On the representation of solutions of anticipating linear partial stochastic differential equations
The unique solution of an anticipating linear partial stochastic differential equation is constructed by means of the Fourier transformation.
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| Cite this: | On the representation of solutions of anticipating linear partial stochastic differential equations / A.V. Ilchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 38–47. — Бібліогр.: 6 назв.— англ. |
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| author | Ilchenko, A.V. |
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| citation_txt | On the representation of solutions of anticipating linear partial stochastic differential equations / A.V. Ilchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 38–47. — Бібліогр.: 6 назв.— англ. |
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Theory of Stochastic Processes
Vol. 13 (29), no. 3, 2007, pp. 38–47
UDC 519.21
ALEXANDR V. ILCHENKO
ON THE REPRESENTATION OF SOLUTIONS OF ANTICIPATING
LINEAR PARTIAL STOCHASTIC DIFFERENTIAL EQUATIONS
The unique solution of an anticipating linear partial stochastic differential equation
is constructed by means of the Fourier transformation.
Denote, by K and L, linear partial differential operators with constant coefficients of
the first and the second order, respectively:
K ≡
n∑
i=1
bi
∂
∂xi
; L ≡
n∑
i,j=1
aij
∂2
∂xi∂xj
.
Operator L is assumed to be elliptic:
n∑
i,j=1
aijλiλj ≥ c|λ|2, |λ|2 =
n∑
i=1
λ2
i , c > 0.
Consider the stochastic differential equation
(1) du(t, x) = Lu(t, x)dt + Ku(t, x)dw(t),
where w(t), 0 ≤ t ≤ 1, is the standard Wiener process, x ∈ R
n.
We will seek for a solution to Eq. (1) that satisfies the initial condition
(2) u(0, x) = αϕ(x),
where α = f(w), w = w(·), is a functional depending on a path of the Wiener process
and ϕ(x) is some real function, x ∈ R
n.
By the solution of problem (1),(2), we mean a solution of the integral equation
(3) u(t, x) = αϕ(x) +
∫ t
0
Lu(s, x)ds +
∫ t
0
Ku(s, x)dw(s),
where the stochastic integral is defined in the extended sense given by A.V. Skorokhod
in [6].
Definition. A function of two variables u(t, x), 0 ≤ t ≤ 1, x ∈ R
n is said to be a
solution of Eq. (3) if the following conditions are fulfilled:
1) u(t, x) is continuous in t and has two continuous derivatives with respect to x for
0 ≤ t ≤ 1, x ∈ R
n;
2) Ku(t, x) and Lu(t, x) are continuous in t for 0 ≤ t ≤ 1, x ∈ R
n;
3) Ku(s, x) belongs to the domain of definition of the extended stochastic integral in
s for 0 ≤ t ≤ 1, x ∈ R
n;
4) relation (3) holds true for all 0 ≤ t ≤ 1, x ∈ R
n with probability 1.
2000 AMS Mathematics Subject Classification. Primary 60H10, 60H15.
Key words and phrases. Anticipating initial condition, extended stochastic integral, partial stochastic
differential equation.
I wish to express my gratitude to A. A. Dorogovtsev for his support for this research.
38
ON THE REPRESENTATION OF SOLUTIONS . . . 39
The purpose of this paper is to formulate the condition for the existence of the solution
of problem (1),(2). In order to solve the problem, it is possible to apply the approach
proposed by A.A. Dorogovtsev in [1]. The solution of the problem (1), (2) can be found
by means of the Fourier transformation (see [4]).
Denote by ϕ̂(λ) the Fourier transform of the function ϕ(x), x ∈ R
n, and ψ̌(x) is the
inverse Fourier transform:
ϕ̂(λ) =
∫
Rn
e−i〈λ,x〉ϕ(x)dx, ψ̌(x) = (2π)−n
∫
Rn
ei〈x,λ〉ψ(λ)dλ.
Here, 〈x, λ〉 =
∑n
i=1 xiλi and |x| =
(∑n
i=1 x2
i
)1/2 are the scalar product and norm of
vectors from R
n, i2 = −1.
Consider the space of rapidly decreasing functions, on which the Fourier transforma-
tion is well defined:
G = {ϕ(·) : ϕ(·) ∈ C∞(Rn), sup
x∈Rn
|xβ∂αϕ(x)| < +∞, α, β ∈ N},
where ∂α = ∂α1
1 . . . ∂αn
n , ∂
αj
j = ∂αj /∂x
αj
j , αj ∈ Z+, j = 1, n.
Let h(λ) and g(λ) be the symbols of the operators L and K, respectively:
h(λ) = −
n∑
i,j=1
aijλiλj ; g(λ) = ip(λ), p(λ) =
n∑
j=1
bjλj .
Apply the Fourier transformation to Eq. (1) and the initial condition (2) in a formal way
assuming that all functions belong to G with respect to x ∈ R
n. We get the following
Cauchy problem:
dy(t, λ) = h(λ)y(t, λ)dt + ip(λ)y(t, λ)dw(t);(4)
y(0, λ) = αϕ̂(λ).(5)
Find the conditions, under which the solution of problem (1),(2) can be obtained as a
result of the inverse Fourier transformation of problem (4),(5).
Theorem 1. Let ϕ̂(λ), y(t, λ) ∈ G with respect to λ for each t : 0 ≤ t ≤ 1. If y(t, λ)
satisfies the equation
y(t, λ) = αϕ̂(λ) +
∫ t
0
h(λ)y(s, λ)ds + i
∫ t
0
p(λ)y(s, λ)dw(s),
then the solution of problem (1),(2) can be found as a result of the inverse Fourier
transformation of the function y(t, λ) : u(t, x) = y̌(t, x).
Proof. If y(t, ·) belongs to G, so do h(λ)y(t, λ) and p(λ)y(t, λ) with respect to λ. Set
u(t, x) = y̌(t, x). Since the Fourier transformation realizes an isomorphism G onto itself,
Lu(t, x) and Ku(t, x), as well as u(t, x), belong to G with respect to x ∈ R
n. In addition,
L̂u(t, λ) = h(λ)y(t, λ); K̂u(t, λ) = ip(λ)y(t, λ).
To complete the proof, it still remains to prove the commutability between the inverse
Fourier transformation and integration operations. Since the function h(λ)y(s, λ) is
integrable in the variable s, has a continuous modification, and belongs to G in λ, the
inverse Fourier transformation commutes with the ordinary integration according to the
Fubini theorem. Really,
(2π)−n
∫
Rn
ei〈x,λ〉
∫ t
0
h(λ)y(s, λ)dsdλ =
∫
Rn
∫ t
0
(2π)−nei〈x,λ〉h(λ)y(s, λ)dsdλ =
40 ALEXANDR V. ILCHENKO
=
∫ t
0
(2π)−n
∫
Rn
ei〈x,λ〉h(λ)y(s, λ)dλds.
For the extended stochastic integral it also suffices to apply the Fubini theorem taking into
account the definition of extended stochastic integral. Denote Vt(s, λ) = 1I[0,t]p(λ)y(s, λ).
For each F ∈ D
1,2, we have
E
(
F
∫ 1
0
(2π)−n
∫
Rn
ei〈x,λ〉Vt(s, λ)dλdw(s)
)
=
= E
( ∫ 1
0
DF (s)(2π)−n
∫
Rn
ei〈x,λ〉Vt(s, λ)dλds
)
=
= (2π)−n
∫
Rn
ei〈x,λ〉
(
E
∫ 1
0
DF (s)Vt(s, λ)ds
)
dλ =
= (2π)−n
∫
Rn
ei〈x,λ〉E
(
F
∫ 1
0
Vt(s, λ)dw(s)
)
dλ =
= E
(
F (2π)−n
∫
Rn
ei〈x,λ〉
∫ 1
0
Vt(s, λ)dw(s)
)
dλ.
Since F is an arbitrary element from D
1,2,∫ 1
0
(2π)−n
∫
Rn
ei〈x,λ〉Vt(s, λ)dλdw(s) = (2π)−n
∫
Rn
ei〈x,λ〉
∫ 1
0
Vt(s, λ)dw(s)dλ
with probability 1.
Consider now the question how to find the solution of problem (4),(5). It should
be noted that problem (4),(5) can be solved under fixed λ. Set a = h(λ), b = p(λ),
z0 = αϕ̂(λ). We have
(6)
{
dz(t) = az(t)dt + ibz(t)dw(t);
z(0) = z0.
In order to get the solution of problem (6), we write the system with respect to the real
and imaginary parts of z(t). Set z(t) = z1(t) + iz2(t), zk(t) ∈ R, k = 1, 2;
z(t) =
(
z1(t)
z2(t)
)
I =
(
1 0
0 1
)
, B =
(
0 −b
b 0
)
, z0 =
(
Re z0
Im z0
)
.
We come to
(7)
{
d z(t) = aI z(t)dt + B z(t)dw(t);
z(0) = z0.
In the case where α has a finite Itô–Wiener expansion, that is,
α =
N∑
k=0
∫
[0,1]k
ak(t1, . . . , tk)dw(t1) . . . dw(tk), N < +∞,
the solution of problem (7) was obtained in [1]. To write this solution, it needs to
introduce the following auxiliary system:
(8)
{
dyt
s = aIyt
sdt + Byt
sdw(t);
ys
s = I, s ≤ t.
The solution of problem (7) can be written in the form (see [1])
(9) z(t) = yt
0
[
z0 +
N∑
k=1
(−1)kBk
∫
Δk(t)
Dk z0(t1, . . . , tk)dt1 . . . dtk
]
,
ON THE REPRESENTATION OF SOLUTIONS . . . 41
where Dk z0(t1, . . . , tk), k ∈ N, are stochastic derivatives of z0; yt
0 is the solution of (8)
when s = 0; Δk(t) = {0 ≤ t1 ≤ . . . ≤ tk ≤ t}. Returning to the complex variable z(t),
we get
z(t) = e
(
a+ 1
2 b2
)
t+ibw(t)ϕ̂(λ)
[
α +
N∑
k=1
(−i)kbk
∫
Δk(t)
Dkα(t1 . . . tk)dt1 . . . dtk
]
.
Since a = h(λ), b = p(λ), the solution of (4), (5) becomes
(10)
y(t, λ) = e
(
h(λ)+ 1
2p2(λ)
)
t+ip(λ)w(t)ϕ̂(λ)
[
α+
+
N∑
k=1
(−i)kpk(λ)
∫
Δk(t)
Dkα(t1 . . . tk)dt1 . . . dtk
]
.
In order to extend the class of the initial conditions α, for which the solution of problem
(4),(5) can be represented in the form (10), we use results from [3]. Suppose that the
map α : C0[0, 1] → R is analytic, i.e. it can be expanded in the Taylor series for w = 0 :
(11) α =
∞∑
n=0
1
n!
α(n)(0)wn, w ∈ C0[0, 1],
where
α(n)(0) ∈ L(
n︷ ︸︸ ︷
X, L(X, . . . , L(X, R)) . . . ), X ≡ C0[0, 1];
α(n)(0)wn ≡ (. . . ((α(n)(0)w)w) . . . )w.
Denote
‖α(n)(0)‖ = sup
‖xj‖≤1,j=1,n
|(. . . ((α(n)(0)x1)x2) . . . )xn|.
It is proved in [3] that, under the condition
(12) lim
n→∞(‖α(n)(0)‖/n!)1/n = 0,
the Cauchy problem (7) in the one-dimensional case has a solution of the form (9) for
N = ∞. That is, this solution can be expanded in an infinite power series. It should
be noted that condition (12) is used only to ensure the convergence of the series, and
the corresponding estimates are independent of the dimension. So that the solution of
problem (7) is identical to (9) for N = ∞. Consequently, we come to the following result.
Theorem 2. Let the functional α : C0[0, 1] → R can be expanded in the infinite power
series (11), and let condition (12) be fulfilled. Then the solution of problem (4),(5) has
the following form:
y(t, λ) = e
(
h(λ)+ 1
2p2(λ)
)
t+ip(λ)w(t)ϕ̂(λ)
[
α+
+
∞∑
k=1
(−i)kpk(λ)
∫
Δk(t)
Dkα(t1 . . . tk)dt1 . . . dtk
]
.
Find the conditions under which the solution y(t, λ) of problem (4),(5) belongs to G
with respect to λ for 0 ≤ t ≤ 1.
42 ALEXANDR V. ILCHENKO
Theorem 3. Suppose that the following conditions are fulfilled:
1) −c + 1
2 (p)2 < 0, where p = max|λ|=1 |p(λ)|;
2)
∣∣∫
[0,t]k
Dkα(t1, . . . , tk)dt1 . . . dtk
∣∣ ≤ Mk, 0 ≤ M < +∞, 0 ≤ t ≤ 1;
3) ϕ ∈ G.
Then y(t, ·) ∈ G, 0 ≤ t ≤ 1.
Proof. The function y(t, λ) has the form
y(t, λ) =
= e
(
h(λ)+ 1
2 p2(λ)
)
teip(λ)w(t)ϕ̂(λ)
[
α +
∞∑
k=0
(−i)k
k!
pk(λ)
∫
[0,t]k
Dkα(t1 . . . tk)dt1 . . . dtk
]
.
The following estimate holds true:∣∣∣∣ ∂r
∂λr
m
∞∑
k=0
(−i)k
k!
pk(λ)
∫
[0,1]k
Dkα(t1 . . . tk)dt1 . . . dtk
∣∣∣∣ =
=
∣∣∣∣
∞∑
k=r
(−i)k
k!
k(k − 1) · . . . · (k − r + 1)pk−r(λ)br
m
∫
[0,t]k
Dkα(t1 . . . tk)dt1 . . . dtk
∣∣∣∣ ≤
≤ |bm|r
∞∑
k=0
|p(λ)|k
k!
∣∣∣∣
∫
[0,t]k
Dkα(t1 . . . tk)dt1 . . . dtk
∣∣∣∣ ≤
≤ |bm|rM reMp|λ|, 1 ≤ m ≤ n, r ∈ Z+.
Now the assertion of the theorem results from the explicit form of y(t, λ), the assumption
of the theorem, and the obtained estimate, because
sup
λ∈Rn
∣∣∣∣λl
j
∂m
∂λm
k
y(t, λ)
∣∣∣∣ < +∞, 0 ≤ t ≤ 1, 1 ≤ k, j ≤ n, l, m ∈ Z+.
Consider now examples of random variables that satisfy the conditions of Theorem 3.
Example 1. Let α = f(w(1)). Suppose that f ∈ C∞(R) and |f (k)(x)| ≤ Mk, 0 ≤ M <
∞, k ∈ Z+, x ∈ R. We have
Dkα(t1, . . . , tk) = f (k)(w(1))1I[0,1](t1) · . . . · 1I[0,1](tk) = f (k)(w(1));∣∣∣∣
∫
[0,1]k
Dkα(t1, . . . , tk)dt1 . . . dtk
∣∣∣∣ ≤ |f (k)(w(1))|tk ≤ Mk, 0 ≤ t ≤ 1.
In this case,
y(t, λ) = e
(
h(λ)+ 1
2p2(λ)
)
t+ip(λ)w(t)ϕ̂(λ)
[
α +
∞∑
k=1
(−i)k
k!
pk(λ)f (k)(w(1))tk
]
, 0 ≤ t ≤ 1.
Example 2. Let
α = f(w(h1), . . . , w(hm)), w(hi) =
∫ 1
0
hi(s)dw(s), hi ∈ L2[0, 1], i = 1, m, m ≥ 1.
Suppose that
f ∈ C∞(Rn), |f (k)
i1,... ,im
(x)| ≤ Mk, 0 ≤ M < +∞, k ∈ Z+, x ∈ R
n.
We have
Dkα(t1, . . . , tk) =
∑
i1+...+im=k
f
(k)
i1,... ,im
(w(h1), . . . , w(hm))h1(t1) · . . . · hm(tk);
ON THE REPRESENTATION OF SOLUTIONS . . . 43∫
[0,1]k
Dkα(t1, . . . , tk)dt1 . . . dtk =
=
∑
i1+...+im=k
f
(k)
i1,... ,im
(w(h1), . . . , w(hm))
∫ t
0
h1(t1)dt1 · . . . ·
∫ t
0
hm(tk)dtk;
∣∣∣∣
∫
[0,1]k
Dkα(t1, . . . , tk)dt1 . . . dtk
∣∣∣∣ ≤ Mk
( m∑
i=1
‖hi‖L2[0,1]
)k
.
The conditions of Theorem 3 hold true. In this case,
y(t, λ) = e
(
h(λ)+ 1
2p2(λ)
)
t+ip(λ)w(t)ϕ̂(λ)
[
α+
+
∞∑
k=1
(−i)k
k!
pk(λ)
∑
i1+...+im=k
f
(k)
i1,... ,im
(w(h1), . . . , w(hm))
∫ t
0
h1(t1)dt1 . . .
∫ t
0
hm(tk)dtk
]
.
Now replace the initial condition (2) by the relation
(13) u(0, x) = η(x, ω) ≡
∞∑
n=0
ϕn(x)In(fn),
where
In(fn) =
∫
[0,1]n
fn(t1, . . . , tn)dw(t1) . . . dw(tn),
ϕn ∈ G. Consider problem (1), (13). Series (13) is assumed to be convergent in L2(Ω) for
every x ∈ R
n. Apply the Fourier transformation to (1),(13), select the real and imaginary
parts, fix λ, and denote ϕn =
(
Re ϕ̂n(λ), Im ϕ̂n(λ)
) . We come to the following Cauchy
problem:
(14)
{
d z(t) = aI z(t)dt + B z(t)dw(t)
z(0) =
∑∞
n=0 ϕnIn(fn).
Suppose that
(15)
∞∑
n=0
ϕnIn(fn) is convergent in L2(Ω).
Let us approach the solution of (14) by the solution of the following problem:
(16)
{
d zM (t) = aI zM (t)dt + B zM (t)dw(t)
zM (0) =
∑M
n=0 ϕnIn(fn).
Find the requirements on ϕn and on fn under which
(17) z(t) = L2 − lim
M→∞
zM (t).
Taking the properties of multiple stochastic integrals and results given in [1] into consid-
eration, we have
zM (t) = yt
0
[
zM (0) +
M∑
k=1
(−1)kBk
M∑
n=k
ϕn
∫
Δk(t)
DkIn(fn)(t1, . . . , tk)dt1 . . . dtk
]
=
= yt
0
M∑
m=0
Im(f̃m),
44 ALEXANDR V. ILCHENKO
where f̃m(·) = ϕmfm(·)+
+(1 − δm,M )
M−m∑
k=1
(−1)k
(
k + m
k
)
Bk ϕk+m
∫
[0,t]k
fk+m(t1, . . . , tk, ·)dt1 . . . dtk,
δm,M is the Kronecker symbol. Since
yt
0 = eaIt
∞∑
m=0
1
m!
Im(gm), gm = (B1I[0,t])⊗m,
the valuable zM (t) can be rewritten as
zM (t) = eaIt
( ∞∑
m=0
1
m!
Im(gm)
)( M∑
k=0
Ik
(
f̃k
))
= eaIt
∞∑
m=0
1
m!
M∑
k=0
Im(gm)Ik
(
f̃k
)
.
The last expression can be changed by means of the relation (see [6])
Ip(fp)Iq(gq) =
p∧q∑
r=0
r!
(
p
r
) (
q
r
)
Ip+q−2r(fp � ⊗rgq), where (fp ⊗r gq)(t1, . . . , tp+q−2r) =
=
∫
[0,t]r
fp(t1, . . . , tp−r, t
1, . . . , tr)gq(tp−r+1, . . . , tp+q−2r, t
1, . . . , tr)dt1 . . . dtr,
where � is the symmetrization operation. As a result, we finally obtain
zM (t) = eaIt
∞∑
m=0
1
m!
M∑
n=0
ϕnIm+n
(
gm � ⊗fn
)
=
= eaIt
[ M∑
l=0
Il
( l∑
n=0
(
gl−n � ⊗( ϕnfn)
)
(l − n)!
)
+
∞∑
l=M+1
Il
( M∑
n=0
(
gl−n � ⊗( ϕnfn)
)
(l − n)!
)]
.
Now we are able to estimate constructively E| zM (t)|2 using the properties of multiple
stochastic integrals. Really,
E| zM (t)|2 =
= e2aIt
[ M∑
l=0
l!
∥∥∥∥
l∑
n=0
(
gl−n � ⊗( ϕnfn)
)
(l − n)!
∥∥∥∥2
l
+
∞∑
l=M+1
l!
∥∥∥∥
M∑
n=0
(
gl−n � ⊗( ϕnfn)
)
(l − n)!
∥∥∥∥2
l
]
≤
≤ e2aIt
[ M∑
l=0
l!
( l∑
n=0
∥∥gl−n � ⊗( ϕnfn)
∥∥
l
(l − n)!
)2
+
∞∑
l=M+1
l!
( M∑
n=0
∥∥gl−n � ⊗( ϕnfn)
∥∥
l
(l − n)!
)2]
.
It still remains to evaluate
∥∥gl−n � ⊗( ϕnfn)
∥∥
l
. We have
∥∥gl−n � ⊗( ϕnfn)
∥∥2
l
≤
∫
[0,1]l
∣∣((B1I[0,t])⊗(l−n) ⊗ ( ϕnfn)
)
(t1, . . . , tl)
∣∣2dt1 . . . dtl =
=
∣∣B∣∣2(l−n)
t(l−n)‖ ϕnfn‖2
n ≤ ∣∣B∣∣2(l−n)| ϕn|2‖fn‖2
n ≤ 2(l−n)
∣∣b∣∣2(l−n)| ϕn|2‖fn‖2
n.
Finally,
E| zM (t)|2 ≤ e2at
√
2
[ M∑
l=0
l!
( l∑
n=0
√
2
(l−n)∣∣b∣∣(l−n)
(l − n)!
| ϕn|‖fn‖n
)2
+
+
∞∑
l=M+1
l!
( M∑
n=0
√
2
(l−n)∣∣b∣∣(l−n)
(l − n)!
| ϕn|‖fn‖n
)2]
.
ON THE REPRESENTATION OF SOLUTIONS . . . 45
Set
ρl = l!
( l∑
n=0
√
2
(l−n)∣∣b∣∣(l−n)
(l − n)!
| ϕn|‖fn‖n
)2
.
Now we are able to formulate the sufficient condition for (17) to be true. In order that
(17) be valid, it is sufficient that
(18)
∞∑
l=0
ρl < +∞.
If (18) holds true then E| z(t)|2 ≤ ∑∞
l=0 ρl < +∞, 0 ≤ t ≤ 1.
Consider the equation
(19) zM (t) = zM (0) +
∫ t
0
aI zM (s)ds +
∫ t
0
B zM (s)dws, 0 ≤ t ≤ 1.
This equation is equivalent to problem (16). Estimate (18) makes it possible to pass to
the limit in (19) under the ordinary integral sign because of the estimate
E
∣∣∣∣
∫ t
0
aI zM (s)ds
∣∣∣∣2 ≤ E
∫ t
0
∣∣aI
∣∣2ds
∫ t
0
∣∣ zM (s)
∣∣2ds ≤
≤ 2|a|2t
∫ t
0
E
∣∣ zM (s)
∣∣2ds ≤ 2|a|2t sup
0≤t≤1
∣∣ zM (s)
∣∣2t < +∞, 0 ≤ t ≤ 1.
The possibility to pass to the limit in L2(Ω) under the stochastic integral sign in (19)
now follows from (19) because the stochastic integral can be expressed in the other terms
of (19), for which the passage to the limit is proved, and acts closely, as it was shown
in [2]. Hence, we can pass to the L2(Ω)-limit in (19) on the whole if condition (18) is
fulfilled. Thus, the following result holds true.
Theorem 4. If conditions (15) and (18) are fulfilled, then problem (14) has a solution
of the form
z(t) = yt
0
[
z(0) +
∞∑
k=1
(−1)kBk
∫
Δk(t)
Dk z(0)(t1, . . . , tk)dt1 . . . dtk
]
.
The correspondent solution of Eq. (4) with the initial condition
(20) y(0, λ) = η̂(λ, ω) ≡
∞∑
n=0
ϕ̂n(λ)In(fn)
has the form
y(t, λ) = e
(
h(λ)+ 1
2p2(λ)
)
t+ip(λ)w(t)
[
η̂(λ, ω)+
+
∞∑
k=1
(−i)kpk(λ)
∫
Δk(t)
Dkη̂(λ, ω)(t1 . . . tk)dt1 . . . dtk
]
.
Remark. Note that, as an example of fulfillment of (18), one can consider such ϕn and
fn that | ϕn| ‖fn‖n ≤ θn/n!. We have
ρl ≤ l!
( l∑
n=0
√
2
(l−n)∣∣b∣∣(l−n)
θn
(l − n)! n!
)2
=
46 ALEXANDR V. ILCHENKO
=
1
l!
( l∑
n=0
(
l
n
)√
2
(l−n)∣∣b∣∣(l−n)
θn
)2
=
(√
2 |b| + θ
)2l
l!
.
∞∑
l=0
ρl ≤
∞∑
l=0
(√
2 |b| + θ
)2l
l!
= e(
√
2 |b|+θ)2 .
E| zM (t)|2 ≤ e2a|I|
∞∑
l=0
(|B| + 1
)2l
l!
≤ e2a|I|e(|B|+1)2 ≤ e2a|I|+(|B|+1)2 .
Clear up a question when the inverse Fourier transformation can be applied to the
solution y(t, λ) of Eq. (4) with the initial condition (20) to get u(t, x). Taking the
arguments from the proof of Theorem 1 into account, it is sufficient to put the condition
(21)
∫
Rn
|Q(λ)y(t, λ)|dλ < +∞ a.s., 0 ≤ t ≤ 1,
on y(t, λ), where Q(λ) is a polynomial. Evidently, condition (21) is fulfilled if
E
∫
Rn
(1 + |λ|m)|y(t, λ)|2dλ < +∞, m ∈ Z+, 0 ≤ t ≤ 1.
Note that
(
Re y(t, λ), Im y(t, λ)
) satisfies (14) for a = h(λ), b = p(λ) and ϕn = ϕn(λ) =(
Re ϕ̂n(λ), Im ϕ̂n(λ)
)
. According to the previous reasoning,(
Re y(t, λ)
Im y(t, λ)
)
= eaIt
∞∑
m=0
1
m!
∞∑
n=0
ϕnIm+n
(
gm � ⊗fn
)
=
=
∞∑
l=0
Il
( l∑
n=0
ψn, l(t, λ)
(l − n)!
(
(1I[0,t])⊗(l−n) � ⊗fn
))
, J =
(
0 −1
1 0
)
,
ψn, l(t, λ) = eh(λ)It
(
p(λ)J
)(l−n)
ϕn(λ).
Suppose that, for each m ∈ Z+, 0 ≤ t ≤ 1,
(22) Δ(m, t) ≡
∞∑
l=0
l!
∫
Rn
(1 + |λ|m)
( l∑
n=0
(
√
2t)(l−n)|ψn, l(t, λ)|
(l − n)!
‖fn‖n
)2
dλ < +∞.
Use the estimates for E|y(t, λ)|2 from the proof of Theorem 4. We have
E
∫
Rn
(1 + |λ|m)|y(t, λ)|2dλ =
∫
Rn
(1 + |λ|m)E|y(t, λ)|2dλ ≤
≤
∫
Rn
(1 + |λ|m)
∞∑
l=0
l!
( l∑
n=0
(
√
2t)(l−n)ψn, l(t, λ)
(l − n)!
‖fn‖n
)2
dλ ≤
≤
∞∑
l=0
l!
∫
Rn
(1 + |λ|m)
( l∑
n=0
(
√
2t)(l−n)ψn, l(t, λ)
(l − n)!
‖fn‖n
)2
dλ = Δ(m, t).
Denote
γn, l(m, t) =
∫
Rn
2(1 + |λ|m)eh(λ)t|p(λ)|(l−n)|ϕ̂n(λ)|dλ.
Since |ψn, l(t, λ)| ≤ 2eh(λ)t|p(λ)|(l−n)|ϕ̂n(λ)| and (
∑l
n=1 an)2 ≤ l
∑l
n=1 a2
n, we come to
(23) Δ(m, t) ≤
∞∑
l=0
l! l
l∑
n=0
(
√
2t)2(l−n)γn, l(m, t)(
(l − n)!
)2 ‖fn‖2
n.
ON THE REPRESENTATION OF SOLUTIONS . . . 47
Hence, the convergence of the series in (23) is the sufficient condition for (22).
So, the following result is proved.
Theorem 5. Let problem (14) have a solution for each λ ∈ R
n. If condition (22) is
fulfilled, then problem (1),(13) has the solution that can be found as
u(t, x) = y̌(t, x),
where y(t, λ) is a solution of problem (4),(20).
Bibliography
1. A.A. Dorogovtsev, Anticipating equations and filtration problem, Theory of Stochastic Processes
v.3(19) (1997), no. 1-2, 154-163.
2. A.A. Dorogovtsev, Stochastic Analysis and Random Maps in Hilbert Space, VSP, Utrecht, 1994,
pp. 110.
3. A.A. Dorogovtsev, One formula for the solution of Itô–Volterra equation with the extended
stochastic integral, in: Random Processes and Infinitely Dimensional Analysis (1992), Inst.
Math., Kyiv, 41-56. (Russian)
4. L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. 1, Springer, Berlin,
1983.
5. D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, 1995.
6. A.V. Skorokhod, One generalization of the stochastic integral, Probability Theory and Appli-
cations 20 (1975), no. 2, 223-237.
E-mail : avi@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4505 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-01T16:06:53Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ilchenko, A.V. 2009-11-19T13:58:39Z 2009-11-19T13:58:39Z 2007 On the representation of solutions of anticipating linear partial stochastic differential equations / A.V. Ilchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 38–47. — Бібліогр.: 6 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4505 519.21 The unique solution of an anticipating linear partial stochastic differential equation is constructed by means of the Fourier transformation. I wish to express my gratitude to A. A. Dorogovtsev for his support for this research. en Інститут математики НАН України On the representation of solutions of anticipating linear partial stochastic differential equations Article published earlier |
| spellingShingle | On the representation of solutions of anticipating linear partial stochastic differential equations Ilchenko, A.V. |
| title | On the representation of solutions of anticipating linear partial stochastic differential equations |
| title_full | On the representation of solutions of anticipating linear partial stochastic differential equations |
| title_fullStr | On the representation of solutions of anticipating linear partial stochastic differential equations |
| title_full_unstemmed | On the representation of solutions of anticipating linear partial stochastic differential equations |
| title_short | On the representation of solutions of anticipating linear partial stochastic differential equations |
| title_sort | on the representation of solutions of anticipating linear partial stochastic differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4505 |
| work_keys_str_mv | AT ilchenkoav ontherepresentationofsolutionsofanticipatinglinearpartialstochasticdifferentialequations |