Canonical decomposition of linear transformations of
two independent Brownian motions1
Hans Follmer
Ching-Tang Wu
Institut fur Mathematik
Humboldt-Universitat zu Berlin
Unter den Linden 6, D-10099 Berlin, Germany
e-mail: foellmer@mathematik.hu-berlin.de
wu@mathematik.hu-berlin.de
Marc Yor
Laboratoire de Probabilit
es
Universit
e Pierre et Marie Curie
4, Place Jussieu, F-75252 Paris, France
Abstract
Motivated by the Kyle-Back model of \insider trading", we consider certain classes of linear transformations of two independent Brownian motions and study their canonical decomposition as semimartingales in their own ltration. In particular we characterize those transformations which generate again a Brownian motion.
Keywords: Brownian motion, canonical decomposition, enlargement of l-
tration, insider trading, stochastic ltering theory, Sturm-Liouville equation,
Volterra kernels.
JEL classication: D82, G14.
AMS 1991 subject classication: 45D05, 60G15, 60G35, 60H05, 60H30,
90A09.
Support of the Deutsche Forschungsgemeinschaft (SFB 373 \Quantikation und Simulation okonomischer Prozesse" and Berliner Graduiertenkolleg \Stochastische Prozesse
und probabilistische Analysis") is gratefully acknowledged.
1
Canonical Decomposition
1
1 Introduction
Consider two independent Brownian motions (Wt)t 0 and (W~ t)t 0. We study
solutions X of stochastic dierential equations
dXt = dWt + Ytdt
(1)
driven by W , where the drift Y depends linearly on X and W~ . Our purpose
is to derive the canonical decomposition of X as a semimartingale in its own
ltration (FtX ) and to characterize those cases where X is again a Brownian
motion.
As a simple example consider the Brownian bridge from 0 to W~ 1 dened
by (1), where the drift is given by
~ ; Xt
(2)
Yt = W11 ;
t :
The process X is a new Brownian motion such that X1 = W~ 1. This example
plays a crucial role in the Kyle-Back model of \insider trading"(see Kyle 14]
and Back 2]). The \insider" knows in advance the nal value W~ 1. He applies
the drift (2) in order to modify the original Brownian motion W in such a way
that (i) the resulting process X ends up in W~ 1, and (ii) the distribution of the
process remains unchanged, i.e., X is again a Brownian motion. Condition
(i) guarantees that the strategy maximizes the insider's expected gain cf.
Back 2]. Condition (ii) corresponds to the notion of equilibrium as dened
in Back 2].
Let us now modify the example as follows. Suppose that the \insider"
cannot anticipate the nal value W~ 1 from the beginning. Instead, his \insider
information" consists in observing the second Brownian motion W~ . This
suggests to replace the anticipating drift (2) by the adapted drift
~ Xt
Yt = W1t ;
(3)
;t :
The resulting process X converges again to W~ 1. But its distribution has
changed: X is no longer a Brownian motion. In section 2 we determine
explicitly its canonical decomposition as a semimartingale in its own ltration
(FtX ).
2
Canonical Decomposition
This example suggests to study a more general class of processes of the
form (1) where the drift is given as a time-dependent linear function in X
and W~ , i.e.,
Yt = f (t)W~ t + h(t)Xt
(4)
for functions f and h in C 1(0 1) satisfying some mild integrability conditions. In section 3, we derive the canonical decomposition of X in its own
ltration. To this end, we consider the Radon-Nikodym density D of the
law of X with respect to that of W . We express D in terms of (W W~ ), and
we compute in closed form E DjW ], its conditional expectation with respect
to W . Applying one more time the Girsanov transformation we obtain the
canonical decomposition of X . In order to formulate the result, we introduce
the fundamental solution (t) of the Sturm-Liouville equation
00(t) = f 2(t)(t)
(5)
with boundary conditions (0) = 0 and 0(0+) = 1. The canonical decomposition of X is given by
Xt = Bt +
Zt
0
(f (u)ku (Xv v u) + h(u)Xu ) du
where (Bt) is an (FtX )-Brownian motion and the functional ku is given by
Zu
ku (Xs s u) = 1(u) 0 (v)(f (v)dXv ; f (v)h(v)Xv dv):
An alternative method consists in applying the stochastic ltering theory
for Gaussian processes. This is explained in section 4. There we consider
stochastic dierential equations (1) where the drift is given by an adapted
linear transformation of X and W~ , i.e.,
0
Zt
dXt = dWt + ( 0 F (t u)dW~ u +
Zt
0
H (t u)dXu )dt
(6)
with some square-integrable Volterra kernels F and H see the denition in
section 3.2. Theorem 4.1 shows that the canonical decomposition of X is of
the form
ZtZs
Xt = Bt + 0 ( 0 GF (s u)dBu +
Zs
0
H (s u)dXu )ds
(7)
Canonical Decomposition
3
where (Bt) is a standard Brownian motion with respect to (FtX ) and GF is
the square-integrable Volterra kernel determined by the equation
Zs
0
F (s u)F (t u)du = GF (t s) +
Zs
0
GF (s u)GF (t u)du:
(8)
In the special cases considered in section 3, the kernel GF can be identied
in terms of the solution of a Sturm-Liouville equation.
In section 5.1 we return to the discussion of properties (i) and (ii) which
appeared in our initial example (2). In the general context of equation (6)
where the drift Y is a linear functional of the past of X and W~ , we characterize those cases which satisfy condition (ii), i.e., the resulting process X is
again a Brownian motion. The criterion is that
H (t u) = ;GF (t u)
where GF is given by (8). But it is not possible to obtain at the same
time condition (i), i.e., to tie such a Brownian motion X to the endpoint of
the Brownian motion W~ . In fact, the simple argument of Proposition 5.1
shows that there is no adapted drift (Yt) such that the solution X of (1) is a
Brownian motion with endpoint X1 = W~ 1.
In section 5.2 we point out the following connection to an enlargement of
ltration. First we note that a Brownian motion X , given as the solution of
equation (7) with H = ;GF , can be expressed directly in terms of W and
W~ :
Zt
Zt
Zt
dXt = dWt + 0 LF (t u)dWu + 0 (F (t u) + u LF (t v)F (v u)dv)dW~ u dt
(9)
where LF denotes the resolvent kernel of GF see (67). In the special case
F (t u) = f (t), this can be reduced to
Wt = Xt ;
Zu
Z t f (u) Z u
0(v )dW
~
(
;
f (v)(v)dWv )du
v
0
0
(u)
0
0
where (t) is the solution of the Sturm-Liouville equation (5). This representation of W in terms of the Brownian motion X can be viewed, after
time reversal, as the decomposition of a Brownian motion in some enlarged
Gaussian ltration.
4
Canonical Decomposition
2 A bridge between two Brownian motions
Let ( F IP ) be a probability space and (Wt)0t1 be a standard Brownian
motion with respect to its canonical ltration (FtW )0t1. Now let (W~ t)0t1
be another standard Brownian motion on the same probability space which
is independent of (Wt)0t1 , and denote by (Ft)0t1 the ltration generated
by these two Brownian motions.
We know that the solution (X~t)0t1 of the stochastic dierential equation
~ ~
(10)
dX~t = dWt + W11 ;; tXt dt
with initial value X~0 = 0, is a standard Brownian motion which converges
to the nal value W~ 1 (cf., for example, Jeulin-Yor 8]). Now we look at
the process (Xt)0t1 starting in X0 = 0 which is dened by the stochastic
dierential equation
~ Xt
(11)
dXt = dWt + W1t ;
; t dt:
Clearly, for any t 2 0 1], Xt is normally distributed, and hX it = t. The
following Lemma shows that Xt approaches W~ 1 as t ! 1. However, we will
see that (Xt )0t1 is no longer a Brownian motion.
Lemma 2.1 Xt ! W~ 1 as t ! 1:
Proof. The explicit solution of (11) is given by
Zt W
Zt 1
~s
Xt = (1 ; t) 0 (1 ; s)2 ds + (1 ; t) 0 (1 ; s) dWs :
(12)
The rst term approaches W~ 1 and the second goes to 0 as t ! 1,1 and this
implies the result. Alternatively, we could note that the process 2; 2 (X ; W~ )
satises the equation of a Brownian bridge tied down to the nal value 0. 2
Lemma 2.2 For 0 s t < 1, we have
E XtW~ t] = t + (1 ; t) log(1 ; t)
(13)
and the covariance function of X is given by
E XsXt ] = s + 2s(1 ; t) + (2 ; s ; t) log(1 ; s):
(14)
Canonical Decomposition
5
Proof. Applying the integration by parts formula in the rst formula in
(12), the solution of (11) is given by
Z t dW
~s
s ; dW
~
1 ; s + Wt:
Since (Wt) and (W~ t) are independent, we establish
ZtW
~
~
~
E XtWt] = t + (1 ; t)E 0 t(dW1 s;;s dWs )
= t + (1 ; t) log(1 ; t)
and
2Z
!2 3
s dWu ; dW
~
u
5
E Xs Xt] = E W~ sW~ t] + (1 ; t)(1 ; s)E 4 0 1 ; u
" Zs
" Zt
~u#
~u#
dW
dW
u ; dW
u ; dW
~
~
+ (1 ; s)E Wt
+ (1 ; t)E Ws
1;u
1;u
0
0
= s + 2s(1 ; t) + (2 ; s ; t) log(1 ; s):
Xt = (1 ; t)
0
2
This Lemma shows that (Xt)0t1 is not a Brownian motion, since its
covariance function diers from t ^ s. But from (11) we see that it is a
semimartingale with respect to (Ft)0t1, and therefore it is obviously a
semimartingale relative to its natural ltration. A natural question is: what
is the explicit form of its canonical decomposition? That is the problem we
want to discuss in this section.
Lemma 2.3 Suppose that the process (Xt)t 0 is given by
Xt = Wt +
Zt
0
Yu du
with a Brownian motion (Wt)t 0 adapted
to a ltration (Ft)t 0 and an (Ft)R
adapted process (Yt )t 0 satisfying 0t E jYu jdu < 1 for all t.
(1) The canonical decomposition of X in its natural ltration (FtX ) is given
by
Zt
Xt = Bt + E Yu jFuX ]du
(15)
0
6
Canonical Decomposition
where the process B dened by (15) is an (FtX )-Brownian motion, which is
often called the innovation process of X . In particular, (Xt ) is a Brownian
motion if and only if
E YujFuX ] = 0
dIP du ; a:s::
(2) Furthermore, if the function s 7;! Ys is L1-continuous on (0 1) and if
(Xt)t 0 is a Gaussian process, then (Xt)t 0 is a Brownian motion if and only
if
for all 0 < s t.
E (Xs Yt) = 0
(16)
Proof of Lemma 2.3. (1) The rst part of the rst assertion is the
Innovation Theorem by Shiryaev and Kailath, the proof of which can be found
in Hida-Hitsuda 5] or Liptser-Shiryaev 15]. The second part is immediate
from the uniqueness of the canonical decomposition of X in (FtX ).
(2) As to the second assertion, suppose (Xt ) is a Brownian motion. Then
E (Xs Wt) = E (XsWs ) = s +
Therefore,
Zt
0
Zs
0
E (Yu Xs )du = s ; E (Xs Wt) = ;
E (YuWs )du:
Zs
0
E (YuWs )du:
Taking derivatives with respect to t on both sides, we obtain (16). Conversely, from Stricker 18] we know that if X is a Gaussian
R t semimartingale,
then its canonical decomposition is Gaussian, i.e., (Xt 0 E YujFuX ]du)t 0 is
a Gaussian process. Since s 7;! Ys is L1-continuous, (Xt E YtjFtX ])t 0 is
Gaussian. From (16) we have E YtjFtX ] = 0, and so X is a Brownian motion
due to (15).
2
Corollary 2.1 Let the process (Xt)0t1 satisfy (11). Then the process B ,
dened as
Bt := Xt ;
Z t E W
~
u jFuX ] ; Xu
1;u
X
is a Brownian motion relative to (Ft )0t1.
0
du
Canonical Decomposition
Proof. Set
7
~
Yu = W1u ;; uXu then from the rst assertion in Lemma 2.3, we obtain the required result. 2
Therefore, we have only to compute the conditional expectation of W~ t
relative to FtX , the goal we want to reach in this section.
p
p
Lemma 2.4 Set A := 12 (1 + 5) and B := 21 (1 ; 5). Then for 0 t < 1,
Zt
;A;1
; t)(1 ; u);B;1 X du
E W~ tjFtX ] = 0 B (1 ; t)(1A;(1u); t)A ;; BA(1
u
(1 ; t)B
(1 ; t)B ; (1 ; t)A X :
(17)
+ A(1
; t)A ; B (1 ; t)B t
Proof. Let us assume the conditional expectation of W~ t with respect to
X
Ft is of the form
E W~ tjFtX ] =
Zt
0
a(u t)Xudu + b(t)Xt:
Apply the projection property of the conditional expectation: E Xs(W~ t ;
E W~ tjFtX ])] = 0 for all 0 s t < 1, as well as the martingale property to
obtain
Zt
E (Xs W~ s) ; b(t)E (XsXt ) = 0 a(u t)E (XsXu )du:
Using (13), (14) and computing explicitly the left hand side (LHS) and the
right hand side (RHS) in this equation, we get
LHS = s1 ; (3 ; 2t)b(t)] + log(1 ; s)1 ; (2 ; t)b(t)] + s log(1 ; s)b(t) ; 1]:
Zs
a(u t)(u + 2u(1 ; s) + (2 ; s ; u) log(1 ; u))du
+ a(u t)(s + 2s(1 ; u) + (2 ; s ; u) log(1 ; s))du:
s
RHS =
Z 0t
Taking the second derivatives with respect to s on both sides implies
1 ; b(t)(t ; s) = ;a(s t) + Z t a(u t)(u ; s) du:
1 ; s (1 ; s)2
(1 ; s)2
s
8
Canonical Decomposition
Multiplication of both sides with (1 ; s)2 leads to:
1 ; s ; b(t)(t ; s) = ;a(s t)(1 ; s) +
2
Zt
s
a(u t)(u ; s)du:
Taking two further derivatives with respect to s on both sides we get
(1 ; s)2a00(s t) ; 4(1 ; s)a0(s t) + a(s t) = 0:
The solution of this dierential equation is given by
a(s t) = c1(t)(1 ; s)(;A;1) + c2(t)(1 ; s)(;B;1):
Substituting this equation in RHS and comparing the coe!cients of s, log(1;
s) and s log(1 ; s) in LHS and RHS, we derive the desired result.
2
Using It^o's product rule, we can rewrite the conditional expectation in
(17) as
E W~ tjFtX ] =
Z t (B + 1)(1 ; s);A ; (A + 1)(1 ; s);B
0
A(1 ; t);B ; B (1 ; t);A
dXs + Xt :
(18)
Therefore, Corollary 2.1 allows us to conclude:
Proposition 2.1 The canonical decomposition of X is given by
Z tZ u
; s);A ; (A + 1)(1 ; s);B dX du
Xt = Bt + 0 0 (B + 1)(1
s
A(1 ; u)A ; B (1 ; u)B
for 0 t < 1.
(19)
Remark 2.1 Using It^o's product rule, the representation (19) can be rewritten in the form:
Z t " (B + 1)(1 ; s)B ; (A + 1)(1 ; s)A
Xt = Bt + 0
A(1 ; t)A ; B (1 ; t)B #
;B ; (B + 1)(1 ; s);A
(
A
+
1)(1
;
s
)
+
dXs :
A(1 ; t);B ; B (1 ; t);A
Canonical Decomposition
9
Remark 2.2 Let (Xt)0t1, a centered Gaussian semimartingale, be of the
form
Xt = Wt +
Zt
0
Ysds:
A result of Hitsuda 7] states that the conditional expectation E YsjFsX ] is
equal to the (orthogonal) projection of Ys to the space Hs (X ), which is the
2
RLs -closure of the set that consists of all stochastic integrals of the form:
0 f (u)dXu with bounded Borel function f . In our situation (11), the explicit form of this projection is given by (19).
3 Canonical decompositions, Sturm-Liouville
equations, and Volterra representations
Now we consider the case where the process (Xt)0t1 satises the stochastic
dierential equation:
dXt = dWt + (f (t)W~ t + h(t)Xt)dt
(20)
with initial value X0 = 0. The functions f and h are assumed to belong
to C 1(0 1) \ A(0 1), where Rthe space A(0 1) is dened as the set of all
measurable functions with 0t s2(s)ds < 1, for all t < 1.
From Girsanov's transformation, the law of (Xs s t), for any t < 1,
may be written in terms of that of (W W~ ) via the following formula:
E F (Xs s t)] = E F (Ws s t) Et]
where F is a measurable functional and
Zt
Z
(21)
t
Et = exp( 0 (f (u)W~ u + h(u)Wu)dWu ; 12 0 (f (u)W~ u + h(u)Wu)2du):
Since (Et)0t<1 is a martingale with respect to (Ft), the natural ltration
generated by (Wt)0t1 and (W~ t)0t1, we know #t := E (EtjFtW ) is also a
martingale with respect to (FtW )0t<1. Once we have obtained in the next
section a closed form of #t, we shall apply again Girsanov's transformation
to get the canonical decomposition of (Xt)0t<1.
10
Canonical Decomposition
3.1 Computation of
t
Obviously, Et may be decomposed as: Et = Et(1)Et(2), where
Zt
Zt
1
(1)
Et := expf 0 h(u)WudWu ; 2 0 h2(u)Wu2dug
and
Zt
Zt
Zt
1
(2)
~
~
Et := expf 0 f (u)Wu dWu ; 0 f (u)h(u)Wu Wudu ; 2 0 f 2(u)W~ u2dug
so that
#t = Et(1)E Et(2)jFtW ]:
Thus, in order to compute #t, it will su!ce to obtain a "robust" formula for
I (t), the expectation of:
Zt
Zt
1
~
E (t) = exp( 0 Ws (ds) ; 2 0 W~ s2(ds))
with the measure (ds) := f 2(s)ds, and (ds) a generic signed measure.
Later, we shall justify the replacement of (ds) by the Gaussian \measure":
W (ds) := f (s)dWs ; f (s)h(s)Ws ds:
In order to present the next \explicit" formula for I (t), we need to
introduce two fundamental solutions $(u) and (u) of the Sturm-Liouville
equation:
(du) = (du)(u)
relative to a measure , which are characterized by:
(i) $(u) is decreasing, and satises $(0) = 1
(ii) (u) satises (0) = 0 and (0+) = 1.
An important relation between $(u) and (u) is
Zu
(22)
(u) = $(u) $21(v) dv:
0
Therefore, we get the important Wronskian relation between $ and :
00
0
$(u) (u) ; (u)$ (u) = 1:
0
0
(23)
Canonical Decomposition
11
Throughout this paper, we discuss the case (ds) = f 2(s)ds. Therefore, the
corresponding Sturm-Liouville equation is given by
(u) = f 2 (u)(u):
(24)
00
Proposition 3.1 The expression for I (t) is equal to
Z t Z t $(s)
Zt
1
2
expf 2 ( $(u) (ds)) du ; (t)( (s) (ds))2g
0
u
0
(t)
q1
0
where
(t) := $ (t)=(2 (t)):
0
0
(25)
(26)
Proof. The proof simply consists in pushing the computation made in
Pitman-Yor 16] a little further. Precisely, changing the Wiener measure with
the Radon-Nikodym density:
Zt
1
1
2
~
^
Zt := expf 2 (F(t)Wt ; F(t)) ; 2 0 W~ s2(ds)g
R
with F(t) = $0(t)=$(t) and F^(t) = 0t F(s)ds = log $(t), it follows that:
Zt
1
2
~
^
~
I (t) = E expf; 2 (F(t)Wt ; F(t)) + 0 Ws (ds)g where, under the new probability measure P , the process (W~ t)0t1 satises:
W~ t = Bt +
Zt
0
F(s)W~ sds
with a P ;Brownian motion (Bt)0t1. Consequently, (W~ t)0t1 is a centered Gaussian process under P . Now we write
Zt
q
I (t) = exp( 12 F^(t))E exp( 0 W~ s (ds) + ;F(t)N W~ t)]
where N is a centered reduced Gaussian random variable, independent of
(W~ s s t). Conditioning with respect to N , we are now facing the computation of
Zt
Zt
E exp( 0 W~ s (ds) + cW~ t)] = exp( 12 E ( 0 W~ s (ds) + cW~ t)2]):
12
Canonical Decomposition
It remains to develop the right-hand side as a second order polynomial
with
q
respect to c, and to integrate in c, relatively to the law of (c =) ;F(t)N .
A little algebra which hinges in particular upon the elementary formula:
2
E exp( a2 N 2 + bN )] = p11; a exp( 2(1b; a) )
(27)
then yields formulas (25) and (26). We use formula (27) to calculate:
q
I (t) = exp( 21 F^ (t))E exp 21 (N 2(;F (t))u(t) + 2N ;F (t)v(t) + w(t))]
q
hence, a = ;F (t)u(t), b = ;F (t)v(t))with the following values for u, v, w:
u(t) = $(t)(t)
Zt
v(t) = $(t) 0 (s) (ds)
w(t) =
Zt
0
1 (Z t $(s) (ds))2:
$2(u) u
2
Corollary 3.1 With the notation:
W (ds) = f (s)dWs ; f (s)h(s)Ws ds
formula (25) with changed into W is the conditional expectation of EW
given W.
Proof. Note that R0t W~ sW (ds) is approximated by
Zt
0
W~ s n(ds) =
X~
Wti f (ti)(Wti+1 ; Wti ) ; h(ti)Wti (ti+1 ; ti)]
n
where (n)n 0 is a sequence of subdivisions of 0 t], whose mesh goes to 0
as n ! 1, and this approximationR holds in the following sense: the limit
occurs
in probability, and exp( 0t W~ s n(ds)) converges in any Lp towards
R t both
exp( 0 W~ s W (ds)). This ensures that formula (25) also holds for W .
2
Canonical Decomposition
13
We now go(2)back to our main object, that is, to compute the conditional
expectation #t := E Et(2)jFtW ]. It follows from (22), (23), It^o's product rule
and Fubini theorem that
1 Z t(Z t $(s) (ds))2du ; (t)(Z t (s) (ds))2]
W
2 0 u $(u) W
0
Zs
Zt
Zt
Zt
1
=
(s)( $(u)W (du))W (ds) ; ($(s) ; 0(s) )( (u)W (du))W (ds)
0
0
s
0
Z
Z
2
t
s
+ 12 (f 0((ss))) 2 ( (u)W (du))2ds
0
0
Zt
Zt
1
=
f (u)ku (Ws s u)dWu ; 2 f 2(u)ku2 (Ws s u)du
0
Zt
0
; 0 f (u)h(u)Wuku (Ws s u)du
where
Zu
1
ku (Ws s u) := (u) 0 (v)W (dv)
Zu
= 1(u) (v)(f (v)dWv ; f (v)h(v)Wv dv): (28)
0
Therefore, we derive
0
0
Zt
#t = expf 0 (f (u)ku (Wv v u) + h(u)Wu)dWu
Zt
1
; 2 0 (f (u)ku (Wv v u) + h(u)Wu)2du:g
(29)
From (21), the denition of #t and Girsanov's transformation, we can get
the main result in this section, i.e., the canonical decomposition of X as a
semimartingale in its own ltration (FtX ).
Theorem 3.1 The canonical decomposition of (Xt )0t1 is given by
Zt
Xt = Bt + 0 (f (u)ku (Xv v u) + h(u)Xu )du
(30)
where (Bt)0t1 is a standard Brownian motion with respect to (FtX )0t1 ,
and the functional (kt (Xv v t))0t1 is of the form (28).
14
Canonical Decomposition
Remark 3.1 (1) Comparing (20) and (30) and recalling the rst assertion
of Lemma 2.3, we see that
E W~ tjFtX ] = kt(Xs s t):
(31)
In particular, we have obtained the explicit form of the projection appearing
in Remark 2.2.
(2) We have obtained that the identication of the conditional expectation in
(31) holds for f h 2 A(0 1). In fact, it is enough to assume that f and h
belong to the set
Z Z
t t
A%(0 1) := f' : f is measurable ( '(s)ds)2du < 1 for all 0 t < 1g:
Note that A(0 1) A%(0 1), since
ZtZt
0
u
0
( '(s)ds) du t
u
2
Zt
0
s'2(s)ds
by the Cauchy-Schwarz inequality. It is not dicult to show that for all s t,
Zt
E Xs(W~ t ; 01(t) 0 (u)(f (u)dXu ; f (u)h(u)Xu du))] = 0
where (t) satises Sturm-Liouville equation (24). This implies (31).
3.2 Volterra representation and canonical decomposition
Hitsuda 6] shows that the law of a Gaussian process (Xt)0t1 with E (Xt) =
0 is equivalent to the Wiener measure if and only if Xt can be represented in
the form:
Z tZ s
Xt = Bt + 0 0 l(s u)dBuds
(32)
where B is a Brownian motion and l(s u) is a square-integrable Volterra
kernel, i.e., a measurable function on (0 1) (0 1) such that
Z tZ s
(i) 0 0 l2(s u)duds < 1 for all t < 1.
(ii) l(s u) = 0 for 0 < s < u < 1.
Canonical Decomposition
15
This representation is unique in the sense that if X has another representation
Xt = B~t +
Z tZ s
0
0
~l(s u)dB~u ds
then B = B~ and l(s u) = ~l(s u) for almost all s u 2 (0 1) see Hida-Hitsuda
5]. We shall call the representation (32) the Volterra representation of X .
Proposition 3.2 The Volterra representation (32) is the canonical decomposition of X as a semimartingale in its own ltration (FtX ). Moreover, we
have (FtX ) = (FtB ).
Proof. Given a square-integrable Volterra kernel l, there is a unique
square-integrable Volterra kernel Rl which satises the equations
8
Zt
>
>
< l(t s) + Rl (t s) + l(t u)Rl(u s)du = 0
Zs t
>
: l(t s) + Rl (t s) + Rl (t u)l(u s)du = 0
(33)
s
for almost all s t We call Rl the resolvent kernel of l see Yosida 21]
Chapter 4 or Hida-Hitsuda 5]. As in Hida-Hitsuda 5] p.136-137, we can
now use the kernel Rl in order to reconstruct B in terms of X :
dXt +
= dBt +
Zt
Z 0t
Z0t
Rl(t u)dXudt
B (t u)dBudt +
Zt
0
Rl (t u) dBu +
= dBt + 0 (l(t u) + Rl(t u) +
= dBt
i.e., we have
and
Xt = Bt +
Bt = Xt +
Zt
u
Z tZ s
0
0
Z tZ s
Zu
0
l(u v)dBv du dt
Rl(t v)l(v u)dv)dBudt
l(s u)dBuds
Rl(s u)dXu ds:
(34)
(35)
Thus, X and B have the same ltration. Hence (32) is the canonical decomposition of X in its own ltration.
2
0
0
16
Canonical Decomposition
Remark 3.2 Let us make explicit the square-integrable Volterra kernel corresponding to our situation (20) in the special case where h(t) = 0, i.e.,
Xt = Wt +
Zt
0
f (u)W~ u du:
We have shown that the canonical decomposition of X is given by
Z t f (s) Z s
Xt = Bt + 0 0(s) 0 f (u)(u)dXu ds:
(36)
Integrating f (t)(t) on both sides and using It^o's product rule, we obtain
Zt
Z t f (v )(v )
0
f (u)(u)dXu = (t) 0 0(v) dBv :
0
Thus, the representation (36) takes the form
Zt
Zs
u)
(37)
Xt = Bt + 0 f (s) 0 f (u)(
0 (u) dBu ds:
This shows that the corresponding square-integrable Volterra kernel is given
by
u) l(s u) = f (s)f (0(uu)(
)
and we read from (36) that
Rl(s u) = ; f 0((ss)) f (u)(u):
(38)
One can verify from (38) that Rl satises indeed the two characteristic equations in (33).
Remark 3.3 The study of the general equation (20) for the pair (f h) can be
reduced to that of (f 0). Suppose a process (Xt ) satises (20) and introduce
the Gaussian process
Zt
t = Wt + 0 f (u)W~ u du:
Remark that (Xt) and (
t ) have the same ltration, since
t = Xt ;
Zt
0
h(u)Xu du
(39)
Canonical Decomposition
and
Xt =
17
Zt
Zt
exp( h(u)du)d
s :
s
Using the canonical decomposition for (
t ) given by (37), we obtain the canonical decomposition
Zt
Zs
u)
(40)
Xt = Bt + 0 (f (s) 0 f (u)(
0 (u) dBu + h(s)Xs )ds
for X . The pathwise solution of this equation leads us to the alternative
representation
Zt
Zt
Zt
Z t f (u)(u) Z t
Xt = 0 exp( s h(u)du)dBs + 0 0(u) u f (v) exp( v h(w)dw)dvdBu
(41)
as a functional of the Brownian motion B , in analogy to (37). From AllingerMitter 1], Davis 4] we know that the ;algebra FtX coincides with FtB for all
t, up to some P null sets. Thus, Ft = FtB = FtX . In the terminology of HidaHitsuda 5], the representation (41) is called \the canonical representation
relative to B ", and B is the innovation process for X .
0
3.3 A class of path dependent transformations
Using the same method as above, we can extend our result in Theorem 3.1
as follows. Consider a process (Xt)0t1 satisfying the stochastic functional
dierential equation
Zt
~
dXt = dWt + (f (t)Wt + h(t s)dXs)dt
(42)
0
11
for some deterministic
R t R s 2 function f 2 C (0 1)\A(0 1), a function h 2 C ((0 1)
(0 1)) with 0 0 h (s u)duds < 1 for all t < 1.
Theorem 3.2 The canonical decomposition of (Xt )0t1 is
1
dXt = dBt + (f (t)kt(Xs s t) +
Zt
0
h(t s)dXs)dt
where (Bt)0t1 is an (FtX )-Brownian motion, and ku (Xs s u) is of the
form
1 Z u (v)f (v)(dX ; Z v h(v r)dX dv):
(43)
v
r
(u) 0
0
0
18
Canonical Decomposition
Proof. We set
W (ds) := f (s)dWs ; f (s)
Zs
0
h(u s)dWuds:
2
The rest of the proof is similar to the proof in section 3.1.
3.4 Some examples
In the rst two examples we look at some processes whose canonical decompositions take a simple form. And in Example 3 we will discuss a special
case of the ltration.
Example 1. Consider a process (Xt )0t1 satisfying the stochastic dierential equation:
dXt = dWt + 1 ;a t (W~ t ; Xt)dt
that is, f (t) = ;h(t) = a=(1 ; t), with a nonzero constant a. Then the
corresponding Sturm-Liouville equation is
$ (u) = (1 ;a u)2 $(u)
2
00
(44)
which, arguably perhaps, is one of the simplest cases where the SturmLiouville equation has elementary solutions. Indeed, it is immediate to check
that a function (1 ; u) solves (44) if and only if ( ; 1) = a2, an equation
which admits the two solutions: + (a) and ; (a), given by
s
(a) := 12 a2 + 41 :
Clearly, ; (a) < 0 < + (a). Thus, the decreasing solution of (44) is
$(u) = (1 ; u)+(a):
And from the denition and the boundary condition of (u) we can get
(a)
+ (a)
(u) = (1 ; u) p ; (1 2; u) :
1 + 4a
;
Canonical Decomposition
19
From (28), the conditional expectation of W~ t relative to FtX is given by
ku (Xs s u)
Zu
2
:= 1(u) (v)( 1 ;a v dXv + (1 ;a v)2 Xv dv)
0
Z u ( (a) + 1)(1 ; v );+ (a) ; ( (a) + 1)(1 ; v ); (a)
;
+
dXv :
= Xu + a
;
(
a
)
+ (a)(1 ; u)
; ; (a)(1 ; u);+(a)
0
In particular, if a = 1, then + (1) = A and ; (1) = B , dened as in Lemma
2.4, and we are led to the same result as in section 2.
2
0
;
;
Example 2. Consider the simple example:
dXt = dWt + a(W~ t ; Xt)dt
with a nonzero constant a. The desired solution of the corresponding SturmLiouville equation is
(t) = 21a (eat ; e;at)
and the conditional expectation of W~ t with respect to FtX is given by
Zt
2
kt(Xs s t) = Xt ; eat + e;at 0 e;au dXu :
Therefore, the canonical decomposition of Xt has the form
Zt
Zu
2
a
Xt = Bt ; 0 eau + e;au ( 0 e;av dXv )du
where (Bt)0t1 is a Brownian motion relative to (FtX )0t1.
2
Example 3. Let (Wt)t 0 and (W~ t)t 0 be two independent Brownian motions,
starting from 0. The process (Xt)t 0 satises the stochastic dierential equation (20) with f (t) = ;k=t with a constant k and h 0. In section 3.2 we
have already discussed a few results about the case with h 0. First, we
have to solve
2
00(s) = ks2 (s)
20
Canonical Decomposition
and then single out a solution
with (0) = 0. It is easy to check that
q
1
1
(s) = s with = 2 + k2 + 4 is the wanted solution. Now formula (37)
gives
2 Z t B
k
(45)
Xt = Bt + 0 uu du
with a Brownian motion (Bt). In Jeulin-Yor (12], Theorem 9) it has been
shown that
ZtB
Xt := Bt ; ss ds
(46)
0
has a strictly smaller ltration than the ltration of B i > 12 . Coming
back to formula (45), and comparing with (46), we nd
p
= 12 ; 12 1 + 4k2:
Thus, in our study, is always negative, hence (Xt)t 0 has the same ltration
as (Bt).
2
4 Application of stochastic ltering theory
for Gaussian processes
In section 3 we have computed the canonical representation of our transformations (20) of two Brownian motions by direct methods. As an alternative,
we can derive them as corollaries of the stochastic ltering theory for Gaussian processes. At the same time, this allows us to extend our results to
a general class of transformations where the drift Y in (1) is given as an
adapted linear functional of X and W~ .
Suppose that the process X satises a stochastic dierential equation of
the form
Zt
Zt
dXt = dWt + ( 0 F (t u)dW~ u + 0 H (t u)dXu )dt
(47)
with X0 = 0, where (W~ t)0t1 (Wt)0t1 are two independent Brownian motions, and F , H are square-integrable Volterra kernels on (0 1) (0 1), i.e.,
they satisfy the conditions in section 3.2. Note that the processes considered
in (20) belong to this class.
Canonical Decomposition
21
Lemma 4.1 There is a unique Brownian motion B and a unique square-
integrable Volterra kernel GF such that
Wt +
Z tZ s
0
0
F (s u)dW~ uds = Bt +
Z tZ s
0
0
GF (s u)dBu ds:
The kernel GF is determined by the equation
Zs
0
F (t v)F (s v)dv = GF (t s) +
Zs
0
GF (t v)GF (s v)dv:
(48)
Moreover the natural ltration of B is identical to that of the left-hand side.
Proof. (1) Let us denote by Z the \signal process"
Zt =
Zt
0
F (t u)dW~ u
and by the \observation process"
t = Wt +
Zt
0
Zs ds = Wt +
Z tZ s
0
0
F (s u)dW~ uds:
(49)
From Lemma 2.3 we know that t can be written as
t = Bt +
Zt
0
E Zs jFs]ds
where (Bt) is an (Ft )-Brownian motion. Since E ZtjFt ](!) can be chosen
(t !)-measurable and (Ft)-adapted, we can write
E ZtjFt ] = (t (!))
where is a non-anticipative functional in the sense of Kallianpur 13], Definition 5.1.1. Furthermore, it follows from
E
that
Zu
0
(E ZujFu])2du E
Zt
0
Zt
0
Zu2du =
2(s (!))ds < 1
Z tZ s
0
0
(F (s u))2duds < 1
IP ; a.s.
22
Canonical Decomposition
for all t < 1. By Kallianpur 13], Theorem 9.4.1 and section 3.2, there is a
unique square-integrable Volterra kernel GF such that the Gaussian process
has the representation
t = Bt +
Z tZ s
0
0
GF (s u)dBuds:
(50)
Moreover we have (Ft ) = (FtB ), due to Proposition 3.2.
(2) Since W and W~ are two independent Brownian motions, we get from
(49), for s t,
Z sZ u
E s
t ] = E WsWt] + E 0
= s+2
+
Z sZ uZ v
Z tZ s Z v
s
0
0
0
0
0
0
F (u r)dW~ r du
Z tZ v
0
F (u r)F (v r)drdvdu
0
F (v q)dW~ qdv]
F (u r)F (v r)drdvdu:
(51)
We can also compute the covariance of from (50):
E s
t] = E (Bs +
= s+2
+2
+
Z sZ u
Z sZ u
0
0
GF (u v)dvdu +
Z s Z0 u Z0 v
Z tZ s Z v
0
s
0
0
0
0
GF (u r)dBr du)(Bt +
Z tZ s
s
0
Z tZ v
0
0
GF (v q)dBqdv)]
GF (u v)dvdu
GF (u r)GF (v r)drdvdu
GF (u r)GF (v r)drdvdu:
(52)
Since the right hand sides of these two equations must coincide, dierentiating rst with respect to t then with respect to s yields (48) for almost every
s t.
2
Remark 4.1 In the notation of Kallianpur 13] p.235, equation (48) can be
viewed as the factorization S = (I + G)(I + G? ) of the integral operator S
dened by I + FF ?, where F , G are integral operators with square-integrable
Volterra kernels F (t s) and GF (t s), respectively, i.e. for all f g 2 L2 (0 1),
h(I + FF ?)f gi = h(I + G)(I + G? )f gi:
Canonical Decomposition
23
In order to see this, let f (u) = I(0s)(u) and g (u) = I(0t)(u) with 0 s t 1. Using the properties of Volterra kernels, we have
=
hZ(I + FF ?)f gi =Z hfZgi + hF ?f F ?gi
1
0
1
1
Z1
f (u)g(u)du + 0 ( 0 F (v u)f (v)dv)( 0 F (r u)g(r)dr)du
= s+
ZsZs
Zt
u
u
0
( F (v u)dv)( F (r u)dr)du
which equals to the right-hand side of (51). On the other hand,
=
=
h(I + G)(I Z+ G?)f gi = h(I + G?)f (I Z+ G?)gi
Z
1
Z 0s
0
(f (u) +
(1 +
Zs
u
1
0
GF (v u)f (v)dv)(g(u) +
GF (v u)dv)(1 +
Zt
u
1
0
GF (v u)g(v)dv)du
GF (v u)dv)du
which is exactly the right-hand side of (52).
Now, we look at the canonical decomposition of the process X given by
(47).
Theorem 4.1 The canonical decomposition of X as a semimartingale in its
own ltration (FtX ) is given by
Zt
dXt = dBt + ( 0 GF (t u)dBu +
Zt
0
H (t u)dXu )dt:
(53)
Moreover we have (FtX ) = (FtB ).
Proof. From (49) and (53) , we have
dXt = d
t +
Zt
0
H (t u)dXudt:
(54)
As in (33), let R;H denote the resolvent kernel of the square-integrable
Volterra kernel ;H . Due to equations (34) and (35), we have
t = Xt ;
Z tZ s
0
0
H (s u)dXu ds
24
Canonical Decomposition
and
Xt = t +
Z tZ s
0
0
R;H (s u)d
uds:
These two equations together with Lemma 4.1 imply FtX = Ft = FtB , for all
t. Hence, (Bt) is also an (FtX )-Brownian motion. Substituting the representation (50) in the equation (54), we obtain (53).
2
Remark 4.2 Comparing (47), (53) and Lemma 2.3, we see that
Zt
E 0 F (t u)dW~ ujFtX ] =
Zt
0
GF (t u)dBu:
Let us now consider the special case when F admits a factorization
F (t s) = f (t)g(s) for some functions f g 2 C 1(0 1) which satisfy
Z tZ u
0
0
f 2(u)g2(v)dvdu < 1
(55)
for all t < 1.
Corollary 4.1 Suppose the process (Xt )0t1 satises
Zt
dXt = dWt + (f (t) 0 g(u)dW~ u +
Zt
0
H (t u)dXu )dt
(56)
with f g 2 C 1 (0 1) satisfying (55), f 6= 0 a.s., and a square-integrable
Volterra kernel H (t s). Then the canonical decomposition of X is of the
form
Zt
Zt
dXt = dBt + (f (t) (u)dBu + H (t u)dXu )dt
(57)
0
0
where the function (t) is the solution of the dierential equation
( f ((tt)) )0 + 2(t) = g2(t)
with boundary condition (0) = 0.
(58)
Canonical Decomposition
25
Proof. We have only to prove that GF (t s) = f (t)(s), where satises
(58) and (0) = 0, is the solution of (48). In fact, the right-hand side of (48)
is equal to
GF (t s) +
Zs
0
GF (t u)GF (s u)du
= f (t)(s) + f (t)f (s)
Zs
2(u)du
0
Zs
= f (t)(s) + f (t)f (s) g2(u)du ; f ((ss)) ]
= f (t)f (s)
Zs
0
0
g (u)du =
2
Zs
which is exactly the left-hand side of (48).
0
F (t u)F (s u)du
2
In order to see the connection with our discussion in the preceding sections, let us set g 1 and H (t u) = h(t). Then equation (58) can be written
as
( f ((tt)) )0 + 2(t) = 1
with (0) = 0. The corresponding solution is given by
t)
(59)
(t) = f (t)(
0(t) where (t) is the solution of the Sturm-Liouville equation (24). Substituting
this result in (57), we see that the result coincides with (40).
Remark 4.3 The discussion of equation (56) can be reduced to the case
g 1. All we need is to consider H = 0. Then, we have
Zt
g(t)dXt = g(t)dWt + f (t)g(t) g(u)dW~ u dt:
0
Let us introduce the (time-changed) processes (X^ u ), (W^ u ) and (W^~ u ) dened
as:
Zt
g(u)dXv = X^G(t) 0
where
Zt
G(t) = g2(u)du
0
26
Canonical Decomposition
etc. We obtain
where
dX^u = dW^ u + '(u)W^~ udu
'(u) = ( fg )(G;1 (u)):
Consequently, we obtain the canonical decomposition of X^ :
X^t = B^t + '(t)
where
Zt
0
^ (u)dB^u
^
^(t) = '(^t)0 (t) (t)
and ^ is our usual notation for the solution of u00 = '2 u. It now remains to
undo the time-change, and relate ^ to , as given in (58).
5 How to get again a Brownian motion
In our initial stochastic dierential equation (10), where the nal value W~ 1
was known in advance, the solution (Xt )0t1 was again a Brownian motion,
and it was tied to the nal value W~ 1. Is it also possible to satisfy both
conditions in our modied situation when at time t we only know the past
of W~ ?
5.1 Characterization of Brownian motions
Consider a process (Xt )0t1 with X0 = 0 which satises the stochastic
functional dierential equation
Zt
dXt = dWt + ( 0 F (t u)dW~ u +
Zt
0
H (t u)dXu )dt
(60)
with square-integrable Volterra kernels F and H .
Theorem 5.1 A process X satisfying (60) is a Wiener process with respect
to its own ltration (FtX ) if and only if H (t s) = ;GF (t s), where GF is the
Canonical Decomposition
27
square-integrable Volterra kernel dened by (48). In other words, X is of the
form
ZtZs
Zs
~
Xt = Wt + ( F (s u)dWu ; GF (s u)dXu )ds:
(61)
0
0
0
Proof. (1) Suppose X is a Wiener process with respect to its own ltration (FtX ). By uniqueness of the Doob-Meyer decomposition in (FtX ), our
representation (53) implies B = X and
Zt
0
GF (t u) + H (t u)]dXu = 0
(62)
IP - a.s. for almost all t. But (62) implies
GF (t u) + H (t u) = 0
for almost all u t, since X is a Brownian motion.
(2) Conversely, assume that X has the form (61). The canonical representation (53) implies
ZtZs
Xt = Bt + 0 ( 0 GF (s u)dBu ;
i.e.,
Xt +
Z tZ s
0
0
GF (s u)dXu ds = Bt +
Zs
0
GF (s u)dXu )ds
Z tZ s
0
0
GF (s u)dBu ds:
We can now apply the reconstruction argument in section 3.2 in order to
2
conclude X = B . In other words, X is a Brownian motion.
Remark 5.1 The last argument shall be taken up again in Lemma 5.1.
Let us look at the special case considered in Corollary 4.1, where F is of
the form F (t s) = f (t)g(s) for some continuously dierentiable functions f
and g satisfying (55).
Corollary 5.1 A process (Xt )0t1 satisfying (56) is a Brownian motion if
and only if
H (t u) = ;f (t)(u)
28
Canonical Decomposition
where (t) is the solution of (58) with boundary condition (0) = 0. In other
words, if (Xt )0t1 is a Brownian motion with respect to its own ltration, it
must be of the form
Zt
dXt = dWt + f (t)( 0 g(u)dW~ u ;
Zt
0
(u)dXu )dt:
In particular, if g 1, then (63) can be written as
Zt
u) dX )dt
dXt = dWt + f (t)(W~ t ; 0 f (u)(
u
0 (u)
where (t) is the solution of the Sturm-Liouville equation
00(t) = f 2(t)(t)
with (0) = 0 and 0(0+) = 1:
(63)
(64)
(65)
Proof. In order to get the characterization for this class of Brownian
motions from Theorem 5.1, we have only to compute GF (t u). From the
proof in Corollary 4.1, we know that GF (t u) = f (t)(u). Substituting this
result in Theorem 5.1, we get the rst assertion. As to the case g 1, simply
substitute (59) in (63).
2
Corollary 5.2 Let X be a process satisfying the stochastic dierential equation (20) with f h 2 C 1(0 1) \ A(0 1). If one of the functions f (t) and h(t)
is not equal to 0, then X cannot be a Brownian motion.
Proof. If X satisfying (20) is a Brownian motion, then it follows from
Corollary 5.1 that f must be of the form
f (t)(t) = c0(t)
(66)
for some non-zero constant c. Substituting (66) in (65), we get
00(t) = cf (t)0(t):
Hence, the corresponding solution of the Sturm-Liouville equation is given
by
Zt
Zu
(t) = exp(c f (v)dv)du:
0
0
Canonical Decomposition
29
Substituting this solution again in (66), and taking derivatives on both sides
with respect to t, we have
cf 0(t) + (1 ; c2)f 2(t) = 0:
This implies
f (t) = 1 ;c c2 1t which does not belong to A(0 1).
2
In the class of adapted linear drift of the form (60), Theorem 5.1 characterizes those cases where the resulting process X in (61) is a new Brownian
motion. Let us now return to the question whether such a Brownian motion
can be tied to the endpoint W~ 1 of the Brownian motion W~ . This turns out
to be impossible as long as we insist on an adapted drift, even if we drop
linearity.
Proposition 5.1 Consider any drift (Yt)0t1 adapted to (Ft) such that
Xt = Wt +
Zt
0
Ysds
is a Brownian motion. If Z is any (Ft)-Brownian motion such that X1 =
Z1 IP -a.s., then we have Zt = Xt = Wt, and in particular, Yt = 0, dt dIP a.s..
Proof. If X1 = Z1, then
Xt = E X1jFtX ] = E Z1jFtX ] = E ZtjFtX ]:
This implies
hence
E XtZt ] = E Xt2] = t
E (Xt ; Zt)2] = E Xt2] + E Zt2] ; 2E XtZt] = 0:
Hence, Zt = Xt is an (Ft)-Brownian motion, and so: Xt = Wt.
2
In our situation the Brownian motion Z = W~ is independent of W , and
so the proposition shows that it is impossible to obtain X1 = W~ 1.
30
Canonical Decomposition
Remark 5.2 (i) Consider a square-integrable (Ft)-adapted process (Xt)t1
which is a martingale in its own ltration (FtX ). Let (Mt) be a squareintegrable (Ft)-martingale such that X1 = M1 and E Xt2 ] = E Mt2] for every
t 1. The proof of Proposition 5.1 shows that this implies Xt = Mt for
every t 1.
(ii) As a special case of (i), assume that (Xt) is a Brownian motion in its
own ltration (FtX ). Then (Xt ) is an (Ft)-Brownian motion if and only if
E X1jFt] is a Brownian motion.
(iii) As an example of (ii), consider a Brownian motion (Bt) with respect to
(Ft). We know that
Zt
Xt := Bt ; Bss ds
0
denes a new Brownian motion (Xt )t1 which is not a Brownian motion
with respect to (Ft ) cf., e.g., Yor 19]. It follows from (ii) that (E X1jFt])t1
cannot be a Brownian motion. In fact, a direct computation shows that
E X1jFt] = Bt ;
ZtB
s
0
s ds ;
= Bt(1 + log t) ;
Z 1 E B jF ]
Z tt Bs
0
s
t
s Z ds
t
s ds = 0 (1 + log s)dBs:
5.2 Enlargement of a Brownian ltration
Consider a Brownian motion X arising as the solution of the linear functional
stochastic dierential equation (61). Let us represent X directly in terms of
the two Wiener processes W and W~ . To this end, we introduce the resolvent
kernel LF of GF , i.e., GF and LF satisfy the following relations:
8
Zt
>
< GF (t s) + LF (t s) + LF (t u)GF (u s)du = 0
Zs t
>
>
: GF (t s) + LF (t s) + GF (t u)LF (u s)du = 0:
(67)
s
Lemma 5.1 The solution of (61) is given by
Xt = Wt+
Zt Zs
0
0
Zs
LF (s u)dWu + 0 (F (s u) +
Zs
u
LF (s v)F (v u)dv)dW~ u ds:
(68)
Canonical Decomposition
31
Proof. By (61), the process dened in (49) has the form
d
t = dXt +
Zt
0
GF (t u)dXudt:
Using the same argument as in the proof in section 3.2, we obtain the representation
Zt
dXt = d
t + 0 LF (t u)d
udt
(69)
Substituting (49) in (69), we obtain (68).
2
In the special case F (t u) = f (t)g(u), we have GF (t s) = f (t)(s), and
the solution LF of (67) is given by
Zt
LF (t s) = ;f (t)(s) exp(; s f (v)(v)dv):
Thus, the solution of (63) can be written as
Zt
Zt
dXt = dWt + f (t)( 0 g(u) exp(; u f (v)(v)dv)dW~ u
Zt
Zt
; 0 (u) exp(; u f (v)(v)dv)dWu)dt:
In particular, the solution of (64) is given by
Zt
Zt
dXt = dWt + f 0((tt)) ( 0 0(u)dW~ u ; 0 f (u)(u)dWu)dt:
(70)
We are now going to show that the decomposition (70) of W as the sum
Z t f (s) Z s
Zs
Wt = Xt + 0 0(s) ( 0 f (u)(u)dWu ; 0 0(u)dW~ u)ds
can be viewed, after time reversal, as the expression of a Brownian motion
in an enlarged Gaussian ltration (see, Jeulin-Chaleyat-Yor 10] or Yor 20]).
We consider an n-dimensional Brownian motion Bt (Bt(1) ::: Bt(n)), and
R
we enlarge its ltration with B () def
= 01((u) dBu ). With respect to the
enlarged ltration, the canonical decomposition of B is given by
Z t (u) R 1 ((s) dB )
s du
u
~
(71)
Bt = Bt + 0
2
(u)
32
Canonical Decomposition
where B~ is again a Brownian motion and is dened as
(u) =
2
Z1
u
j(t)j dt 2
Z1X
n
u
( (i (t))2)dt
i=1
see Jeulin-Chaleyat-Yor 10] or Yor 20]. In order to make sure that formula
(71) is meaningful (as a semimartingale decomposition), it is necessary and
su!cient that
Z t j(u)j
du < 1:
0 (u)
On the other hand, (70) yields
Bt(1) = B~t(1) ;
Z
Z t f (1 ; u) Z 1
0 (1 ; v )dB (2) ; 1 f (1 ; v )(1 ; v )dB (1))du
(
v
v
0
0
(1 ; u)
u
u
(72)
(1)
(2)
~
~
~
with Bt = W1 ; W1;t, Bt = X1 ; X1;t and Bt = W1 ; W1;t. We want
to view the representation (72) as an enlargement formula (71) for a suitable
choice of . Thus, we would like to nd a pair of functions (1 2) such that
(1)
8
>
1 (u)1(s) = f (1 ; u)f (1 ; s)(1 ; s)
>
>
>
0(1 ; u)
< 2 (u)
>
>
1 (u)2(s) = ; f (1 ; u)0(1 ; s) :
>
>
: 2 (u)
0(1 ; u)
(73)
The problem is now to retrieve 1 and 2 from the system (73). The solution
is given by
8
>
< 1 (s) = cf (1 ; s)(1 ; s)
>
: 2 (s) = ;c0 (1 ; s)
with some nonzero constant c.
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Canonical Decomposition
33
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34
Canonical Decomposition
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