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 = (ξ_{t})_{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} = (ξ_{t∧s}, s ∈ [0, 1])

has a Markov property, i.e., for any Borel set B in D[0, 1]

P [ξ^{t} ∈ B|ξ^{t_{1}}, ξ^{t_{2}}, …, ξ^{t_{n}}] = P [ξ^{t} ∈ B|ξ^{t_{n}}], 0 ≤ t_{1} ≤ ⋯ ≤ t_{n} ≤ 1, (1.1)

since σ-fields σ(ξ^{t}) = σ(ξ_{s}, 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.)

dξ_{t} = a(t, ξ)dw_{t} + b(t, ξ)dt,  (1.2)

where a(t, x), b(t, x) are nonanticipative functionals and w = (w_{t})_{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 Itô’s formula for

363 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 Itô’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].

1991 Mathematics Subject Classification. 60H05, 60H07.

Key words and phrases. Markov dilation, infinitesimal operator, Itô’s formula, replicating portfolio.

1072-947X/99/0700-0363$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

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, L_{m}^{2}[0, 1] a space of square integrable functions w.r.t. measure m, i_{t} = 1_{[0,t)}, j_{t} = 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:

C_{t}x = xi_{t} + x(t)j_{t}, C_{t} : D[0, 1] → D[0, 1] ⊂ L_{m}^{2}[0, 1],

R_{t}x = x(t)j_{t}, R_{t} : D[0, 1] → D[0, 1] ⊂ L_{m}^{2}[0, 1],

L_{t}^{ψ}x(s) ≡ ψ ∘_{t} x(s) = \begin{cases}ψ(s) − ψ(t) + x(t) & if s > t, \\ x(s) & if s < t, \end{cases}

ψ ∈ D[0, 1], L_{t}^{ψ} : D[0, 1] → D[0, 1] ⊂ L_{m}^{2}[0, 1],

Q_{t}x(s) = \begin{cases}x(0) & if s > t, \\ x(t) − x(s) + x(0) & if s ≤ t, \end{cases}

Q_{t} : D[0, 1] → D[0, 1] ⊂ L_{m}^{2}[0, 1].

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 ∥x − y∥ =

∑_{k=1}^{∞} 2^{−k} sup_{s≤k}{|x(s) − y(s)|, 1} and spaces CB(R_{+} × W), CB^{i,k}(R_{+} × W)

defined in the same way.

2. Repesentation of a Diffusion Type Process in the Space L_{m}^{2}[0, 1]

Let (Ω, F, P) be a probabilist space with filtration (F_{t})_{t∈[0,1]}. Let (w_{t}, F_{t})_{t∈[0,1]} be a Wiener process and (β_{t}, F_{t})_{t∈[0,1]} a random process with paths from the space D[0, 1]∩V [0, 1]. Denote by I_{β}(g)_{t} and I_{w}(g)_{t} the integral ∫_{0}^{t} g_{s}dβ_{s} and the stochastic integral ∫_{0}^{t} g_{s}dw_{s} respectively, for a process (g_{t}, F_{t}) with values in the Hilbert space satisfy standart condition, which guarantees the existence of those integrales. If β_{t} = t, then I_{β}(g) is shortly denoted by I(g). Then the following theorem is valid.

Theorem 2.1. Let (a_{t}, F_{t}), (b_{t}, F_{t}) be real processes such that

P ( ∫_{0}^{1} |b_{t}|dβ_{t} < ∞) = 1, E ∫_{0}^{1} |a_{t}|^{2}dt < ∞.

Then the identities take place

C(I_{β}(B)) = I_{β}(R(b)), C(I_{w}(a)) = I_{w}(R(a)).

In other words, if ξ_{t} = ∫_{0}^{t} a_{s}dw_{s} + ∫_{0}^{t} b_{s}dβ_{s}, then C_{t}ξ = ∫_{0}^{t} R_{s}(a)dw_{s} + ∫_{0}^{t} R_{s}(b)dβ_{s}.

Proof. Let ψ ∈ L_{m}^{2}[0, 1]. Then

(ψ, C_{t}(I_{β}(b))) = ∫_{0}^{t} ψ_{s} ∫_{0}^{s} b_{u}dβ_{u}dm_{s} + ∫_{t}^{1} ψ_{s} ∫_{0}^{t} b_{u}dβ_{u}dm_{s} =
= ∫_{0}^{t} ∫_{0}^{t} ψ_{s}b_{u}i_{s}(u)dβ_{u}dm_{s} + ∫_{t}^{1} ∫_{0}^{t} ψ_{s}b_{u}dβ_{u}dm_{s} =

= ∫_{0}^{t} ∫_{0}^{t} ψ_{s}b_{u}1_{[u,1]}(s)dβ_{u}dm_{s} + ∫_{t}^{1} ∫_{0}^{t} ψ_{s}b_{u}dβ_{u}dm_{s} =

= ∫_{0}^{t} b_{u} ∫_{0}^{t} ψ_{s}j_{u}(s)dm_{s}dβ_{u} + ∫_{0}^{t} b_{u} ∫_{t}^{1} ψ_{s}dm_{s}dβ_{u} =
= ∫_{0}^{t} b_{u} ∫_{u}^{1} ψ_{s}dm_{s}dβ_{u} = ∫_{0}^{t}(ψ, R_{u}(b))dβ_{u} =
= (ψ, ∫_{0}^{t} R_{u}(b)dβ_{u}).

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

∫_{0}^{t} dm_{s} ∫_{0}^{t} ψ_{s}a_{u}j_{s}(u)dw_{u} = ∫_{0}^{t} ∫_{0}^{t} ψ_{s}a_{u}j_{s}(u)dm_{s}dw_{u}

is true by Fubini’s theorem for stochastic integrals [3, p. 217]. In particular, we must take

Ω̃ = [0, 1], F̃_{t} = B[0, 1], p̃ = m, g_{s}(w, w̃) = ψ_{u}a_{s}(w)1_{[0,s]}(w̃)

in the equality

∫Ω̃ ∫_{0}^{1} g_{s}(w, w̃)dw_{s}dP̃ = ∫_{0}^{1} ∫Ω̃ g_{s}(w, w̃)dP̃dw_{s}.

Corollary. Let f : [0, 1] × L_{m}^{2}[0, 1] → R be a bounded function with bounded and continuous Frechet derivative f_{t}, ∇_{x}f, ∇_{x}^{2}f and let ξ_{t} = ∫_{0}^{t} a_{s}dw_{s} + ∫_{0}^{t} b_{s}dβ_{s} be the Itô process. Then

f(t, C_{t}ξ) − f(t_{0}, C_{t_{0}}ξ) = ∫_{t_{0}}^{t} [\frac{∂}{∂s}f(s, C_{s}ξ) + b_{s} \frac{∂}{∂j_{s}} f(s, C_{s}ξ) +

+\frac{1}{2}a_{s}^{2} \frac{∂^{2}}{∂j_{s}^{2}} f(s, C_{s}ξ)]ds + ∫_{t_{0}}^{t} a_{s} \frac{∂}{∂j_{s}} f(s, C_{s}ξ)dw_{s},  (2.1)

where \frac{∂}{∂j_{t}} denotes the Gateux derivative in the direction j_{t}.

Proof. Using Itô’s formula for the L_{m}^{2}[0,1]-valued Itô process η_{t} =∫_{0}^{t}a_{s}j_{s}dw_{s}+ ∫_{0}^{t} b_{s}j_{s}ds [4] and taking into account

\begin{aligned}(∇_{x}f(t, x), b_{t}j_{t}) & = b_{t} \frac{∂}{∂j_{t}} f(t, x), \\ ∇_{x}^{2}f(t, x)(a_{t}j_{t}, a_{t}j_{t}) & = a_{t}^{2} \frac{∂^{2}}{∂j_{s}^{2}} f(t, x), \end{aligned}  (2.2)

we obtain (2.1).

About C_{t}ξ, Q_{t}ξ or L_{t}^{ψ}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] × L_{m}^{2}[0, 1] → R be Lipschitz function, satisfying linearly growth condition. Suppose that functions a^{ψ}, b^{ψ} : [0, 1] × W [0, 1] → R defined by

a^{ψ}(t, x) = A(t, L_{t}^{ψ}x), b^{ψ}(t, x) = B(t, L_{t}^{ψ}x)

and ξ_{s}^{t,ψ}, θ_{s}^{t,ψ} denote solutions of the S.D.E.

ξ_{s} = ψ_{s} + j_{t}(s) ∫_{t}^{s} a^{ψ}(u, ξ)dw_{u} + j_{t}(s) ∫_{t}^{s} b^{ψ}(u, ξ)du,  (2.3)
θ_{s} = ψ + ∫_{t}^{s} A(u, θ_{u})j_{u}dw_{u} + ∫_{t}^{s} B(u, θ_{u})j_{u}du,  (2.4)

respectively. Then θ_{s}^{t,ψ} = L_{s}^{ψ}(ξ^{t,ψ}).

Proof. Equation (2.3) has a unique strong solution ([3], [4]) and equation (2.4) has a unique strong solution only in the space L_{m}^{2}[0, 1] [5]. By Theorem 2.1 equation (2.3) gives

C_{s}ξ = ψ_{s} + ∫_{t}^{s} a^{ψ}(u, ξ)j_{u}dw_{u} + ∫_{t}^{s} b^{ψ}(u, ξ)j_{u}du.

Since a_{u}^{ψ}(ξ) = A_{u}(L_{u}^{ψ}ξ), we have

L_{s}^{ψ}ξ = ψ + ∫_{t}^{s} A(u, L_{u}^{ψ}ξ)j_{u}dw_{u} + ∫_{t}^{s} B(u, L_{u}^{ψ}ξ)j_{u}du,

θ_{s}^{t,ψ} = L_{s}^{ψ}(ξ^{t,ψ}) because of the solution is unique.

Corollary 1. Suppose that in addition to the conditions of Theorem 2.2 the functions A, B belong to C^{1,2}([0, 1] × L_{m}^{2}) and η ∈ C^{2}(L_{m}^{2}). Then the Cauchy problem

(\frac{∂}{∂t} + A(t))u(t, φ) ≡ (\frac{∂}{∂t} + b(t, φ)\frac{∂}{∂j_{t}} + \frac{1}{2}a(t, φ)^{2} \frac{∂^{2}}{∂j_{t}^{2}} )u(t, φ) = 0, (2.5)
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).

Corollary 2. ξ_{s} = ξ_{s}^{0,ψ(0)j_{0}} satisfies the S.D.E.

ξ_{s} = ψ(0) + ∫_{0}^{s} a(u, ξ)dw_{u} + ∫_{0}^{s} b(u, ξ)du,

where a(u, ξ) = A(u, C_{u}ξ), b(u, ξ) = B(u, C_{u}ξ), and we have

E[η(ξ)|F_{t}^{ξ}] = Eη(ξ^{t,ψ})|_{ψ=C_{t}ξ}.

Remark 1. If we suppose

ψ = ψ_{0}j_{0}, m = δ_{1}, A(t, x) = Ã(t, x(t)), B(t, x) = B̃(t, x(t)),

then a(t, x) = A(t, C_{t}x) = Ã(t, x(t)), b(t, x) = B(t, C_{t}x) = B̃(t, x(t)).

Theorem 2.3. Suppose that a continuous bounded function

f : [0, 1] × L_{m}^{2}[0, 1] → R

possesses the continuous bounded derivatives \frac{∂}{∂t}f(t,ψ), \frac{∂}{∂j_{t}} f(s,ψ), \frac{∂^{2}}{∂j_{t}^{2}} f(s,ψ) for each t, s ∈ [0, 1]. Then (2.1) holds.

Proof. By a standard way the proof of formula (2.1) reduces to the case ξ_{t} = x_{t_{0}} + (t − t_{0})R_{t_{0}}(b) + (w(t) − w(t_{0}))R_{t_{0}}(a). Evidently, ξ_{t} − x_{t_{0}} = η_{t}j_{t}, g(t, η_{t}) = f(t, ξ_{t}), where g(t, q) = f(t, qx_{t_{0}}j_{t_{0}}), η_{t} = (t − t_{0})b_{t_{0}} + (w(t) − w(t_{0}))a_{t_{0}}. By the Itô formula for scalar case we obtain

g(t, η_{t}) − g(t_{0}, 0) = ∫_{t_{0}}^{t} [\frac{∂}{∂s}g(s, η_{s}) + b_{t_{0}} \frac{∂}{∂q} g, η_{s}) +

+\frac{1}{2}a_{t_{0}}^{2} \frac{∂^{2}}{∂q^{2}} g(s, η_{s})]ds + ∫_{t_{0}}^{t} a_{t_{0}} \frac{∂}{∂q} g(s, η_{s})dw_{s}.

Then the Itô formula is obtained for ξ_{t}, since \frac{∂^{k}}{∂q^{k}} g(t, q) = \frac{∂^{k}}{∂j_{t_{0}}^{k}} f(t, x_{t_{0}} +qj_{t_{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 function, then by the results of [2] both \frac{∂}{∂j_{t}} f(t, C_{t}x), \frac{∂^{2}}{∂j_{t}^{2}} f(t, C_{t}x) are r.c.l.l. in t.

Theorem 2.4. Suppose that f ∈ CB^{1,2}([0, 1] × W [0, 1]), f^{0}(t, x) = \frac{∂}{∂t}f(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 η_{t} = ψ+∫_{t_{0}}^{t} a_{s}j_{s}dw_{s}+∫_{t_{0}}^{t} b_{s}j_{s}ds, where (a_{t}, F_{t}), (b_{t}, F_{t}) satisfy the conditions of Theorem 2.1. Then

f(t, η_{t}) − f(t_{0}, η_{t_{0}}) = ∫_{t_{0}}^{t} [f(s, η_{s}) + b_{s}f^{1}(s, η_{s}) + \frac{1}{2}a_{s}^{2}f^{2}(s, η_{s})]ds +

+ ∫_{t_{0}}^{t} a_{s}f^{1}(s, η_{s})dw_{s}.  (2.8)

Proof. For each h ∈ L^{2}[0, 1] we denote

h_{n}(t) = n ∫_{0}^{t} e^{−n(t−s)}h(s)ds.  (2.9)

Evidently, the mapping L^{2}[0, 1] ∈ h → h_{m} ∈ W [0, 1] is a lineary continuous function and h_{m} → h, m → ∞ in L^{2}[0, 1]. The mapping f^{(n)}(t, x) = f(t, x) is a differentiable w.r.t. t and twice Frechet differetiable w.r.t. x. For h ∈ L^{2}[0, 1] we have

\frac{∂}{∂h}f^{(n)}(t, x) = ∫_{0}^{1} f_{x}(t, x^{n}, ds)h_{n}(s),
\frac{∂^{2}}{∂h^{2}} F^{(n)}(t, x) = ∫_{0}^{1} ∫_{0}^{1} f_{xx}(t, x^{n}, dudv)h_{n}(u)h_{n}(v)

for h = j_{s}, h_{n}(t) = e^{−nt}(e^{nt} − e^{ns}), i.e.,

\frac{∂}{∂j_{t}} f^{(n)}(t, x) → f^{1}(t, x), n → ∞.

Similarly,

\frac{∂^{2}}{∂j_{t}^{2}} f^{(n)}(t, x) → f^{2}(t, x), n → ∞.

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 that f(t, x) = f^{0}(t, x) + ∫_{0}^{t} f^{1}(s, x)dx_{s} for x ∈ V [0, 1]. If f(t, x) = F (t, C_{t}x) then the derivatives f^{0}(t, x), f^{1}(t, x) can be calculated as

∫_{0}^{t} \frac{∂F}{∂s} (s, C_{s}x)ds, \frac{∂F}{∂j_{t}} (t, C_{t}x),

respectively.

Remark 4. It is possible to define the Markov dilation by the operator Q_{t}. Then the infinitesimal operator will have the form \frac{∂}{∂t} + b(t, x)\frac{∂}{∂i_{t}} +

\frac{1}{2}a(t, x)^{2} \frac{∂^{2}}{∂i_{t}^{2}} .

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.

ξ_{s} = ψ_{s} + ∫_{t}^{s} a(u, ξ)dw_{u} + ∫_{t}^{s} b(u, ξ)du, t ≤ s,  (3.1)

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

\begin{matrix}|a(t, x) − a(t, y)|^{2} + |b(t, x) − b(t, y)|^{2} ≤ K∥x − y∥_{t}^{2}, \\ |a(t, x)|^{2} + |b(t, x)|^{2} ≤ K(1 + ∥x∥^{2}). \end{matrix}  (3.2)

Lemma 3.1. Let ξ_{s}^{t,ψ}, s ≥ t be a solution of (3.1). Then

ξ_{τ}^{t,ψ} = ξ_{τ}^{ψ∘_{s}ξ^{t,ψ}}, t ≤ s ≤ τ.

Proof. By inserting φ = ξ^{t,ψ} in the equality

ξ_{τ}^{ψ∘_{s}φ} = ψ ∘_{s} φ(τ) + ∫_{t}^{τ} a(u, ξ^{ψ∘_{s}φ})dw_{u} + ∫_{t}^{τ} b(u, ξ^{ψ∘_{s}φ})du, ψ ∈ W,

and taking into account that

∫_{s}^{τ} g(u, r)dw_{u}|_{r=η} = ∫_{s}^{τ} g(u, η)dw_{u}, η ∈ F_{s},

we have

ξ_{τ}^{s,ψ∘_{s}ξ^{t,ψ}} = ψ(τ) − ψ(s) + ξ_{s}^{t,ψ} +
+ ∫_{s}^{τ} a(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})dw_{u} + ∫_{s}^{τ} b(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})du =
= ξ_{s}^{t,ψ} − ψ(s) − ∫_{t}^{s} a(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})dw_{u} − ∫_{t}^{s} b(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})du +
+ ψ(τ) + ∫_{t}^{τ} a(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})dw_{u} + ∫_{t}^{τ} b(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})du =
= ψ(τ) + ∫_{t}^{τ} a(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})dw_{u} + ∫_{t}^{τ} b(u, ξ^{s,ψ∘_{s}ξ^{t,ψ}})du.

This equality is valid, since ξ_{u}^{t,ψ} = ξ_{u}^{s,ψ∘_{s}ξ^{t,ψ}} for t ≤ u ≤ s. By the uniqueness of a solution it follows that ξ_{τ}^{t,ψ} = ξ_{τ}^{ψ∘_{s}ξ^{t,ψ}}.

Theorem 3.1. θ_{s}^{t,ψ} ≡ ψ ∘_{s} ξ^{t,ψ} = L_{s}^{ψ}(ξ^{t,ψ}) is a W -valued Markov family of processes.

Proof. Suppose f ∈ CB(W). By Lemma 3.1 ψ ∘_{s} ξ^{t,ψ} = ψ ∘_{s} ξ^{s,ψ∘_{s}ξ_{s}^{t,ψ}}, i.e., θ_{τ}^{t,ψ} = ψ ∘_{s} ξ^{s,ψ∘_{s}ξ_{s}^{t,ψ}}, t ≤ s ≤ τ, and

E[f(θ_{τ}^{t,ψ})|θ_{s}^{t,ψ}] = E[f(θ_{τ}^{s,θ_{s}^{t,ψ}})|θ_{s}^{t,ψ}] = E[f(θ_{τ}^{s,ψ})]_{ψ=θ_{s}^{s,ψ}}.

On the other hand, (1.1) is fulfilled, i.e., E[f(θ_{τ}^{t,ψ})|θ_{s}^{t,ψ}] = Ef(θ_{τ}^{s,ψ})_{ψ=θ_{s}^{s,ψ}}, since

σ(θ_{s}^{t,ψ}) = σ(ξ_{u}^{t,ψ}, t ≤ u ≤ s) = σ(θ_{u}^{t,ψ}, t ≤ u ≤ s).

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

Theorem 3.2. Suppose that the bounded function f : R_{+} ×L^{2}(R_{+}) → R has the continuous bounded derivatives \frac{∂}{∂t}f(t, ψ), \frac{∂}{∂j_{s}} f(t, ψ), \frac{∂^{2}}{∂j_{s}^{2}} f(t, ψ) for each t, s ∈ R_{+}. Then

f(s, θ_{s}^{t,ψ}) − f(t, ψ) = ∫_{t}^{s} [\frac{∂}{∂u} f(u, θ_{u}^{t,ψ}) + A(u)F (u, θ_{u}^{t,ψ})]du +

+ ∫_{t}^{s} a(u, θ_{u}^{t,ψ}) \frac{∂}{∂j_{u}} f(u, θ_{u}^{t,ψ})dw_{u}.

Proof. It can be obtained immediately by the theorem 2.3 and the representation

θ_{s}^{t,ψ} = ψ_{s} + ∫_{t}^{s} j_{u}a(u, ψ)dw_{u} + ∫_{t}^{s} j_{u}b(u, ψ)du.

Let us denote by CB_{J}^{1,2}(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, x^{n}), where x^{n} 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 x_{n}(t) → x(t) uniformly on each segment [a, b], i.e., x_{n} → x in W. By the continuity of f we have f^{(n)}(t, x) → f(t, x), f(t, x) ≤ ∥f∥_{∞}.

Corollary. CB_{J}^{1,2}(R_{+} × W) dense in CB_{J}(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

η_{s}^{t,ψ} = (t + s, θ_{t+s}^{t,ψ}), s ≥ 0.

Evidently, by Theorem 2.3 for each f ∈ CB_{J}^{1,2}(R_{+} × W), we have the following decomposition

f(η_{s}^{t,ψ}) = f(t, ψ) + ∫_{0}^{s} Lf(u + t, θ_{u+t}^{t,ψ})du +

+ ∫_{0}^{s} \frac{∂f}{∂j_{u+t}} (u + t, θ_{u+t}^{t,ψ})a(u + t, θ_{u+t}^{t,ψ})dw_{u}.

Therefore CB_{J}^{1,2} belongs to the domain of the generator of the Markov family \{η_{s}^{t,ψ}\} and the equation Lf = (\frac{∂}{∂t} + A(t))f holds for any f ∈ CB_{J}^{1,2}(R_{+} × W). We will show the solvability of this equatiion.

Lemma 3.3. Let a_{n}(t, u), b_{n}(t, u), c_{n}(t), (t, u) ∈ R_{+} × R, n = 0, 1, …, be the processes adapted with filtration (F_{t}) for each fixed u. Furtermore, a_{n}(t, ·), b_{n}(t, ·) are functions of finite variation. Assume that there exist K such that

∥a_{n}(t, ·)∥ ≤ K, ∥b_{n}(t, ·)∥ ≤ K, E sup_{s≤t} |c_{n}(s)| ≤ K,

and

∥a_{n}(t, ·) − a_{0}(t, ·)∥_{V [0,t]} → 0, ∥b_{n}(t, ·) − b_{0}(t, ·)∥_{V [0,t]} → 0,

E sup_{s≤t} |c_{n}(s) − c_{0}(s)| → 0

in probability. Let ζ_{n}, n = 0, 1, …, be solutions of

ζ_{n}(t) = c_{n}(t) + ∫_{0}^{t} ∫_{0}^{s} a_{n}(s, du)ζ_{n}(u)dw_{s} + ∫_{0}^{t} ∫_{0}^{s} b_{n}(s, du)ζ_{n}(u)ds.

Then E∥ζ_{n} − ζ_{0}∥_{t}^{2} → 0.

Proof. We shall use the proof from [9]. Using the inequality (a + b + c)^{2} ≤ 3(a^{2} + b^{2} + c^{2}) we obtain

E sup_{s≤t} |ζ_{n}(τ) − ζ_{0}(τ)|_{t}^{2} ≤

≤ 3E sup_{τ≤t} ∫_{0}^{τ} | ∫_{0}^{s}(a_{n}(s, du)ζ_{n}(du) − a_{0}(s, du)ζ_{0}(u))dw_{s}|^{2} +

+ 3E sup_{τ≤t} ∫_{0}^{τ} | ∫_{0}^{s}(b_{n}(s, du)ζ_{n}(du)−b_{0}(s, du)ζ_{0}(u))ds|^{2} +

+ 3E sup_{τ≤t} |c_{n}(t)−c_{0}(t)|^{2}.

By Doob’s inequality we have

E sup_{τ≤t} ∫_{0}^{τ} | ∫_{0}^{s}(a_{n}(s, du)ζ_{n}(du) − a_{0}(s, du)ζ_{0}(u))dw_{s}|^{2} ≤

≤ 4E ∫_{0}^{t} | ∫_{0}^{s}(a_{n}(s, du)ζ_{n}(du) − a_{0}(s, du)ζ_{0}(u))|^{2}ds,

i.e.,

E sup_{τ≤t} |ζ_{n}(τ) − ζ_{0}(τ)|^{2} ≤ 3E sup_{τ≤t} |c_{n}(t) − c_{0}(t)|^{2} +

+ 12E ∫_{0}^{t} | ∫_{0}^{s}(a_{n}(s, du)ζ_{n}(u) − a_{0}(s, du)ζ_{0}(u))|^{2}ds +

MARKOV DILATION OF DIFFUSION TYPE PROCESSES 373

+ 3tE ∫_{0}^{t} | ∫_{0}^{s}(b_{n}(s, du)ζ_{n}(u) − b_{0}(s, du)ζ_{0}(u))|^{2}ds ≤

≤ 3E sup_{τ≤t} |c_{n}(τ) − c_{0}(τ)|^{2} +

+ 24E ∫_{0}^{t} | ∫_{0}^{s}[(a_{n}(s, du) − a_{0}(s, du)]ζ_{0}(u))|^{2}ds + + 6tE ∫_{0}^{t} | ∫_{0}^{s}[b_{n}(s, du) − b_{0}(s, du)]ζ_{0}(u))|^{2}ds + + 24E ∫_{0}^{t} | ∫_{0}^{s} a_{n}(s, du)[ζ_{n}(u) − ζ_{0}(u)]|^{2}ds + + 6tE ∫_{0}^{t} | ∫_{0}^{s} b_{n}(s, du)(ζ_{n}(u) − ζ_{0}(u))|^{2}ds ≤

≤ 3E sup_{τ≤t} |c_{n}(τ) − c_{0}(τ)|^{2} +

+ 24E ∫_{0}^{t} | ∫_{0}^{s} a_{n}(s, du)[ζ_{n}(u) − ζ_{0}(u)]|^{2}ds + + 6tE ∫_{0}^{t} | ∫_{0}^{s} b_{n}(s, du)(ζ_{n}(u) − ζ_{0}(u))|^{2}ds ≤

≤ δ_{n}(t) + 6K^{2}t ∫_{0}^{t} E sup_{τ≤s} |ζ_{n}(u) − ζ_{0}(u)|^{2}ds +

+ 24K^{2} ∫_{0}^{t} E sup_{τ≤s} |ζ_{n}(u) − ζ_{0}(u)|^{2}ds,

where

δ_{n}(t) = 3E sup_{s≤t} |c_{n}(s) − c_{0}(s)| +

+ 24E sup_{s≤t} |ζ_{0}(s)|^{2}E ∫_{0}^{t} ∥a_{n}(s, ·) − a_{0}(s, ·)∥^{2}ds +

+ 6tE sup_{s≤t} |ζ_{0}(s)|^{2}E ∫_{0}^{t} ∥b_{n}(s, ·) − b_{0}(s, ·)∥^{2}ds.

Evidently, δ_{n}(t) → 0, t ≥ 0. By Gronwall’s lemma

E|ζ_{n}(s) − ζ_{0}(s)|^{2} ≤ sup_{s≤t} |δ_{n}(s)|^{2}e^{Ht},

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) = \frac{∂}{∂j_{u}} ξ_{s}^{t,ψ} satisfies the equation

d_{s}Y (s, u) = (∇a(s, ξ), Y (·, u))dw_{s} + (∇b(s, ξ), Y (·, u))ds, s ≥ u,
Y (u, u) = 1,  (3.3)
Y (s, u) = 0, s < u.

Proof. Let t, ψ be fixed. Denote ζ_{s}^{φ,ϵ} = \frac{1}{ϵ}[ζ_{s}^{t,ψ+ϵφ} − ζ_{s}^{t,ψ}] for any direction φ. Then

ζ_{s}^{φ,ϵ} =ψ_{s} + j_{t} ∫_{t}^{s} \frac{1}{ϵ}[a(u, ξ^{t,ψ+ϵφ}) − a(u, ξ^{t,ψ})]dw_{u} +

+ j_{t} ∫_{t}^{s} \frac{1}{ϵ}[b(u, ξ^{t,ψ+ϵφ}) − b(u, ξ^{t,ψ})]du.

Using the mean value theorem we obtain

ζ_{s}^{ψ,ϵ} =ψ_{s} + j_{t} ∫_{t}^{s}(∇a(u, ξ^{t,ψ} + ϵθζ^{φ,ϵ}), ζ^{φ,ϵ})dw_{u} +
+ j_{t} ∫_{t}^{s}(∇b(u, ξ^{t,ψ} + ϵθζ^{φ,ϵ}), ζ^{φ,ϵ})du

for some θ, 0 ≤ θ ≤ 1. Introduce the notation a_{n} = ∇a(u, ξ^{t,ψ} + ϵ_{n}ζ^{φ,ϵ_{n}}), b_{n} = ∇b(u, ξ^{t,ψ} + ϵ_{n}ζ^{φ,ϵ_{n}}), where ϵ_{n} → 0. By Lemma 3.3 it follows that ζ^{φ,ϵ_{n}} → \frac{∂}{∂φ}ξ^{t,ψ} and consequently sup_{s≤t}[ϵ_{n}ζ_{s}^{φ,ϵ_{n}}]→0. Thus a_{n} →∇a(u, ξ^{t,ψ}), b_{n} → ∇b(u, ξ^{t,ψ}) as n → ∞. Evidently (\frac{∂}{∂φ}ξ^{t,ψ})(s) = φ(s), s < t. It remains to take φ = j_{u}. 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

(\frac{∂}{∂t} + A(t))u(t, ψ) = 0, lim_{t→∞} u(t, ψ) = η(ψ).

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=ξ^{t,ψ}}] = Eu(s, ψ ∘_{s} ξ^{t,ψ}),

y ∈ W, i.e., u(t, ψ) = Eu(s, θ_{s}^{t,ψ}). Let s = t + h. By Itô’s formula

u(t + h, θ_{t+h}^{t,ψ}) − u(t + h, ψ) =

= ∫_{t}^{t+h} [b(s, θ_{s}^{t,ψ})\frac{∂}{∂j_{s}} + \frac{1}{2}a(s, θ_{s}^{t,ψ})^{2} \frac{∂^{2}}{∂j_{s}^{2}} ]u(t + h, θ_{s}^{t,ψ})ds +

+ ∫_{t}^{t+h} a(s, θ_{s}^{t,ψ})\frac{∂}{∂j_{s}} u(t + h, θ_{s}^{t,ψ})dw_{s}.

Consequently,

u(t, ψ) − u(t + h, ψ) = Eu(t + h, θ_{t+h}^{t,ψ}) − u(t + h, ψ) =

+E[b(s′, θ_{s′}^{t,ψ})\frac{∂}{∂j_{s}} u(t + h, θ_{s′}^{t,ψ}) + \frac{1}{2}a(s′, θ_{s′}^{t,ψ})^{2} \frac{∂^{2}}{∂j_{s}^{2}} u(t + h, θ_{s′}^{t,ψ})]h,

for some s′, t ≤ s′ ≤ t + h, i.e.,

\frac{1}{h}(u(t, ψ) − u(t + h, ψ)) → A(t)u(t, ψ), h → 0,

and (\frac{∂}{∂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

(\frac{∂}{∂t} + A(t) − r(t, φ))u(t, φ) = 0, lim_{t→∞} u(t, φ) = η(φ),

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 (ξ_{t})_{t∈[0,1]} is a solution of (1.2). Then

E[η(ξ)|F_{t}] = E[η(ξ)] + ∫_{0}^{t} E[ ∫_{u}^{1} ∇η(ξ, ds)Y (s, u)|F_{u}]a(u, ξ)dw_{u}, (4.1)

where Y satisfies (3.3).

Proof. By the Markov property we have E[η(ξ)|F_{t}] = V (t, C_{t}ξ), where V (t, ψ) = Eη(ξ^{t,ψ}). By Theorem 3.4 and Itô’s formula for V (t, C_{t}ξ) we have

V (t, C_{t}ξ) = V (0, ξ_{0}) + ∫_{0}^{t} \frac{∂}{∂j_{u}} V (u, C_{u}ξ)a(u, ξ)dw_{u}.

By Theorem 3.3

\frac{∂}{∂j_{u}} V (u, ψ) = E(∇η(ξ), Y (·, u)) = E ∫_{u}^{1} ∇η(ξ, ds)Y (s, u).

Using Corollary 2 of Theorem 2.2 we can write

\frac{∂}{∂j_{u}} V (u, C_{u}ξ) = E[ ∫_{u}^{1} ∇η(ξ, ds)Y (s, u)|F_{u}].

Thus relation (4.1) is valid.

The other application refers to financial mathematics. Suppose that the stock price process S_{t} satisfies

dS_{t} = μ(t, S)dt + σ(t, S)dw_{t}

and the bond price process satisfies

dB_{t} = r(t, S)B_{t}dt.

Also suppose that g(S) is a contingent claim under the stock S_{t} 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)S_{t} + β(t, S)B_{t} can be represented as

h(t, S) = h(0, S_{0}) + ∫_{0}^{t} α(s, S)dS_{s} + ∫_{0}^{t} β(s, S)dB_{s}.

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 (α_{t}, β_{t}) whose wealth process is a solution of the Cauchy problem

\begin{matrix}\frac{∂}{∂t}h(t, ψ) + ψ_{t}r(t, ψ)\frac{∂}{∂j_{t}} h(t, ψ) + \\ +\frac{1}{2}σ(t, ψ)^{2} \frac{∂^{2}}{∂j_{t}^{2}} h(t, ψ) − r(t, ψ)h(t, ψ) = 0, \\ h(1, ψ) = g(ψ). \end{matrix}  (4.2)

Moreover, α(t, ψ) = \frac{∂}{∂j_{t}} 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

α(t, S) = \frac{∂}{∂j_{t}} h(t, S), β(t, S) = \frac{1}{B_{t}} (h(t, S) − S_{t} \frac{∂}{∂j_{t}} h(t, S)).

This strategy replicates the contingent claim, since h(1, S) = g(S). To verify the self-financing property we perform the transformation

∫_{0}^{t} α_{u}dS_{u} + ∫_{0}^{t} β_{u}dB_{u} =

= ∫_{0}^{t} \frac{∂}{∂j_{u}} h(u, ψ)σ(u, S)dw_{u} + ∫_{0}^{t} μ(u, S)\frac{∂}{∂j_{u}} h(u, ψ)du +

+ ∫_{0}^{t}(h(u, S) − S_{u} \frac{∂}{∂j_{u}} h(u, ψ))r(u, S)du =

= ∫_{0}^{t} σ(u, S)\frac{∂}{∂j_{u}} h(u, ψ)dw_{u},

∫_{0}^{t}[h(u, S)r(u, S) + (μ(u, S) − S_{u}r(u, S))\frac{∂}{∂j_{u}} h(u, ψ)]du =

= ∫_{0}^{t} σ(u, S)\frac{∂}{∂j_{u}} h(u, ψ)dw_{u} +

+ ∫_{0}^{t}[h(u, S)r(u, S) + (A^{μ}(u) − A^{r}(u))h(u, S)]du =

= ∫_{0}^{t} σ(u, S)\frac{∂}{∂j_{u}} h(u, ψ)dw_{u} + ∫_{0}^{t} A^{μ}(u)h(u, S)du =

= h(t, S) − h(0, S_{0}),

where we use the notation

A^{μ}(s) = μ(u, S)\frac{∂}{∂j_{u}} + \frac{1}{2}σ(u, S)^{2} \frac{∂^{2}}{∂j_{u}^{2}} ,
A^{r}(s) = r(u, S)S_{u} \frac{∂}{∂j_{u}} + \frac{1}{2}σ(u, S)^{2} \frac{∂^{2}}{∂j_{u}^{2}} .

The equality is valid by Itô’s formula.

Acknowledgement

This research was supported by the Grant 1.26 of the Georgian Academy of Sciences.

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(Received 21.08.1997)

Institute of Cybernetics Georgian Academy of Sciences 5, S. Euli St., Tbilisi 380086 Georgia