Smoothing Problem in Anticipating Scenario
We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration.
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| author | Dorogovtsev, A. A. Дороговцев, А. А. |
| author_facet | Dorogovtsev, A. A. Дороговцев, А. А. |
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| description | We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration. |
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UDC 519.21
A. A. Dorogovtsev (Inst. Math. Nat. Acad. Sci Ukraine, Kyiv)
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO*
ZADAÇA INTERPOLQCI} DLQ NEUZHODÛENYX ÍUMIV
We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with
Wiener processes which can have not a semimartingale property with respect to the joint filtration.
Rozhlqda[t\sq zadaça interpolqci] dlq vypadkovyx procesiv, wo zadovol\nqgt\ stoxastyçni dyferen-
cial\ni rivnqnnq z vinerovymy procesamy, qki ne [ semimartynhalamy vidnosno spil\no] fil\traci].
0. Introduction. This article is devoted to the stochastic anticipating equations with the
extended stochastic integral with respect to the Gaussian processes of a special type and
its application to the smoothing problem in the case when noise is represented by the two
jointly Gaussian Wiener processes, which can have not a semimartingale property with
respect to the joint filtration. In order to describe the objects of our consideration more
explicitly, consider the following example.
Example 0.1. Consider the ordinary stochastic differential equation in R
dx(t) = a(x(t))dt+ b(x(t))dw(t)
with the smooth enough coefficients a and b. Denote by x(r, s, t) the solution which
starts at the moment s from the point r. Let the function ϕ ∈ C2(R) have bounded
derivatives. For r ∈ R, s ∈ [0;T ], define
Φ(r, s) = Γ(ϕ(r, s, T )), (0.1)
where Γ is the certain operator of the second quantization [1, 2]. In particular, Γ can be
a mathematical expectation. Then it can be proved that Φ satisfies the following partial
stochastic differential equation:
dΦ(s, r) = −
[
1
2
b2(r)
∂2
∂r2
Φ(r, s) + a(r)
∂
∂r
Φ(r, s)
]
ds+
+ b(r)
∂
∂r
Φ(r, s)dγ(s).
Here, γ(s) = Γw(s) and the last differential is treated in the sense of anticipating
stochastic integration. When Γ is the mathematical expectation, the last term vanishes.
This example shows the main goal of this article. Namely, there exist situations
when the naturally arising Wiener functionals satisfy the anticipating stochastic differ-
ential equations and can be described with the using of stochastic calculus. Here we
propose the appropriate machinery and derive the correspondent equations.
We will consider the second quantization transformation of the different Wiener func-
tionals. For such transformed functionals we will get the anticipating stochastic equations
with the extended stochastic integral. Accordingly to this aim the article is organized as
follows. The first part contains the properties of the second quantization operators in con-
nection with the extended stochastic integral or, more generally, with the Gaussian strong
random operators [3]. Sections 2 and 3 are devoted to the following pair of equations:
∗ Research is supported in part by the Ministry of Education and Science of Ukraine (Grant of the Presi-
dent), project GP/F8/0086.
c© A. A. DOROGOVTSEV, 2005
1218 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1219
dx1(t) = a1(x1(t)) + dw1(t),
dx2(t) = a2(x1(t)) + dw2(t),
where w1, w2 are jointly Gaussian Wiener processes, which can have not a semimartin-
gale property with respect to the joint filtration. Here we will look for the equation for
E(f(x1(t))/x2).
1. Second quantization and integrators. The material of this section is partially
based on the works [3 – 5]. Corresponding facts are placed here for the completeness of
the exposition but their proofs are omitted. New claims are presented with the proofs.
We will start here with the abstract picture, when the “white noise” generated by the
Wiener process is substituted by the generalized Gaussian random element in the Hilbert
space. Let H be a separable real Hilbert space with the norm ‖ · ‖ and inner product
(· , ·). Suppose that ξ is the generalized Gaussian random element in H with zero mean
and identical covariation. In other words ξ is the family of jointly Gaussian random
variables denoted by (ϕ, ξ), ϕ ∈ H with the properties:
1) (ϕ, ξ) has the normal distribution with zero mean and variance ‖ϕ‖2 for every
ϕ ∈ H;
2) (ϕ, ξ) is linear with respect to ϕ.
During this section we suppose that all the random variables and elements are mea-
surable with respect to σ(ξ) = σ((ϕ, ξ), ϕ ∈ H.) If the random variable α has the finite
second moment, then α has an Ito – Wiener expansion [6]
α =
∞∑
k=0
Ak(ξ, . . . , ξ). (1.1)
Here, for every k ≥ 1, Ak(ξ, . . . , ξ) is the infinite-dimensional generalization of the
Hermite polinomial from ξ, correspondent to the k -linear symmetric Hilbert – Schmidt
form Ak on H. Moreover, now the following relation holds:
Eα2 =
∞∑
k=0
k!‖Ak‖2
k. (1.2)
Here ‖ · ‖k is the Hilbert – Schmidt form in H⊗k. The same expansion for H-valued
random elements will be necessary. Let x be a random element in H such that
E‖x‖2 < +∞.
Then, for every ϕ ∈ H,
(x, ϕ) =
∞∑
k=0
Ak(ϕ; ξ, . . . , ξ). (1.3)
It can be easily checked using (1.2) that now Ak is the (k + 1)-linear (not necessary
symmetric) Hilbert – Schmidt form. So one can write now
x =
∞∑
k=0
Ãk(ξ, . . . , ξ), (1.4)
where
Ãk(ϕ1, . . . , ϕk) :=
∞∑
j=1
Ak(ej ;ϕ1, . . . , ϕk)ej
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1220 A. A. DOROGOVTSEV
for the arbitrary orthonormal basis {ej ; j ≥ 1} in H and the series (1.4) converges in
H in the square mean. The relation (1.2) remains to be true
E‖x‖2 =
∞∑
k=0
k!‖Ãk‖2
k, (1.5)
where ‖Ãk‖k is the Hilbert – Schmidt norm in the space of H-valued k-linear forms
on H.
Now recall the definition of the operators of the second quantization. Let C be a
continuous linear operator in H. Suppose that the operator norm ‖C‖ ≤ 1. Then for α
and x from (1.1) and (1.4) define
Γ(C)α =
∞∑
k=0
Ak(Cξ, . . . , Cξ),
Γ(C)x =
∞∑
k=0
Ãk(Cξ, . . . , Cξ),
(1.6)
where for k ≥ 1Ak(C·, C·, . . . , C·) and Ãk(C·, C·, . . . , C·) are new Hilbert – Schmidt
forms.
Using the estimation
‖Ak(C·, C·, . . . , C·)‖k ≤ ‖C‖k‖Ak‖k,
it is easy to prove [2], that Γ(C) is a continuous linear operator in the space of square
integrable random variables or elements in H.
Definition 1.1 [2]. Operator Γ(C) is the operator of the second quantization corre-
sponding to the operator C.
Before to consider some examples, we will present the useful representation of the
second quantization operators. Let ξ′ be the generalized Gaussian random element in
H, independent and equidistributed with ξ.
Consider the following generalized Gaussian random element in H :
η =
√
1 − CC∗ξ′ + Cξ. (1.7)
This element can be properly defined by the formula
∀ϕ ∈ H : (ϕ, η) := (
√
1 − CC∗ϕ, ξ′) + (C∗ϕ, ξ). (1.8)
Note that η has zero mean and identity covariation. In order to check this, it is sufficient
to note the relation
‖ϕ‖2 = ‖
√
1 − CC∗ϕ‖2 + ‖C∗ϕ‖2.
For every random variable α with an expansion (1.1), define
α(η) :=
∞∑
k=0
Ak(η, . . . , η).
The following representation will be useful.
Lemma 1.1. For arbitrary α ∈ L2(Ω, σ(ξ),P) and operator C in H with ‖C‖ ≤ 1,
Γ(C)α = E(α(η)/ξ). (1.9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1221
Proof. Note that the both parts of (1.9) are continuous with respect to α in the square
mean. So, it is enough to check (1.9) for the following random variables
e(ϕ,ξ)− 1
2‖ϕ‖2
, ϕ ∈ H. (1.10)
Really, the random variable of this kind has the Ito – Wiener expansion of the form
e(ϕ,ξ)− 1
2‖ϕ‖2
=
∞∑
k=0
1
k!
ϕ⊗k(ξ, . . . , ξ).
Here, ϕ⊗k is k-th tensor power of ϕ which acts on H by the rule
ϕ⊗k(ψ1, . . . , ψk) =
k∏
j=1
(ϕ,ψj).
So, for α, which has an expansion (1.1),
Eαe(ϕ,ξ)− 1
2‖ϕ‖2
=
∞∑
k=0
Ak(ϕ, . . . , ϕ).
Hence, the set of all linear combinations of the variables (1.10) is dense in L2. Now [2]
the following equality holds:
Γ(C)e(ϕ,ξ)− 1
2‖ϕ‖2
= e(C
∗ϕ,ξ)− 1
2‖C∗ϕ‖2
. (1.11)
On the other hand,
e(ϕ,η)− 1
2‖ϕ‖2
= e(C
∗ϕ,ξ)− 1
2‖C∗ϕ‖2
e(
√
1−CC∗ϕ,ξ′)− 1
2‖
√
1−CC∗ϕ‖2
.
In order to finish the proof, it is sufficient now to note that
Ee(
√
1−CC∗ϕ,ξ′)− 1
2‖
√
1−CC∗ϕ‖2
= 1
and that ξ′ and ξ are independent. It follows from here that
E
(
e(ϕ,η)− 1
2‖ϕ‖2
/ξ
)
= e(C
∗ϕ,ξ)− 1
2‖C∗ϕ‖2
.
The lemma is proved.
This lemma has the following useful application for us.
Corollary 1.1. Let Γ(C) be an operator of the second quantization. Let x be a
random element in the complete separable metric space X measurable with respect to ξ.
Then there exists the random probability measure µ on X such that for every bounded
measurable function f : X → R the following equality holds:∫
X
fdµ = Γ(C)f(x).
Proof. Let us define µ as a conditional distribution of x(η) with respect to ξ. Then,
for every bounded measurable f : X → R,∫
X
fdµ = E(f(x(η))/ξ) = Γ(C)f(x).
The unique difficulty on this way lies in the proper definition of x(η) (remind that ξ
and η are not usual random elements). In order to break this difficulty, we will use the
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1222 A. A. DOROGOVTSEV
following analog of the Levy theorem. Let {ej ; j ≥ 1} be an orthonormal basis in H.
Define the sequences of random elements in H by the rule
ξn =
n∑
j=1
(ej , ξ)ej ,
ηn =
n∑
j=1
(ej , η)ej , n ≥ 1.
Note that the sequences {ξn;n ≥ 1} and {ηn;n ≥ 1} are equidistributed. Now for ev-
ery n ≥ 1 consider the random measure νn in X, which is constructed in the following
way:
νn(∆) = E{1I∆(x)/ξn}.
Here, ∆ is an arbitrary Borel subset of X. This random measures have two important
properties. First of all, for every n ≥ 1, νn can be viewed as ν̃n(ξn), where ν̃n is a
Borel function from H to the space of all probability measures on X equipped with the
distance of weak convergence. Secondly, with probability one, νn weakly converge to
δx as n tends to infinity. The last assertion follows from the usual Levy theorem [7].
More precisely, for arbitrary continuous bounded function f : X → R,
f(x) = E(f(x)/ξ) = lim
n→∞
E(f(x)/ξn) =
= lim
n→∞
∫
X
f(u)νn(du) a.s.
Taking f from the countable set which define the weak convergence [8], we get the
required statement. Now note that the sequence of random measures {ν̃n(ηn);n ≥ 1}
is equidistributed with {ν̃n(ξn);n ≥ 1}. Hence, with probability one, there exists the
weak limit of ν̃n(ηn) which is a delta-measure concentrated in the certain random point
y. This random point y is by definition x(η). The correctness of this definition can be
easily checked.
The lemma is proved.
Consider the examples of the random measures, which arise in the application of the
Corollary 1.1 and will be important for us.
Example 1.1. Suppose that H = L2([0;T ],Rd). Define the generalized Gaussian
random element ξ in H with the help of the d-dimensional Wiener process W on
[0;T ]. Namely, for ϕ = (ϕ1, . . . , ϕd) ∈ L2([0; 1],Rd), define
(ϕ, ξ) :=
d∑
j=1
T∫
0
ϕj(s)dWj(s). (1.12)
Now consider the functions a : R
d → R
d and b : R
d → R
d×d which satisfy the
Lipschitz condition and the domain G in R
d with the C1-boundary Γ . Let for every
s ∈ [0;T ] and u ∈ R
d x(u, s, T ) denote the solution at time T of the following
Cauchy problem:
dx(t) = a(x(t))dt+ b(x(t))dW (t),
x(s) = u.
(1.13)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1223
Denote by νu,s the random measure obtained from x(u, s, T ) via Corollary 1.1 with
the help of the certain operator of the second quantization Γ(C). In the next section, we
will obtain the stochastic variant of the Kolmogorov equation for νu,s. Note that in the
case C = 0 measures νu,s became to be deterministic and satisfy the usual Kolmogorov
equation [1].
Now let us define for every u ∈ G the random moment
τu,s = inf{t, t ≤ T : x(u, s, t) ∈ Γ}.
Let µu,s be the random measure obtained from x(u, s, τu,s) via Corollary 1.1. It occurs
that measures µu,s satisfy certain anticipating boundary-value problem.
In order to describe the anticipating SPDE for the random measures from above men-
tioned example, we need in the relation between the operators of the second quantization
and extended stochastic integral. We will study this connection in the more general situ-
ation when the extended stochastic integral is substituted by the general Gaussian strong
random operator (GSRO in the sequel). Let us recall the following definition.
Definition 1.2 [3]. The Gaussian strong random linear operator (GSRO) A in H
is the mapping, which maps every element x of H into the jointly Gaussian with ξ
random element in H and is continuous in the square mean.
As an example of GSRO the integral with respect to Wiener process can be considered.
Example 1.2. Consider H and ξ from Example 1.1. Let for simplicity d = 1.
Define GSRO A in the following way
∀ϕ ∈ H : (Aϕ)(t) =
t∫
0
ϕ(s)dw(s), t ∈ [0;T ].
It can be easily seen that Aϕ now is a Gaussian random element in H, and A is con-
tinuous in square mean.
In order to include in this picture the integration with respect to another Gaussian
processes (for example, with respect to the fractional Brownian motion), consider more
general GSRO. Suppose that K is a bounded linear operator, which acts from L2([0;T ])
to L2([0;T ]2). Define
∀ϕ ∈ H : (Aϕ)(t) =
T∫
0
(Kϕ)(t, s)dw(s).
It can be checked that A is GSRO in H. Making an obvious changes, one can define
the GSRO acting from the different Hilbert space H1 into H. For example, consider for
α ∈
(
1
2
; 1
)
the covariation function of the fractional Brownian motion [9] with Hurst
parameter α
R(s, t) =
1
2
(t2α + s2α − |t− s|2α).
Define the space H1 as a completion of the set of step functions on [0;T ] with respect
to the inner product under which
(1I[0;s], 1I[0;t]) = R(s, t).
Consider the kernel Kα from the integral representation of the fractional Brownian mo-
tion Bα [10]
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1224 A. A. DOROGOVTSEV
Bα(t) =
t∫
0
Kα(t, s)dw(s)
and
∂Kα
∂t
(t, s) = cα
(
α− 1
2
)
(t− s)α − 3
2
(s
t
) 1
2−α
.
Define for ϕ ∈ H1
(Kϕ)(t, s) =
t∫
s
ϕ(r)
∂Kα
∂r
(r, s)dr1I[0;t](s).
Now let
(Aϕ)(t) =
T∫
0
(Kϕ)(t, s)dw(s) =
t∫
0
(Kϕ)(t, s)dw(s).
Then
(Aϕ)(t) =
t∫
0
ϕ(s)dBα(s).
We will consider the action of GSRO on the random elements in H. The correspond-
ing definition was proposed in [3, 11]. Consider arbitrary GSRO A in H. Then for every
ϕ ∈ H the Ito – Wiener expansion of Aϕ contains only two terms:
Aϕ = α0ϕ+ α1(ϕ)(ξ). (1.14)
Here, α0 is a continuous linear operator in H and α1 is a continuous linear operator
from H to the space of Hilbert – Schmidt operators in H. Now let x be a random
element in H with the finite second moment. Then α1(x) has a finite second moment
in the space of Hilbert – Schmidt operators. So, for every ϕ ∈ H,
α1(x)(ϕ) =
∞∑
k=0
Bk(ϕ; ξ, . . . , ξ).
It can be easily verified that Bk is (k + 1)-linear H-valued Hilbert – Schmidt form on
H. Define ΛBk as a symmetrization of Bk with respect to all k + 1 variables.
Definition 1.3 [3, 11]. The random element x belongs to the domain of definition of
GSRO A if the series
∞∑
k=0
ΛBk(ξ, . . . , ξ)
converges in H in the square mean and in this case
Ax = α0x+
∞∑
k=0
ΛBk(ξ, . . . , ξ). (1.15)
In the partial cases, this definition gives us the definition of the extended stochastic
integral [6, 12, 13]. We will define this integral for the special class of Gaussian processes.
Suppose that H = L2([0;T ]) and ξ is generated by the Wiener process w as above.
Consider the jointly Gaussian with w process {γ(t); t ∈ [0;T ]} with zero mean.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1225
Definition 1.4 [14]. Process γ is an integrator if there exists the constant C such
that, for every step function ϕ on [0;T ] of the form
ϕ =
n−1∑
k=0
ak1I[tk;tk+1),
the following unequality holds:
E
(
n−1∑
k=0
ak(γ(tk+1) − γ(tk))
)2
≤ C
n−1∑
k=0
a2
k(tk+1 − tk). (1.16)
The good examples of the integrators can be obtained via the following simple state-
ment.
Lemma 1.2. Let Γ(C) be an operator of the second quantization. Then
γ(t) = Γ(C)w(t), t ∈ [0;T ],
is an integrator.
The proof of this lemma easily follows from the properties of Γ(C).
Note that the integrator can have the unbounded quadratic variation and consequently
can have not the semimartingale properties (see [14]). It is easy to see from (1.16) that,
for every integrator γ and ϕ ∈ L2([0;T ]), the stochastic integral
t∫
0
ϕdγ
exists as a limit of the integrals from the step functions and
E
t∫
0
ϕdγ
2
≤ C
t∫
0
ϕ2(s)ds.
So, one can define GSRO Aγ associated with the integrator γ by the rule
∀ϕ ∈ L2([0;T ]) : (Aγϕ)(t) =
t∫
0
ϕdγ.
In this situation, Definition 1.3 became to be a definition of the extended stochastic inte-
gral with respect to γ. Note that in the case γ = w it will be a usual extended integral.
Now let us consider the relation between the action of GSRO and the operators of the
second quantization.
Theorem 1.1 [4]. Let A be a GSRO in H and Γ(C) be an operator of the second
quantization. Suppose that the random element x lies in the domain of definition of A
in the sence of Definition 1.3. Then Γ(C)x belongs to the domain of definition of GSRO
Γ(C)A and the following equality holds:
Γ(C)(Ax) = Γ(C)A(Γ(C)x). (1.17)
Here, Γ(C)A is the GSRO which acts by the rule
∀ϕ ∈ H : Γ(C)Aϕ = Γ(C)(Aϕ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1226 A. A. DOROGOVTSEV
The proof of this theorem is placed in [4] and so is omitted. Instead the proof con-
sider the following important example of application of Theorem 1.1 to the stochastic
integration.
Example 1.3. Consider in the situation of Example 1.2 GSRO of integration with
respect to Wiener process w. Suppose that random function x in L2([0;T ]) with the
finite second moment is adapted to the flow of σ -fields generated by w. It is well known
[12, 13] that, in this case, the extended stochastic integral
t∫
0
x(s)dw(s), t ∈ [0;T ],
exists and coincides with the Ito integral. Now the Theorem 1.1 says us that
Γ(C)
t∫
0
x(s)dw(s)
=
t∫
0
Γ(C)x(s)dγ(s),
where γ is an integrator of the type γ(t) = Γ(C)w(t) and the integral in the right part
is an extended stochastic integral.
2. Smoothing problem. The last sections of the article are devoted to the following
problem. Let (w1, w2) be the pair of jointly Gaussian one-dimensional Wiener processes.
Let the processes x1, x2 are obtained via the relations
dx1(t) = a1(x1(t))dt+ dw1(t),
dx2(t) = a2(x1(t))dt+ dw2(t),
x1(0) = x2(0) = 0.
(2.1)
Note that the second equality is just a definition of x2 but not an equation. The
problem is to find the conditional distribution of x1(t) for t ∈ [0; 1] under given
{x2(s); s ∈ [0; 1]}. We will try to get the equation for
E(f(x1(t))/x2)
for the appropriate functions f.
First, let us study the joint distribution of (w1, w2) . Note that there exists the bounded
linear operator V : L2([0; 1]) → L2([0; 1]) such that
∀ϕ1, ϕ2 ∈ L2([0; 1]) : E
1∫
0
ϕ1dw1
1∫
0
ϕ2dw2 =
∫ 1
0
ϕ1V ϕ2ds.
This fact follows from the reason that the left part of the above formula is the contin-
uous bilinear form with respect to ϕ1 and ϕ2. Moreover, the operator norm ‖V ‖ ≤ 1.
In this section, we consider the density of the distribution (x1, x2) with respect to the
distribution of (w1, w2) and study its properties under the conditional expectation. The
problem is that the distribution of (w1, w2) is not a Wiener measure in C([0; 1],R2). So,
in order to get the density, we need to adapt the general Gaussian measure setup [15] to the
our case. For the future let us denote C([0; 1]) as C and identify the space C([0; 1],R2)
with the direct sum C ⊕ C, which is furnished by the sum of the norms. Denote also by
H the space
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1227
L2([0; 1],R2) = L2([0; 1]) ⊕ L2([0; 1])
with the scalar product defined by the formula
(ϕ,ψ) =
1∫
0
ϕ1ψ1ds+
1∫
0
ϕ2ψ2ds.
With the pair (w1, w2) we can associate the generalized Gaussian random element ξ in
H by the rule
(ϕ, ξ) =
1∫
0
ϕ1dw1 +
1∫
0
ϕ2dw2.
Note that ξ has not an identity covariation operator. Really,
E(ϕ, ξ)(ψ, ξ) =
1∫
0
ϕ1ψ1ds+
1∫
0
ϕ2ψ2ds+
+
1∫
0
ϕ1V ψ2ds+
1∫
0
ψ1V ϕ2ds.
Here, V is described above bounded linear operator in L2([0; 1]). Denote by S the
operator in H which acts by the rule
Sϕ = (ϕ1 + V ϕ2, V
∗ϕ1 + ϕ2).
Then
E(ϕ, ξ)(ψ, ξ) = (Sϕ, ψ).
Our aim is to describe the transformations of the pair (w1, w2) in the terms of ξ. Let us
start with the deterministic admissible shifts. Denote by i the canonical embedding of
H into C2, i.e.,
i(ϕ)(t) =
t∫
0
ϕ1ds,
t∫
0
ϕ2ds
.
Lemma 2.1. Let the operator norm ‖V ‖ < 1, then for every h ∈ H, i(h) is
admissible shift for µw1,w2 and the corresponding density has the form
ph(ξ) = exp
{
(S−1h, ξ) − 1
2
(S−1h, h)
}
. (2.2)
Remark 2.1. Due to the condition ‖V ‖ < 1, the operator S−1 is bounded on H
and can be written in the form
S−1ϕ = ϕ+ (Q11ϕ1 +Q12ϕ2, Q21ϕ1 +Q22ϕ2) = ϕ+Qϕ,
where ‖Q‖ < 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1228 A. A. DOROGOVTSEV
Proof. Note that, by the definition, the operator S is nonnegative. Define
ξ′ = S− 1
2 ξ.
Then the shift of the distribution of (w1, w2) on the vector i(h) is related to the shift of
ξ′ on the vector S− 1
2h. Now the statement of the lemma follows from the well-known
formula for the density in terms of ξ′:
p(ξ′) = exp
{
(ξ′, S− 1
2h) − 1
2
(S− 1
2h, S− 1
2h)
}
if we rewrite it in terms of ξ.
The lemma is proved.
Remark 2.2. Formula (2.2) can be rewritten in terms of w1, w2. Really, by the
definition, (
S−1h, ξ
)
− 1
2
(
S−1h, h
)
=
=
1∫
0
(S−1h)1dw1 +
1∫
0
(S−1h)2dw2−
−1
2
1∫
0
(S−1h)1h1ds−
1
2
1∫
0
(S−1h)2h2ds =
=
1∫
0
(h1 +Q11h1 +Q12h2)dw1 +
1∫
0
(h2 +Q21h1 +Q22h2)dw2−
−1
2
1∫
0
(h1 +Q11h1 +Q12h2)h1ds−
1
2
1∫
0
(h2 +Q21h1 +Q22h2)h2ds. (2.3)
Using the same method, one can find the density of µx1,x2 with respect to µw1,w2 .
First, define the stochastic derivatives of the functionals from w1, w2 with respect to ξ
and the extended stochastic integral in terms of ξ. Let ϕ be a differentiable bounded
function on C ⊕ C. Define the stochastic derivative of the random variable ϕ(w1, w2)
by the formula
Dϕ(w1, w2) := i∗∇ϕ(w1, w2).
By this definition, for every t ∈ [0; 1],
Dw1(t) = (1I[0;t], 0),
Dw2(t) = (0, 1I[0;t]).
Note that ϕ(w1, w2) can be regarded as a functional from the generalized random el-
ement ξ′ which was introduced in the proof of Lemma 2.1. Since ξ′ has an identity
covariation operator, the stochastic derivatives and extended stochastic integral for the
functionals from ξ′ are connected by the usual relation. Now we will define the extended
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1229
stochastic integral with respect to ξ. It can be done in the following way. Consider the
Gaussian random functional on H of the kind
J(ϕ) = (ϕ, ξ).
Then, in terms of ξ′, J can be rewritten as
J(ϕ) = (S
1
2ϕ, ξ′).
So, the action of J on the random element x in H via Definition 1.3 has the form
J(x) = I(S
1
2x). (2.4)
Here, I is the extended stochastic integral with respect to ξ′. Note also that for the
stochastic derivatives with respect to ξ′ and ξ, we have the obvious relation
Dξ′α = S− 1
2Dξα.
Hence, on the domain of definition,
E(Dξα, x) = E(S− 1
2Dξα, S
1
2x) =
= E(Dξ′α, S
1
2x) = EαI(S
1
2x) = EαJ(x). (2.5)
Thus, the relation between the stochastic derivative and extended stochastic integral with
respect to ξ is the same as for ξ′. Now let us turn to the nonadapted shifts of the distri-
bution of (w1, w2). Consider the pair of random processes x1, x2 which are defined by
Equations (2.1).
The next lemma is standard.
Lemma 2.2. Let the functions a1, a2 be continuously differentiable and have bounded
derivatives. Then:
1) for every t ∈ [0; 1], the random variables x1(t), x2(t) have the stochastic deriva-
tives Dx1(t), Dx2(t);
2) the random element (a1(x1(·)), a2(x1(·))) in H has the stochastic derivative
and
D(a1(x1(s)), a2(x1(s)))(t) =
= (a′1(x1(s))Dx1(s)(t), a′2(x1(s))Dx1(s)(t));
3) the stochastic derivative of x1 with respect to w1 (i.e., the first coordinate of
Dx1) satisfies the equation
D1x1(s)(t) = 1 +
s∫
t
a′1(x1(r))D1x1(r)(t)dr, 0 ≤ t ≤ s ≤ 1,
D1x1(s)(t) = 0, t > s,
and
Dx1(s)(t) = (D1x1(s)(t), 0).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1230 A. A. DOROGOVTSEV
It follows from Lemma 2.2 that ‖D(a1(x1(·)), a2(x1(·)))‖H can be made small if we
take a′1 and a′2 small enough. Since the operator S− 1
2 is bounded in H, we have due to
Theorem 3.2.2 from [15] that the distribution of (x1, x2) is absolutely continuous with
respect to the distribution (w1, w2) for sufficiently small a′1, a′2. The corresponding
density will be denoted by p. Accordingly to [15], p has the form
p = ζ exp
{
I(S− 1
2h) − 1
2
(S−1h, h)
}
, (2.6)
where
h(t) = (a1(w1(t)), a2(w1(t))),
and ζ is the corresponding Carleman – Fredholm determinant. Due to (2.4), (2.6) can be
rewritten as
p = ζ exp
{
J(S−1h) − 1
2
(S−1h, h)
}
. (2.7)
This expression allows us to conclude that up to the term ζ, p has the stochastic deriva-
tive. We will suppose that this is so in the next section, where the formulas for the con-
ditional expectation and extended stochastic integral will be obtained in non-Gaussian
case.
3. Conditional expectation. For the processes (x1, x2) from (2.1), let us search
for the conditional distribution of x1(t) under fixed {x2(s); s ∈ [0; 1]}. First note that
under our conditions, the distribution of (x1, x2) is absolutely continuous with respect
to the distribution of (w1, w2) and, consequently, the distribution of x2 is absolutely
continuous with respect to the distribution of w2.
Denote for a moment by µ the distribution of the pair (w1, w2) on C([0; 1]) ⊕
⊕ C([0; 1]) and by µ1 and µ2 the distributions of w1 and w2 (surely, these are the
Wiener measures but on the different copies of C([0; 1]). It follows from the general
theory of integration that the measure µ can be desintegrated with respect to µ2, i.e.,
µ(∆) =
∫
C([0;1])
ν(u,∆u)µ2(du),
for arbitrary Borel ∆ in C([0; 1]) ⊕ C([0; 1]). Here, ν is a measurable family of the
probability measures and ∆u = {v ∈ C([0; 1]) : (v, u) ∈ ∆}.
Define for the measurable bounded function ϕ : C([0; 1]) → R the function ψ by
the rule
C([0; 1]) � u �→ ψ(u) =
∫
C([0;1])
ϕ(v)p(v, u)ν(u, dv)×
×
∫
C([0;1])
p(v, u)ν(u, dv)
−1
. (3.1)
The following variant of the Bayes formula holds.
Lemma 3.1.
E(ϕ(x1)/x2) = ψ(x2).
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SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1231
Proof. First note that ψ(x2) is correctly defined because the function ψ is defined
up to the set of Wiener measure zero, and the distribution of x2 is equivalent to this
measure. Now, for arbitrary bounded and measurable function γ : C([0; 1]) → R,
Eϕ(x1)γ(x2) = Eϕ(w1)γ(w2)p(w1, w2) =
= Eγ(w2)E(ϕ(w1)p(w1, w2)/w2) =
= Eγ(w2)
E(ϕ(w1)p(w1, w2)/w2)
E(p(w1, w2)/w2)
E(p(w1, w2)/w2) =
= Eγ(w2)ψ(w2)p(w1, w2) = Eγ(x2)ψ(x2).
This finishes the proof.
For arbitrary t ∈ [0; 1], denote by πt the random measure on R whose pairing with
the bounded measurable function f is defined by the formula∫
R
f(r)πt(dr) = E(f(w1(t))p(w1, w2)/w2).
In view of the previous lemma, it is sufficient to get the equation for πt. The next
lemma contains the necessary facts from the theory of extended stochastic integral.
Lemma 3.2. Let H be the separable Hilbert space, ξ be a generalized Gaussian
random element in H with zero mean and identity covariation. Suppose that the random
element x in H has two stochastic derivatives and let I and D be the symbols of the
extended stochastic integral and stochastic derivative correspondingly. Then for arbitrary
h ∈ H and stochastically differentiable bounded random variable α, the following
formulas hold:
1) αI(x) = I(αx) + (x,Dα);
2) (DI(x), h) = (x, h) + I((Dx, h)).
Proof. The first statement is the well-known relation [12]. Let us check 2). Use the
integration by part formula. Consider the random variable β which is twice stochastically
differentiable. Then, using 1),
E(DI(x), h)β = E(DI(x), βh) =
= EI(x)I(βh) = EI(x)(βI(h) − (Dβ, h)) =
= E(x,D(βI(h) − (Dβ, h))) =
= E[(x, βh) + (x,Dβ)I(h) − (x, (D2β, h))] =
= E(x, h)β + E(x, I(Dβh)) =
= Eβ((x, h) + I((Dx, h))).
The lemma is proved.
Remark 3.1. Note that the statement of the lemma remains to be true in the case
when α is not bounded but all terms are well-defined. Also, due to the formula (2.5), the
lemma holds in the case when the initial generalized Gaussian random element has not
identity covariation.
Now let us turn to our filtration problem.
Take the function f ∈ C2
0 (R). Then for arbitrary r ∈ R from the Ito formula
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1232 A. A. DOROGOVTSEV
f(r + w1(t))p(w1, w2) =
= f(r)p(w1, w2) +
t∫
0
f ′(r + w1(s))dw1(s)p(w1, w2)+
+
1
2
t∫
0
f ′′(r + w1(s))p(w1, w2)ds.
Consider the second summand. It contains the Ito integral which coincides with the ex-
tended stochastic integral as it was mentioned before. So, we can apply the formula 1)
from Lemma 3.2:
p(w1, w2)
t∫
0
f ′(r + w1(s))dw1(s) =
=
t∫
0
f ′(r + w1(s))p(w1, w2)dw1(s)+
+
t∫
0
f ′(r + w1(s))(SDp(w1, w2))1(s)ds.
Here, the index 1 symbolizes the first coordinate of correspondent element from H. Now
note that the conditional expectation with respect to w2 is an operator of the second
quantization. So, if we denote
γ1(t) = E(w1(t)/w2),
then, due to Theorem 1.1,
E(f(r + w1(t))p(w1, w2)/w2) =
= E(f(r)p(w1, w2)/w2) +
t∫
0
E(f ′(r + w1(s))p(w1, w2)/w2)dγ(t)+
+
1
2
t∫
0
E(f ′′(r + w1(s))p(w1, w2)/w2)ds+
+
t∫
0
E(f ′(r + w1(s)(SDp(w1, w2))1)(s)/w2)ds, (3.2)
where the integral with respect to γ is an extended stochastic integral. In order to get the
stochastic differentiability of p, let us consider the case when the Carleman – Fredholm
determinant ζ is equal to one. Denote by Pt the orthogonal projector in L2([0; 1]) ⊕
⊕ L2([0; 1]) on the subspace L2([0; t]) ⊕ L2([0; t]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SMOOTHING PROBLEM IN ANTICIPATING SCENARIO 1233
Lemma 3.3. Suppose that the operator S has the property
∀t ∈ [0; 1] : PtS = PtSPt.
Then ζ = 1.
Proof. The value ζ is the Carleman – Fredholm determinant of the operator SDh,
where
h = (a1(w1(·)), a2(w1(·))).
Now
Dh(t, s) =
(
a′1(w1(t))1I[0;t](s) 0
a′2(w1(t))1I[0;t](s) 0
)
. (3.3)
In order to prove that
det2(Id+ SDh) = 1,
we will use Theorem 3.6.1 from [15]. Due to this theorem, it is sufficient to check that
the operator SDh is quasinilpotent, i.e., that
lim
n→∞
‖(SDh)n‖ 1
n = 0. (3.4)
It follows from representation (3.3) that
∀ϕ ∈ H ∀t ∈ [0; 1] : ‖PtDhϕ‖2 ≤ c
t∫
0
‖Psϕ‖2ds,
where c depends on sup
R
(|a′1| + |a′2|).
Consequently,
‖(SDh)n(ϕ)‖2 ≤ ‖S‖2c
1∫
0
‖Pt1(SDh)n−1(ϕ)‖2dt1 ≤
≤ ‖S‖2c
1∫
0
‖Pt1SPt1Pt1Dh(SDh)n−2(ϕ)‖2dt1 ≤
≤ ‖S‖4c2
1∫
0
t1∫
0
‖Pt2(SDh)n−2(ϕ)‖2dt2dt1 ≤ . . .
. . . ≤ ‖S‖2ncn
1∫
0
t1∫
0
. . .
tn−1∫
0
‖Ptn
ϕ‖2dtn . . . dt1 ≤
≤ ‖ϕ‖2 ‖S‖2ncn
n!
.
This means that (3.4) holds and ζ = 1.
The lemma is proved.
Now one can conclude that p has the stochastic derivative and (3.1) is correct. The
further concretization of (3.2) can be possible due to the special form of p. As a conse-
quence of (3.2) and (3.4), we have the following theorem.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1234 A. A. DOROGOVTSEV
Theorem 3.1. Suppose that the coefficients a1, a2 and the operator V satisfy the
conditions of Lemma 3.3. Then the random function
U(r, t) = E(f(r + w1(t))p(w1, w2)/w2)
satisfies relation
dU(r, t) =
1
2
∂2
∂r2
U(r, t)dt+
+
∂
∂r
U(r, t)γ(dt) + Ef ′(r + w1(t))(SDp(w1, w2))1(t)dt. (3.5)
In some particular case, the last term can be written in a simple form. For example,
when a2 = 0, then (3.5) transforms into
dU(r, t) =
1
2
∂2
∂r2
U(r, t)dt+
∂
∂r
U(r, t)γ(dt)+
+ a1(r)
∂
∂r
U(r, t)dt.
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Received 08.07.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
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| id | umjimathkievua-article-3680 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:59Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/85/64900410667d543f39e96ec25da58885.pdf |
| spelling | umjimathkievua-article-36802020-03-18T20:02:00Z Smoothing Problem in Anticipating Scenario Задача інтерполяції для неузгоджених шумів Dorogovtsev, A. A. Дороговцев, А. А. We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration. Розглядається задача інтерполяції для випадкових процесів, що задовольняють стохастичні диференціальні рівняння з вінеровими процесами, які не є семімартингалами відносно спільної фільтрації. Institute of Mathematics, NAS of Ukraine 2005-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3680 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 9 (2005); 1218–1234 Український математичний журнал; Том 57 № 9 (2005); 1218–1234 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3680/4086 https://umj.imath.kiev.ua/index.php/umj/article/view/3680/4087 Copyright (c) 2005 Dorogovtsev A. A. |
| spellingShingle | Dorogovtsev, A. A. Дороговцев, А. А. Smoothing Problem in Anticipating Scenario |
| title | Smoothing Problem in Anticipating Scenario |
| title_alt | Задача інтерполяції для неузгоджених шумів |
| title_full | Smoothing Problem in Anticipating Scenario |
| title_fullStr | Smoothing Problem in Anticipating Scenario |
| title_full_unstemmed | Smoothing Problem in Anticipating Scenario |
| title_short | Smoothing Problem in Anticipating Scenario |
| title_sort | smoothing problem in anticipating scenario |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3680 |
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