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GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 363-378

MARKOV DILATION OF DIFFUSION TYPE PROCESSES
AND ITS APPLICATION TO THE FINANCIAL
MATHEMATICS
R. TEVZADZE

Abstract. The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for
the obtained Markov processes are described. Some applications to
the integral representation for functionals of diffusion type processes
and to the construction of a replicating portfolio for a non-terminal
contingent claim are considered.

1. Introduction
Let ξ = (ξt )t∈[0,1] be a stochastic process in the metric space X with
the sample paths from the space D([0, 1], X) of functions which are right
continuous with left limit (r.c.l.l.). It is easy to see that the D([0, 1], X)valued process defined for each t ∈ [0, 1] by
ξ t = (ξt∧s , s ∈ [0, 1])
has a Markov property, i.e., for any Borel set B in D[0, 1]
P [ξ t ∈ B|ξ t1 , ξ t2 , . . . , ξ tn ] = P [ξ t ∈ B|ξ tn ], 0 ≤ t1 ≤ · · · ≤ tn ≤ 1, (1.1)
since σ-fields σ(ξ t ) = σ(ξs , s ≤ t) increase as t increases.

Consider the case X = R and suppose that ξ is a diffusion type process,i.e., it satisfies a stochastic differential equation (S.D.E.)
dξt = a(t, ξ)dwt + b(t, ξ)dt,

(1.2)

where a(t, x), b(t, x) are nonanticipative functionals and w = (wt )t∈[0,1] is
the Wiener process. We want to find a S.D.E. and infinitesimal operators
for a random process ξ t . This will allow us to write a parabolic equation
for functionals of diffusion type processes and to derive Itˆo’s formula for
1991 Mathematics Subject Classification. 60H05, 60H07.
Key words and phrases. Markov dilation, infinitesimal operator, Itˆ
o’s formula, replicating portfolio.

363
c 1999 Plenum Publishing Corporation
1072-947X/99/0700-0363$15.00/0 

364

R. TEVZADZE

nonanticipative functionals. In Section 2 we represent ξ t as a solution of a
S.D.E. in the space of square integrable paths. For this case the infinitesimal
operator can be defined using the results of infinite dimensional stochastic
analysis but one needs strong conditions on the coefficients a and b. In
Section 3 more general and complicated case of the continuous path space
is studied. Finally, we shall obtain a new proof of Clark’s formula for
Itˆo’s processes [1] and we bring some applications to financial mathematics.
When ξ is a homogeneous Markov process, our results have some intersection
with the recent results of [2].
Here we use the following notation: W [0, 1] is the space of continuous
functions, D[0, 1] the space of r.c.l.l. functions, L2m [0, 1] a space of square
integrable functions w.r.t. measure m, it = 1[0,t) , jt = 1[t,1] . CB([0, 1] ×
W [0, 1]) denotes class of bounded continuous functions and CB i,k ([0, 1] ×
W [0, 1]) denote the subclasses of functions from CB([0, 1] × W [0, 1]) with
continuous and bounded derivatives w.r.t. the first variable up to order i
and continuous bounded Frechet derivatives w.r.t. the second variable up
to order k.
For any t ∈ [0, 1] we shall consider the operators:
Ct x = xit + x(t)jt , Ct : D[0, 1] → D[0, 1] ⊂ L2m [0, 1],

2
Rt x = x(t)jt , Rt : D[0, 1] → D[0, 1] ⊂ Lm
[0, 1],
(
ψ(s) − ψ(t) + x(t) if s > t,
Ltψ x(s) ≡ ψ ◦t x(s) =
x(s)
if s < t,
2
ψ ∈ D[0, 1], Lψ
t : D[0, 1] → D[0, 1] ⊂ Lm [0, 1],
(
x(0)
if s > t,
Qt x(s) =
x(t) − x(s) + x(0) if s ≤ t,
2
[0, 1].
Qt : D[0, 1] → D[0, 1] ⊂ Lm

Their restrictions on the space W [0, 1] will be denoted by the same
symbols. We shall also use the space W [0, ∞) with metric kx − yk =
∞
P
2−k sup{|x(s) − y(s)|, 1} and spaces CB(R+ × W ), CB i,k (R+ × W )

k=1

s≤k

defined in the same way.

2. Repesentation of a Diffusion Type Process in the Space
L2m [0, 1]
Let (Ω, F, P ) be a probabilist space with filtration (Ft )t∈[0,1] . Let
(wt , Ft )t∈[0,1] be a Wiener process and (βt , Ft )t∈[0,1] a random process with
paths from the space D[0, 1] ∩ V [0, 1]. Denote by Iβ (g)t and Iw (g)t the inteRt
Rt
gral 0 gs dβs and the stochastic integral 0 gs dws respectively, for a process

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

365

(gt , Ft ) with values in the Hilbert space satisfy standart condition, which
guarantees the existence of those integrales. If βt = t, then Iβ (g) is shortly
denoted by I(g). Then the following theorem is valid.
Theorem 2.1. Let (at , Ft ), (bt , Ft ) be real processes such that
’Z 1
“
Z 1
P
|bt |dβt < ∞ = 1, E
|at |2 dt < ∞.
0

0

Then the identities take place
C(Iβ (B)) = Iβ (R(b)), C(Iw (a)) = Iw (R(a)).

Rt
Rt
Rt
In other words, if ξt = 0 as dws + 0 bs dβs , then Ct ξ = 0 Rs (a)dws +
Rt
Rs (b)dβs .
0
Proof. Let ψ ∈ L2m [0, 1]. Then
Z
Z t Z s
(ψ, Ct (Iβ (b))) =
ψs
bu dβu dms +
0

=

0

Z tZ

0

=

bu

t

ψs bu is (u)dβu dms +

Z

1

t

t

ψs bu 1[u,1] (s)dβu dms +

0

Z

t

bu dβu dms =

0

t

0

t

Z

Z

ψs

0

Z tZ
0

=

1

t

ψs ju (s)dms dβu +
0

Z

Z

Z

t

0
1

t

t

bu
0

ψs bu dβu dms =
t

Z

ψs bu dβu dms =

0

Z

1

ψs dms dβu =

t

Z 1
Z t
(ψ, Ru (b))dβu =
ψs dms dβu =
bu
0
u
0
“
’ Z t
Ru (b)dβu .
= ψ,

=

t

Z

0

By the arbitrariness of ψ the first relation is established. Identitities for
stochastic integrals may be derived in similar way. For instance, the transition
Z
Z
Z Z
t

t

t

0

0

t

ψs au js (u)dwu =

dms

ψs au js (u)dms dwu

0

0

is true by Fubini’s theorem for stochastic integrals [3, p. 217]. In particular,
we must take
˜ = [0, 1], F˜t = B[0, 1], p˜ = m, gs (w, w)
˜
˜ = ψu as (w)1[0,s] (w)
Ω
in the equality
Z Z
˜
Ω

0

1

˜
gs (w, w)dw
˜
s dP =

Z

1
0

Z

˜
Ω

gs (w, w)d
˜ P˜ dws .

366

R. TEVZADZE

Corollary. Let f : [0, 1] × L2m [0, 1] → R be a bounded function with
bounded and continuous Frechet derivative ft , ∇x f , ∇2x f and let ξt =
Rt
Rt
a dws + 0 bs dβs be the Itˆ
o process. Then
0 s
Z th
∂
∂
f (s, Cs ξ) +
f (s, Cs ξ) + bs
∂js
t0 ∂s
Z t
i
∂
1 ∂2
as
+ a2s 2 f (s, Cs ξ) ds +
f (s, Cs ξ)dws ,
2 ∂js
∂j
s
t0

f (t, Ct ξ) − f (t0 , Ct0 ξ) =

where

∂
∂jt

(2.1)

denotes the Gateux derivative in the direction jt .

Rt
Proof. Using Itˆo’s formula for the L2m [0,1]-valued Itˆo process ηt = 0as js dws+
Rt
b j ds [4] and taking into account
0 s s
∂
f (t, x),
∂jt
∂2
∇2x f (t, x)(at jt , at jt ) = a2t 2 f (t, x),
∂js
(∇x f (t, x), bt jt ) = bt

(2.2)

we obtain (2.1).
About Ct ξ, Qt ξ or Lψ
t ξ we shall say that each of them is the Markov
dilation of ξ, since each of them has the Markov property.
Theorem 2.2. Let A, B : [0, 1] × L2m [0, 1] → R be Lipschitz function,
satisfying linearly growth condition. Suppose that functions aψ , bψ : [0, 1] ×
W [0, 1] → R defined by
aψ (t, x) = A(t, Ltψ x), bψ (t, x) = B(t, Lψ
t x)
and ξst,ψ , θst,ψ denote solutions of the S.D.E.
Z s
Z s
bψ (u, ξ)du,
aψ (u, ξ)dwu + jt (s)
ξs = ψs + jt (s)
t
Z s t
Z s
A(u, θu )ju dwu +
B(u, θu )ju du,
θs = ψ +
t

(2.3)
(2.4)

t

t,ψ
).
respectively. Then θst,ψ = Lψ
s (ξ

Proof. Equation (2.3) has a unique strong solution ([3], [4]) and equation
(2.4) has a unique strong solution only in the space L2m [0, 1] [5]. By Theorem
2.1 equation (2.3) gives
Z s
Z s
bψ (u, ξ)ju du.
C s ξ = ψs +
aψ (u, ξ)ju dwu +
t

t

MARKOV DILATION OF DIFFUSION TYPE PROCESSES
ψ
Since aψ
u (ξ) = Au (Lu ξ), we have
Z
Z s
ψ
=
ψ
+
ξ
+
Lψ
A(u,
dw
L
ξ)j
u
u
s
u

=

Lsψ (ξ t,ψ )

s

B(u, Lψ
u ξ)ju du,

t

t

θst,ψ

367

because of the solution is unique.

Corollary 1. Suppose that in addition to the conditions of Theorem 2.2
2
the functions A, B belong to C 1,2 ([0, 1] × L2m ) and η ∈ C 2 (Lm
). Then the
Cauchy problem
‘
∂
∂
∂
1
∂2 ‘
+ A(t) u(t, ϕ) ≡
+ b(t, ϕ)
+ a(t, ϕ)2 2 u(t, ϕ) = 0, (2.5)
∂t
∂t
∂jt
2
∂jt
u(1, ϕ) = η(ϕ)
(2.6)
has a solution which can be represented as
u(t, ϕ) = Eη(ξ t,ϕ ),

(2.7)

where ξ t,ϕ is a solution of (2.3).
Proof. It follows from the results of [5, pp. 322, 325] taking into account
(2.2).
0,ψ(0)j0

Corollary 2. ξs = ξs

satisfies the S.D.E.
Z s
Z s
b(u, ξ)du,
a(u, ξ)dwu +
ξs = ψ(0) +
0

0

where a(u, ξ) = A(u, Cu ξ), b(u, ξ) = B(u, Cu ξ), and we have
E[η(ξ)|Ftξ ] = Eη(ξ t,ψ )|ψ=Ct ξ .
Remark 1. If we suppose
˜ x(t)), B(t, x) = B(t,
˜ x(t)),
ψ = ψ0 j0 , m = δ1 , A(t, x) = A(t,
˜ x(t)), b(t, x) = B(t, Ct x) = B(t,
˜ x(t)).
then a(t, x) = A(t, Ct x) = A(t,
Theorem 2.3. Suppose that a continuous bounded function
2
f : [0, 1] × Lm
[0, 1] → R

possesses the continuous bounded derivatives
for each t, s ∈ [0, 1]. Then (2.1) holds.

∂
∂2
∂
∂t f (t,ψ), ∂jt f (s,ψ), ∂jt2 f (s,ψ)

368

R. TEVZADZE

Proof. By a standard way the proof of formula (2.1) reduces to the case
ξt = xt0 + (t − t0 )Rt0 (b) + (w(t) − w(t0 ))Rt0 (a). Evidently, ξt − xt0 = ηt jt ,
g(t, ηt ) = f (t, ξt ), where g(t, q) = f (t, qxt0 jt0 ), ηt = (t − t0 )bt0 + (w(t) −
w(t0 ))at0 . By the Itˆo formula for scalar case we obtain
Z th
∂
∂
g(s, ηs ) + bt0 g, ηs ) +
∂s
∂q
t0
Z t
i
∂
1 2 ∂2
at0 g(s, ηs )dws .
+ at0 2 g(s, ηs ) ds +
2
∂q
∂q
t0

g(t, ηt ) − g(t0 , 0) =

Then the Itˆo formula is obtained for ξt , since

∂k
g(t, q)
∂q k

=

∂k
∂jtk

f (t, xt0 +qjt0 ),

0

k = 1, 2.
The first and second Frechet derivatives ∇F , ∇2 F for the functions F :
W [0, 1] → R by Riesz’ theorem are represented as a Borel measure on [0, 1]
and a symmetric (w.r.t. the inversion (u, v) → (v, u)) Borel measure on
[0, 1]2 , respectively [6, p. 68]. These measures are denoted by ∇F (x, du)
and ∇2 F (x, dudv).
Remark 2. If f : W [0, 1] → R is a twice continuous Frechet differentiable
∂2
function, then by the results of [2] both ∂j∂ t f (t, Ct x), ∂j
2 f (t, Ct x) are r.c.l.l.
t
in t.
Theorem 2.4. Suppose that f ∈ CB 1,2 ([0, 1] × W [0, 1]), f 0 (t, x) =
f 1 (t, x) = ∇f (t, x, [t, 1]), f 2 (t, x, [t, 1]2 ) = ∇2 f (t, x, [t, 1]2 ) and
Rt
t
ηt = ψ+ t0 as js dws + t0 bs js ds, where (at , Ft ), (bt , Ft ) satisfy the conditions
of Theorem 2.1. Then
∂
∂t f (t, x),
R

f (t, ηt ) − f (t0 , ηt0 ) =

Z th

i
1
f (s, ηs ) + bs f 1 (s, ηs ) + a2s f 2 (s, ηs ) ds +
2
t0
Z t
(2.8)
as f 1 (s, ηs )dws .
+
t0

Proof. For each h ∈ L2 [0, 1] we denote
hn (t) = n

Z

t

e−n(t−s) h(s)ds.

(2.9)

0

Evidently, the mapping L2 [0, 1] ∈ h → hm ∈ W [0, 1] is a lineary continuous
function and hm → h, m → ∞ in L2 [0, 1]. The mapping f (n) (t, x) = f (t, x)

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

369

is a differentiable w.r.t. t and twice Frechet differetiable w.r.t. x. For
h ∈ L2 [0, 1] we have
Z 1
∂ (n)
fx (t, xn , ds)hn (s),
f (t, x) =
∂h
0
Z 1Z 1
∂ 2 (n)
fxx (t, xn , dudv)hn (u)hn (v)
F
(t,
=
x)
∂h2
0
0
for h = js , hn (t) = e−nt (ent − ens ), i.e.,
∂ (n)
f (t, x) → f 1 (t, x),
∂jt

n → ∞.

Similarly,
∂ 2 (n)
f (t, x) → f 2 (t, x), n → ∞.
∂jt2
By Lebesgue’s theorem we can pass to the limit in formula (2.1).
Remark 3. In [7] it was proposed to define derivatives of anticipative
functions as follows. f (t, x) called differentiable if there exist f 0 and f 1 such
Rt
that f (t, x) = f 0 (t, x) + 0 f 1 (s, x)dxs for x ∈ V [0, 1]. If f (t, x) = F (t, Ct x)
then the derivatives f 0 (t, x), f 1 (t, x) can be calculated as
Z t
∂F
∂F
(t, Ct x),
(s, Cs x)ds,
∂s
∂jt
0
respectively.
Remark 4. It is possible to define the Markov dilation by the operator
∂
Qt . Then the infinitesimal operator will have the form ∂t
+ b(t, x) ∂i∂t +
1
2 ∂2
2 a(t, x) ∂i2t .

3. Markov dilation of a Diffusion Type Process in the Space
W [0, ∞)
Let a, b : R × W [0, ∞) → R be nonanticipative continuous functions such
that the S.D.E.
Z s
Z s
ξs = ψs +
b(u, ξ)du, t ≤ s,
(3.1)
a(u, ξ)dwu +
t

t

being defined as ψ(s) for s < t has a unique strong solution for any t. For
this it must satisfies the following condition [5]: There exists K > 0 such
that
|a(t, x) − a(t, y)|2 + |b(t, x) − b(t, y)|2 ≤ Kkx − ykt2 ,
|a(t, x)|2 + |b(t, x)|2 ≤ K(1 + kxk2 ).

(3.2)

370

R. TEVZADZE

Lemma 3.1. Let ξst,ψ , s ≥ t be a solution of (3.1). Then
ξτt,ψ = ξτψ◦s ξ

t,ψ

,

t ≤ s ≤ τ.

Proof. By inserting ϕ = ξ t,ψ in the equality
Z
Z τ
a(u, ξ ψ◦s ϕ )dwu +
ξτψ◦s ϕ = ψ ◦s ϕ(τ ) +

τ

b(u, ξ ψ◦s ϕ )du, ψ ∈ W,

t

t

and taking into account that
Z τ
Z
g(u, r)dwu |r=η =
s

τ

g(u, η)dwu ,

η ∈ Fs ,

s

we have
ξτs,ψ◦s ξ

t,ψ

= ψ(τ ) − ψ(s) + ξst,ψ +
Z τ
Z τ
t,ψ
t,ψ
a(u, ξ s,ψ◦s ξ )dwu +
+
b(u, ξ s,ψ◦s ξ )du =
s
s
Z s
Z s
t,ψ
t,ψ
s,ψ◦s ξ t,ψ
= ξs − ψ(s) −
)dwu −
a(u, ξ
b(u, ξ s,ψ◦s ξ )du +
Z τ t
Z τ t
t,ψ
t,ψ
b(u, ξ s,ψ◦s ξ )du =
a(u, ξ s,ψ◦s ξ )dwu +
+ ψ(τ ) +
Z τt
Z τt
t,ψ
s,ψ◦s ξ t,ψ
b(u, ξ s,ψ◦s ξ )du.
)dwu +
= ψ(τ ) +
a(u, ξ
t

t

t,ψ

This equality is valid, since ξut,ψ = ξus,ψ◦s ξ for t ≤ u ≤ s. By the uniqueness
t,ψ
of a solution it follows that ξτt,ψ = ξτψ◦s ξ .
t,ψ
Theorem 3.1. θst,ψ ≡ ψ ◦s ξ t,ψ = Lψ
) is a W -valued Markov family
s (ξ
of processes.
t,ψ

Proof. Suppose f ∈ CB(W ). By Lemma 3.1 ψ ◦s ξ t,ψ = ψ ◦s ξ s,ψ◦s ξs , i.e.,
t,ψ
θτt,ψ = ψ ◦s ξ s,ψ◦s ξs , t ≤ s ≤ τ , and
s,θst,ψ

E[f (θτt,ψ )|θst,ψ ] = E[f (θτ

)|θst,ψ ] = E[f (θτs,ψ )]ψ=θss,ψ .

On the other hand, (1.1) is fulfilled, i.e., E[f (θτt,ψ )|θst,ψ ] = Ef (θτs,ψ )ψ=θss,ψ ,
since
σ(θst,ψ ) = σ(ξut,ψ ,

t ≤ u ≤ s) = σ(θut,ψ , t ≤ u ≤ s).

Now we shall describe infinitesimal operators of Markov family of processes. Here jt will denote 1[t,∞) .

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

371

Theorem 3.2. Suppose that the bounded function f : R+ ×L2 (R+ ) → R
∂
∂2
has the continuous bounded derivatives ∂t
f (t, ψ), ∂j∂s f (t, ψ), ∂j
2 f (t, ψ) for
s
each t, s ∈ R+ . Then
Z sh
i
∂
t,ψ
f (s, θs ) − f (t, ψ) =
f (u, θut,ψ ) + A(u)F (u, θut,ψ ) du +
∂u
Z s t
∂
a(u, θut,ψ )
+
f (u, θut,ψ )dwu .
∂j
u
t
Proof. It can be obtained immediately by the theorem 2.3 and the representation
Z s
Z s
ju a(u, ψ)dwu +
θst,ψ = ψs +
ju b(u, ψ)du.
t

t

Let us denote by CBJ1,2 (R+ × W ) a class of function satisfying the conditions of Theorem 3.2.
Lemma 3.2. Suppose f ∈ CB(R+ × W ), f (n) (t, x) = f (t, xn ), where xn
is defined by (2.9). Then f (n) (t, x) → f (t, x), (t, x) ∈ R+ × W , and {f n } is
the uniformly bounded family.
Proof. The results of [8] imply that xn (t) → x(t) uniformly on each segment
[a, b], i.e., xn → x in W . By the continuity of f we have f (n) (t, x) → f (t, x),
f (t, x) ≤ kf k∞ .
Corollary. CBJ1,2 (R+ × W ) dense in CBJ (R+ × W ) in the topology of
bounded pointwise convergence.
Proof. It is sufficient to recall that Gateux differentiable functions in the
Hilbert space H dense in CB(H) [5].
Now we introduce the R+ × W -valued homogeneous Markov family
t,ψ
ηst,ψ = (t + s, θt+s
),

s ≥ 0.

Evidently, by Theorem 2.3 for each f ∈ CBJ1,2 (R+ × W ), we have the
following decomposition
Z s
t,ψ
t,ψ
Lf (u + t, θu+t
)du +
f (ηs ) = f (t, ψ) +
0
Z s
∂f
t,ψ
t,ψ
)dwu .
(u + t, θu+t
)a(u + t, θu+t
+
∂j
u+t
0
Therefore CBJ1,2 belongs to the domain of the generator of the Markov
∂
family {ηst,ψ } and the equation Lf = ( ∂t
+ A(t))f holds for any f ∈
1,2
CBJ (R+ × W ). We will show the solvability of this equatiion.

372

R. TEVZADZE

Lemma 3.3. Let an (t, u), bn (t, u), cn (t), (t, u) ∈ R+ × R, n = 0, 1, . . . ,
be the processes adapted with filtration (Ft ) for each fixed u. Furtermore,
an (t, ·), bn (t, ·) are functions of finite variation. Assume that there exist K
such that
kan (t, ·)k ≤ K,

kbn (t, ·)k ≤ K, E sup |cn (s)| ≤ K,
s≤t

and
kan (t, ·) − a0 (t, ·)kV [0,t] → 0, kbn (t, ·) − b0 (t, ·)kV [0,t] → 0,
E sup |cn (s) − c0 (s)| → 0
s≤t

in probability. Let ζn , n = 0, 1, . . . , be solutions of
Z tZ
Z tZ s
ζn (t) = cn (t) +
an (s, du)ζn (u)dws +
0

0

0

s

bn (s, du)ζn (u)ds.

0

Then Ekζn − ζ0 kt2 → 0.
Proof. We shall use the proof from [9]. Using the inequality (a + b + c)2 ≤
3(a2 + b2 + c2 ) we obtain
E sup |ζn (τ ) − ζ0 (τ )|2t ≤
s≤t

ŒZ s
Œ
Œ
Œ2
Œ
Œ
(an (s, du)ζn (du) − a0 (s, du)ζ0 (u))dws Œ +
≤ 3E sup
Œ
τ ≤t 0
0
Œ2
Z τ ŒZ s
Œ
Œ
Œ
Œ
+ 3E sup
Œ (bn (s, du)ζn (du)−b0 (s, du)ζ0 (u))dsŒ +
Z

τ ≤t

τ

0

0

+ 3E sup |cn (t)−c0 (t)|2 .
τ ≤t

By Doob’s inequality we have
Œ2
Z τ ŒZ s
Œ
Œ
Œ
Œ
E sup
(a
du)ζ
(u))dw
(s,
(s,
(du)
−
a
du)ζ
0
n
n
0
sŒ ≤
Œ
τ ≤t

0

0

Œ2
Z t ŒZ s
Œ
Œ
Œ
(an (s, du)ζn (du) − a0 (s, du)ζ0 (u))ŒŒ ds,
≤ 4E
Œ
0

i.e.,

0

E sup |ζn (τ ) − ζ0 (τ )|2 ≤ 3E sup |cn (t) − c0 (t)|2 +
τ ≤t

τ ≤t

Œ2
Z tŒZ s
Œ
Œ
Πds +
Œ
(s,
(u)
(a
(s,
du)ζ
−
a
(u))
du)ζ
+ 12E
0
0
n
n
Œ
Œ
0

0

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

+ 3tE

373

Œ
Z tŒZ s
Œ
Œ2
Œ
Œ
(u)
b
(b
(s,
du)ζ
−
(s,
du)ζ
(u))
n
0
n
0
Œ
Œ ds ≤
0

0

≤ 3E sup |cn (τ ) − c0 (τ )|2 +
τ ≤t

Œ2
Z tŒZ s
Œ
Œ
Œ
Πds +
(s,
a
du)
−
[(a
(s,
du)]ζ
(u))
n
0
0
Œ
Œ
0
0
Œ
Œ
Z t Z s
Œ2
Œ
Πds +
Œ
(u))
−
du)]ζ
(s,
du)
b
(s,
+ 6tE
[b
0
0
n
Œ
Œ
0
0
Œ2
Z tŒZ s
Œ
Œ
Œ
+ 24E
an (s, du)[ζn (u) − ζ0 (u)]ŒŒ ds +
Œ
+ 24E

0

Z tŒZ
Œ
Œ
+ 6tE
Œ
0

0
s

0

Œ2
Œ
Œ
bn (s, du)(ζn (u) − ζ0 (u))Œ ds ≤

≤ 3E sup |cn (τ ) − c0 (τ )|2 +
τ ≤t

Œ
Z tŒZ s
Œ
Œ2
Œ
Πds +
a
du)[ζ
(u)]
(s,
(u)
−
ζ
n
n
0
Œ
Œ
0
0
Œ2
Z tŒZ s
Œ
Œ
Œ
+ 6tE
bn (s, du)(ζn (u) − ζ0 (u))ŒŒ ds ≤
Œ
0
0
Z t
2
≤ δn (t) + 6K t
E sup |ζn (u) − ζ0 (u)|2 ds +
+ 24E

0

+ 24K 2

Z

τ ≤s

t

E sup |ζn (u) − ζ0 (u)|2 ds,
τ ≤s

0

where
δn (t) = 3E sup |cn (s) − c0 (s)| +
s≤t
2

+ 24E sup |ζ0 (s)| E
s≤t

+ 6tE sup |ζ0 (s)|2 E
s≤t

Z

t

0
Z t

kan (s, ·) − a0 (s, ·)k2 ds +
kbn (s, ·) − b0 (s, ·)k2 ds.

0

Evidently, δn (t) → 0, t ≥ 0. By Gronwall’s lemma
E|ζn (s) − ζ0 (s)|2 ≤ sup |δn (s)|2 eHt ,
s≤t

where H = 24K 2 + 6K 2 t.
Theorem 3.3. Let a, b ∈ CB 0,2 (R+ ×W ). Then the solution of equation
(3.1) is Frechet differeniable w.r.t. ψ for each t, s. Moreover, Y (s, u) =

374

R. TEVZADZE

∂ t,ψ
∂ju ξs

satisfies the equation

ds Y (s, u) = (∇a(s, ξ), Y (·, u))dws + (∇b(s, ξ), Y (·, u))ds,

s ≥ u,

Y (u, u) = 1,
Y (s, u) = 0,

(3.3)

s < u.

Proof. Let t, ψ be fixed. Denote ζsϕ,ǫ = 1ǫ [ζst,ψ+ǫϕ − ζst,ψ ] for any direction
ϕ. Then
Z s
1
ϕ,ǫ
[a(u, ξ t,ψ+ǫϕ ) − a(u, ξ t,ψ )]dwu +
ζs =ψs + jt
ǫ
Z st
1
+ jt
[b(u, ξ t,ψ+ǫϕ ) − b(u, ξ t,ψ )]du.
ǫ
t
Using the mean value theorem we obtain
Z s
(∇a(u, ξ t,ψ + ǫθζ ϕ,ǫ ), ζ ϕ,ǫ )dwu +
ζsψ,ǫ =ψs + jt
Z st
(∇b(u, ξ t,ψ + ǫθζ ϕ,ǫ ), ζ ϕ,ǫ )du
+ jt
t

for some θ, 0 ≤ θ ≤ 1. Introduce the notation an = ∇a(u, ξ t,ψ + ǫn ζ ϕ,ǫn ),
bn = ∇b(u, ξ t,ψ + ǫn ζ ϕ,ǫn ), where ǫn → 0. By Lemma 3.3 it follows that
∂ t,ψ
ξ and consequently sups≤t [ǫn ζsϕ,ǫn ] → 0. Thus an → ∇a(u, ξ t,ψ ),
ζ ϕ,ǫn → ∂ϕ
∂ t,ψ
bn → ∇b(u, ξ t,ψ ) as n → ∞. Evidently ( ∂ϕ
ξ )(s) = ϕ(s), s < t. It remains
to take ϕ = ju . The existence of second derivatives at ψ is proved in a
similar way [9].
Theorem 3.4. Let the conditions of Theorem 3.3 hold and η ∈ C 2 (W ).
Then u(t, ψ) = Eη(ξ t,ψ ) belongs to C 1,2 (R+ × W ) and satisfies
∂
‘
+ A(t) u(t, ψ) = 0, lim u(t, ψ) = η(ψ).
t→∞
∂t
Proof. The differentiability of the functions u(t, ψ) w.r.t. ψ follows from
Theorem 3.2. We have
u(t, ψ) = Eη(ξ t,ψ ) = Eη(ξ s,ψ◦s ξ

t,ψ

) = E[η(ξ s,ψ◦s y )|y=ξt,ψ ] = Eu(s, ψ ◦s ξ t,ψ ),

y ∈ W , i.e., u(t, ψ) = Eu(s, θst,ψ ). Let s = t + h. By Itˆo’s formula
t,ψ
u(t + h, θt+h
) − u(t + h, ψ) =
Z t+h h
1
∂2 i
∂
=
+ a(s, θst,ψ )2 2 u(t + h, θst,ψ )ds +
b(s, θst,ψ )
∂js
2
∂js
t
Z t+h
∂
a(s, θst,ψ )
+
u(t + h, θst,ψ )dws .
∂j
s
t

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

375

Consequently,
t,ψ
) − u(t + h, ψ) =
u(t, ψ) − u(t + h, ψ) = Eu(t + h, θt+h

+E[b(s′ , θst,ψ
′ )

2
1
∂
2 ∂
u(t + h, θst,ψ
a(s′ , θst,ψ
u(t + h, θst,ψ
′ ) +
′ )]h,
′ )
∂js
2
∂js2

for some s′ , t ≤ s′ ≤ t + h, i.e.,
1
(u(t, ψ) − u(t + h, ψ)) → A(t)u(t, ψ),
h

h → 0,

∂
and ( ∂t
+ A(t))u(t, ψ) = 0. Evidently, ξ t,ψ → ψ when t tends to infinity.

Remark 5. Applying a similar reasoning, one can prove the solvability of
the Cauchy problem
∂
‘
+ A(t) − r(t, ϕ) u(t, ϕ) = 0, lim u(t, ϕ) = η(ϕ),
t→∞
∂t

where r ∈ C 0,2 (R+ × W ).

4. Applications
The obtained results allow us to derive a representation formula for functionals of a diffusion type process (Clark’s formula) [1]. However our conditions will be stronger than those in [1] and more general than those in the
recent work [2].
Theorem 4.1. Let a(t, ψ), b(t, ψ), η(ψ) be the functions with bounded
continuous Frechet derivatives of first and second order w.r.t. ψ ∈ W [0, 1].
Suppose (ξt )t∈[0,1] is a solution of (1.2). Then
E[η(ξ)|Ft ] = E[η(ξ)] +

Z

t

E
0

”Z

1
u

•
∇η(ξ, ds)Y (s, u)|Fu a(u, ξ)dwu , (4.1)

where Y satisfies (3.3).
Proof. By the Markov property we have E[η(ξ)|Ft ] = V (t, Ct ξ), where
V (t, ψ) = Eη(ξ t,ψ ). By Theorem 3.4 and Itˆo’s formula for V (t, Ct ξ) we
have
Z t
∂
V (u, Cu ξ)a(u, ξ)dwu .
V (t, Ct ξ) = V (0, ξ0 ) +
0 ∂ju
By Theorem 3.3

∂
V (u, ψ) = E(∇η(ξ), Y (·, u)) = E
∂ju

Z

1

u

∇η(ξ, ds)Y (s, u).

376

R. TEVZADZE

Using Corollary 2 of Theorem 2.2 we can write
•
”Z 1
∂
∇η(ξ, ds)Y (s, u)|Fu .
V (u, Cu ξ) = E
∂ju
u
Thus relation (4.1) is valid.
The other application refers to financial mathematics. Suppose that the
stock price process St satisfies
dSt = µ(t, S)dt + σ(t, S)dwt
and the bond price process satisfies
dBt = r(t, S)Bt dt.
Also suppose that g(S) is a contingent claim under the stock St with delivery
time 1.
The portfolio process (α(t, S), β(t, S)), where α denotes the number of
stocks and β the number of bonds, is called a self-financing if the wealth
process h(t, S) = α(t, S)St + β(t, S)Bt can be represented as
Z t
Z t
h(t, S) = h(0, S0 ) +
β(s, S)dBs .
α(s, S)dSs +
0

0

The process (α(t, S), β(t, S)), is called a replicating portfolio for the contingent claim g(S), if, additionaly, h(1, S) = g(S) [10].
Theorem 4.2. Suppose that σ(t,ψ), r(t,ψ) belongs to C 0,2 ([0,1]×W [0,1])
and g(ψ) belongs to BC 2 (W [0, 1]). Then there exists a replicating portfolio
process (αt , βt ) whose wealth process is a solution of the Cauchy problem
∂
∂
h(t, ψ) +
h(t, ψ) + ψt r(t, ψ)
∂t
∂jt
1
∂2
+ σ(t, ψ)2 2 h(t, ψ) − r(t, ψ)h(t, ψ) = 0,
2
∂jt
h(1, ψ) = g(ψ).
Moreover, α(t, ψ) =

(4.2)

∂
∂jt h(t, ψ).

Proof. Let h(t, ψ) be a solution of problem (4.2) existing by virtue of Theorem 3.4. Define the portfolio process by
‘
1 
∂
∂
α(t, S) =
h(t, S), β(t, S) =
h(t, S) .
h(t, S) − St
∂jt
Bt
∂jt

MARKOV DILATION OF DIFFUSION TYPE PROCESSES

377

This strategy replicates the contingent claim, since h(1, S) = g(S). To
verify the self-financing property we perform the transformation
Z t
Z t
βu dBu =
αu dSu +
0
0
Z t
Z t
∂
∂
µ(u, S)
h(u, ψ)σ(u, S)dwu +
h(u, ψ)du +
=
∂j
∂j
u
u
0
0
Z t
∂
h(u, ψ))r(u, S)du =
(h(u, S) − Su
+
∂j
u
0
Z t
∂
h(u, ψ)dwu ,
σ(u, S)
=
∂j
u
0
Z t
∂
[h(u, S)r(u, S) + (µ(u, S) − Su r(u, S))
h(u, ψ)]du =
∂j
u
0
Z t
∂
h(u, ψ)dwu +
=
σ(u, S)
∂j
u
0
Z t
+
[h(u, S)r(u, S) + (Aµ (u) − Ar (u))h(u, S)]du =
0
Z t
Z t
∂
σ(u, S)
h(u, ψ)dwu +
Aµ (u)h(u, S)du =
=
∂ju
0
0
= h(t, S) − h(0, S0 ),
where we use the notation
1
∂2
∂
+ σ(u, S)2 2 ,
∂ju
2
∂ju
∂
1
∂2
+ σ(u, S)2 2 .
Ar (s) = r(u, S)Su
∂ju
2
∂ju

Aµ (s) = µ(u, S)

The equality is valid by Itˆo’s formula.
Acknowledgement
This research was supported by the Grant 1.26 of the Georgian Academy
of Sciences.

References
1. U. G. Haussman, Functionals of Itˆo’s processes as stochastic integrals.
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(1997), No. 2, 995–1010.

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(Received 21.08.1997)
Institute of Cybernetics
Georgian Academy of Sciences
5, S. Euli St., Tbilisi 380086
Georgia