflned by
1 X
e
(1:5) f(x ;:::;x ) = f(x ;:::;x )
1 n n! (cid:190)1 (cid:190)n
(cid:190)
e
where the sum is taken over all permutations (cid:190) of (1;:::;n). Note that f = f if and only
if f is symmetric. For example if
f(x ;x ) = x2 +x sinx
1 2 1 2 1
then
1
fe(x ;x ) = [x2 +x2 +x sinx +x sinx ]:
1 2 2 1 2 2 1 1 2
1.1
Note that if f is a deterministic function deflned on S (n ‚ 1) such that
n
Z
kfk2 := f2(t ;:::;t )dt ¢¢¢dt < 1;
L2(Sn) 1 n 1 n
Sn
then we can form the (n-fold) iterated Ito integral
ZTZtn Zt3 Zt2
(1:6) Jn(f):= ¢¢¢ ( f(t1;:::;tn)dW(t1))dW(t2)¢¢¢dW(tn¡1)dW(tn);
0 0 0 0
because at each Ito integration with respect to dW(t ) the integrand is F -adapted and
i t
square integrable with respect to dP £dt , 1 • i • n.
i
Moreover, applying the Ito isometry iteratively we get
ZT Ztn Zt2
E[J2(h)] = E[f ( ¢¢¢ h(t ;:::;t )dW(t )¢¢¢)dW(t )g2]
n 1 n 1 n
0 0 0
ZT Ztn Zt2
= E[( ¢¢¢ h(t1;:::;tn)dW(t1)¢¢¢dW(tn¡1))2]dtn
0 0 0
ZTZtn Zt2
(1.7) = ¢¢¢ = ¢¢¢ h2(t ;:::;t )dt ¢¢¢dt = khk2 :
1 n 1 n L2(Sn)
0 0 0
Similarly, if g 2 L2(S ) and h 2 L2(S ) with m < n, then by the Ito isometry applied
m n
iteratively we see that
E[J (g)J (h)]
m n
ZT Zsm Zs2
= E[f ( ¢¢¢ g(s ;:::;s )dW(s )¢¢¢dW(s )g
1 m 1 m
0 0 0
ZT Zsm Zt2
f ( ¢¢¢ h(t1;:::;tn¡m;s1;:::;sm)dW(t1)¢¢¢)dW(sm)g]
0 0 0
ZT Zsm Zs2
= E[f ¢¢¢ g(s1;:::;sm¡1;sm)dW(s1)¢¢¢dW(sm¡1)g
0 0 0
Zsm Zt2
f ¢¢¢ h(t1;:::;sm¡1;sm)dW(t1)¢¢¢dW(sm¡1)g]dsm
0 0
ZT Zsm Zs2 Zs1 Zt2
= ¢¢¢ E[g(s1;s2;:::;sm) ¢¢¢ h(t1;:::;tn¡m;s1;:::;sm)
0 0 0 0 0
dW(t1)¢¢¢dW(tn¡m)]ds1;¢¢¢dsm
(1.8) = 0
because the expected value of an Ito integral is zero.
1.2
We summarize these results as follows:
(
0 if n 6= m
(1:9) E[J (g)J (h)] =
m n (g;h) if n = m
L2(Sn)
where
Z
(1:10) (g;h) = g(x ;:::;x )h(x ;:::;x )dx ¢¢¢dx
L2(Sn) 1 n 1 n 1 n
Sn
is the inner product of L2(S ).
n
Note that (1.9) also holds for n = 0 or m = 0 if we deflne
J (g) = g if g is a constant
0
and
(g;h) = gh if g;h are constants:
L2(S0)
If g 2 Lb2([0;T]n) we deflne
Z
(1:11) I (g):= g(t ;:::;t )dW›n(t):= n!J (g)
n 1 n n
[0;T]n
Note that from (1.7) and (1.11) we have
(1:12) E[I2(g)] = E[(n!)2J2(g)] = (n!)2kgk2 = n!kgk2
n n L2(Sn) L2([0;T]n)
for all g 2 Lb2([0;T]n).
Recall that the Hermite polynomials h (x); n = 0;1;2;::: are deflned by
n
dn
(1:13) h (x) = (¡1)ne12x2 (e¡12x2); n = 0;1;2;:::
n dxn
Thus the flrst Hermite polynomials are
h (x) = 1; h (x) = x; h (x) = x2 ¡1; h (x) = x3 ¡3x;
0 1 2 3
h (x) = x4 ¡6x2 +3; h (x) = x5 ¡10x3 +15x;:::
4 5
There is a useful formula due to Ito [I] for the iterated Ito integral in the special case
when the integrand is the tensor power of a function g 2 L2([0;T]):
ZTZtn Zt2 (cid:181)
(1:14) n! ¢¢¢ g(t )g(t )¢¢¢g(t )dW(t )¢¢¢dW(t ) = kgknh ( );
1 2 n 1 n n kgk
0 0 0
where
ZT
kgk = kgk and (cid:181) = g(t)dW(t):
L2([0;T])
0
1.3
For example, choosing g · 1 and n = 3 we get
ZTZt3Zt2 W(T)
6¢ dW(t )dW(t )dW(t ) = T3=2h ( ) = W3(T)¡3T W(T):
1 2 3 3 T1=2
0 0 0
THEOREM 1.1. (The Wiener-Ito chaos expansion) Let ’ be an F -measurable
T
random variable such that
k’k2 := k’k2 := E [’2] < 1:
L2(›) L2(P) P
Thenthereexistsa(unique)sequenceff g1 of(deterministic)functionsf 2 Lb2([0;T]n)
n n=0 n
such that
X1
(1:15) ’(!) = I (f ) (convergence in L2(P)):
n n
n=0
Moreover, we have the isometry
X1
(1:16) k’k2 = n!kf k2
L2(P) n L2([0;T]n)
n=0
Proof. By the Ito representation theorem there exists an F -adapted process ’ (s ;!),
t 1 1
0 • s • T such that
1
ZT
(1:17) E[ ’2(s ;!)ds ] • k’k2
1 1 1 L2(P)
0
and
ZT
(1:18) ’(!) = E[’]+ ’ (s ;!)dW(s )
1 1 1
0
Deflne
(1:19) g = E[’] (constant):
0
For a.a. s • T we apply the Ito representation theorem to ’ (s ;!) to conclude that
1 1 1
there exists an F -adapted process ’ (s ;s ;!); 0 • s • s such that
t 2 2 1 2 1
Zs1
(1:20) E[ ’2(s ;s ;!)ds ] • E[’2(s )] < 1
2 2 1 2 1 1
0
and
Zs1
(1:21) ’ (s ;!) = E[’ (s )]+ ’ (s ;s ;!)dW(s ):
1 1 1 1 2 2 1 2
0
1.4
Substituting (1.21) in (1.18) we get
ZT ZT Zs1
(1:22) ’(!) = g + g (s )dW(s )+ ( ’ (s ;s ;!)dW(s )dW(s )
0 1 1 1 2 2 1 2 1
0 0 0
where
(1:23) g (s ) = E[’ (s )]:
1 1 1 1
Note that by the Ito isometry, (1.17) and (1.20) we have
ZT Zs1 ZT Zs1
(1:24) E[f ( ’ (s ;s ;!)dW(s ))dW(s )g2]= ( E[’2(s ;s ;!)]ds )ds • k’k2 :
2 1 2 2 1 2 1 2 2 1 L2(P)
0 0 0 0
Similarly, for a.a. s • s • T we apply the Ito representation theorem to ’ (s ;s ;!) to
2 1 2 2 1
get an F -adapted process ’ (s ;s ;s ;!); 0 • s • s such that
t 3 3 2 1 3 2
Zs2
(1:25) E[ ’2(s ;s ;s ;!)ds ] • E[’2(s ;s )] < 1
3 3 2 1 3 2 2 1
0
and
Zs2
(1:26) ’ (s ;s ;!) = E[’ (s ;s ;!)]+ ’ (s ;s ;s ;!)dW(s ):
2 2 1 2 2 1 3 3 2 1 3
0
Substituting (1.26) in (1.22) we get
ZT ZT Zs1
’(!) = g + g (s )dW(s )+ ( g (s ;s )dW(s ))dW(s )
0 1 1 1 2 2 1 2 1
0 0 0
ZT Zs1 Zs2
(1.27) + ( ( ’ (s ;s ;s ;!)dW(s ))dW(s ))dW(s );
3 3 2 1 3 2 1
0 0 0
where
(1:28) g (s ;s ) = E[’ (s ;s )]; 0 • s • s • T:
2 2 1 2 2 1 2 1
By the Ito isometry, (1.17), (1.20) and (1.25) we have
ZTZs1Zs2
(1:29) E[f ’ (s ;s ;s ;!)dW(s )dW(s )dW(s )g2] • k’k2 :
3 3 2 1 3 2 3 L2(P)
0 0 0
By iterating this procedure we obtain by induction after n steps a process ’ (t ;t ;:::,
n+1 1 2
t ;!); 0 • t • t • ¢¢¢ • t • T and n + 1 deterministic functions g ;g ;:::;g
n+1 1 2 n+1 0 1 n
with g constant and g deflned on S for 1 • k • n, such that
0 k k
Z
Xn
(1:30) ’(!) = J (g )+ ’ dW›(n+1);
k k n+1
k=0
Sn+1
1.5
where
Z ZT tZn+1 Zt2
(1:31) ’ dW›(n+1) = ¢¢¢ ’ (t ;:::;t ;!)dW(t )¢¢¢dW(t )
n+1 n+1 1 n+1 1 n+1
Sn+1 0 0 0
is the (n+1)-fold iterated integral of ’ . Moreover,
n+1
Z
(1:32) E[f ’ dW›(n+1)g2] • k’k2 :
n+1 L2(›)
Sn+1
In particular, the family
Z
ˆ := ’ dW›(n+1); n = 1;2;:::
n+1 n+1
Sn+1
is bounded in L2(P). Moreover
(1:33) (ˆ ;J (f )) = 0 for k • n, f 2 L2([0;T]k):
n+1 k k L2(›) k
Hence by the Pythagorean theorem
Xn
(1:34) k’k2 = kJ (g )k2 +kˆ k2
L2(›) k k L2(›) n+1 L2(›)
k=0
In particular,
Xn
kJ (g )k2 < 1
k k L2(›)
k=0
P1
and therefore J (g ) is strongly convergent in L2(›). Hence
k k
k=0
lim ˆ =:ˆ exists (limit in L2(›))
n!1 n+1
But by (1.33) we have
(1:35) (J (f );ˆ) = 0 for all k and all f 2 L2([0;T]k)
k k L2(›) k
In particular, by (1.14) this implies that
(cid:181)
E[h ( )¢ˆ] = 0 for all g 2 L2([0;T]), all k ‚ 0
k kgk
RT
where (cid:181) = g(t)dW(t).
0
But then, from the deflnition of the Hermite polynomials,
E[(cid:181)k ¢ˆ] = 0 for all k ‚ 0
1.6