On the Markov property of strong solutions to sde with generalized coefficients
We present the complete proof of the Markov property of the strong solution to a multidimensional stochastic differential equation, whose coefficients involve the local time on a hyperplane of the unknown process.
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irk-123456789-44352009-11-10T12:00:35Z On the Markov property of strong solutions to sde with generalized coefficients Zaitseva, L.L. We present the complete proof of the Markov property of the strong solution to a multidimensional stochastic differential equation, whose coefficients involve the local time on a hyperplane of the unknown process. 2005 Article On the Markov property of strong solutions to sde with generalized coefficients / L.L. Zaitseva // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 140–146. — Бібліогр.: 11 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4435 519.21 en Інститут математики НАН України |
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We present the complete proof of the Markov property of the strong solution to a
multidimensional stochastic differential equation, whose coefficients involve the local
time on a hyperplane of the unknown process. |
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Article |
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Zaitseva, L.L. |
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Zaitseva, L.L. On the Markov property of strong solutions to sde with generalized coefficients |
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Zaitseva, L.L. |
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Zaitseva, L.L. |
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On the Markov property of strong solutions to sde with generalized coefficients |
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On the Markov property of strong solutions to sde with generalized coefficients |
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On the Markov property of strong solutions to sde with generalized coefficients |
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On the Markov property of strong solutions to sde with generalized coefficients |
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On the Markov property of strong solutions to sde with generalized coefficients |
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on the markov property of strong solutions to sde with generalized coefficients |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/4435 |
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On the Markov property of strong solutions to sde with generalized coefficients / L.L. Zaitseva // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 140–146. — Бібліогр.: 11 назв.— англ. |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 140–146
UDC 519.21
LUDMILA L. ZAITSEVA
ON THE MARKOV PROPERTY OF STRONG SOLUTIONS
TO SDE WITH GENERALIZED COEFFICIENTS
We present the complete proof of the Markov property of the strong solution to a
multidimensional stochastic differential equation, whose coefficients involve the local
time on a hyperplane of the unknown process.
1. Introduction.
We consider one class of stochastic differential equations (SDE) with the local time
on a hyperplane of the unknown process included into the drift and martingale part. It
is shown that this equation has the strong unique solution, and this solution is a Markov
process. The processes constructed may be described as a generalized diffusion process
in the sense of Portenko (see, for example, [1]) with the drift vector and the diffusion
matrix of the general form.
One of the most known examples of a process described by an SDE with its coefficients
containing the local time of the unknown process is the skew Brownian motion introduced
in [2] (Section 4.2, Problem 1). In 1981, J.M. Harrisson and L.A. Shepp (see [3]) proved
that the skew Brownian motion may be constructed as the strong solution to the SDE
of the form dx(t) = qdηt + dw(t), where q ∈ [−1, 1] is a given parameter and {ηt} is
the local time at 0 of the process {x(t)}. One-dimensional equations with the drift of a
more general form were considered by J.-F. Le Gall, M. Barlow, K. Burdzy, H. Kaspi,
A. Mandelbaum (see [4,5]). The main result of those papers is the theorem on the
existence and uniqueness of a strong solution to the corresponding SDE.
The characterization of the obtained strong solution (say, as a Markov process or a
generalized diffusion process) is a separate complicated problem. J.M. Harrisson and
L.A. Shepp in their paper, while stating that the skew Brownian motion obtained as the
strong solution to the SDE is a Markov process, referred to the corresponding results for
the standard SDE in [6]. Since this question is rather delicate, let us discuss it in more
details. The scheme of proof in [6] (or, in more generality, in [7], Chapter 6) consists
of two steps. The first step contains the construction of a modification of the strong
solution, jointly measurable w.r.t. starting point, and the Wiener noise. The second one,
namely the proof of the Markov property, essentially uses the fact that the increments of
a Wiener process are independent. The first step requires the proof of the additional fact
that the solution is continuous in probability as a function of the starting point. This
proof is non-trivial even in the simplest case of a skew Brownian motion (see [8] for a
general result), and the arguments of J.M. Harrisson and L.A. Shepp are incomplete at
this point. Notice that A.M. Kulik (see [9]) gave a sketch of the proof of the Markov
property in the case of the one-dimensional equation considered by J.-F. Le Gall, which
takes this difficulty into account. In the multidimensional case, we meet another difficulty
2000 AMS Mathematics Subject Classification. Primary 60G20, 60J55.
Key words and phrases. Generalized diffusion process, skew Brownian motion, local time, strong
solution to an SDE.
140
ON THE MARKOV PROPERTY OF STRONG SOLUTIONS TO SDE 141
caused by the fact that the martingale noise does not necessarily possess independent
increments. The main purpose of this paper is to give the complete proof of the Markov
property for the strong solution to an SDE involving the local time of the unknown
process in the most general multidimensional case.
2. Construction of the process.
Let S be a hyperplane in �d orthogonal to a fixed unit vector ν ∈ �d : S ={
x ∈ �d|(x, ν) = 0
}
. By πS and L, we denote, respectively, the operator of orthogonal
projection on S and the one-dimensional subspace of �d, generated by ν.
We consider a Wiener process {w(t)} in �d and a filtration Fw
t = σ {w(u), 0 ≤ u ≤ t},
t ≥ 0. We denote w(t) = (w1(t), wS(t)), where w1(t) = (w(t), ν), wS (t) = πSw(t). For a
given parameter q ∈ [−1, 1] and the initial point x1
0 ∈ L, we construct the skew Brownian
motion (see [3]), i.e. a pair of {Fw
t }-adapted processes {(x1(t), ηt)}, where {ηt} is the
local time at 0 of {x1(t)},
ηt = lim
ε↓0
1
2ε
∫ t
0
1I{|x1(τ)|≤ε}dτ,
such that {x1(t)} and {ηt} satisfy the equality
x1(t) = x1
0 + qηt + w1(t)
for all t ≥ 0.
Let us assume that there exists one more Wiener process {w̃(t)} in S that does not
depend on {w(t)}. Consider a new process ζ(t) = w̃(ηt), t ≥ 0. We shall deal with the
SDE involving the processes {w(t)} and {ζ(t)} as a martingale part of the equation.
Therefore, we have to construct a filtration such that {ζ(t)} is the square integrable
martingale and {w(t)} is the Wiener process with respect to this filtration. Remark that
the process {ηt} is not a stopping time w.r.t. {Fw
t }, and therefore the classical results
cannot be applied here.
For t ≥ 0, we consider
F̃t = σ
{
{w̃(s) ∈ Γ}
⋂
{ηt ≥ s} , s ≥ 0, Γ ∈ BS
}
,
where BS is the Borel σ-algebra on S. We put
Mt = Fw
t
∨
F̃t.
The following three lemmas show that {Mt} is the required filtration.
Lemma 1. {Mt} is a filtration.
Proof. We have to prove that Mt1 ⊆ Mt2 , when t1 ≤ t2. For fixed s ≥ 0 and Γ ∈ BS let
us consider the set At1 = {w̃(s) ∈ Γ}⋂ {ηt1 ≥ s} . If At1 = ∅ then At1 ∈ Mt2 .
Let At1 �= ∅. Then At1
⋂ {ηt2 ≥ s} �= ∅, because {ηt} is the increasing process. Also
At1 = At2
⋂{ηt1 ≥ s}. Since At2 ∈ F̃t2 and {ηt1 ≥ s} ∈ Fw
t1 ⊆ Fw
t2 , At1 ∈ Mt2 .
The lemma is proved.
Lemma 2. The process {ζ(t)} is a square integrable martingale w.r.t. {Mt}, and its
characteristic is equal to {ηt}.
Proof. Firstly, we show that the process ζ(t) is Mt-measurable for all t ≥ 0. We can
approximate ζ(t) by step functions in the following way
ζ(t) = lim
n→∞
∞∑
k=1
w̃
(
k − 1
n
)
1I{ k−1
n ≤ηt<
k
n}.
142 LUDMILA L. ZAITSEVA
We can write
w̃
(
k − 1
n
)
1I{ k−1
n ≤ηt<
k
n} =
[
w̃
(
k − 1
n
)
1I{ k−1
n ≤ηt}
] [
1I{ηt<
k
n}
]
,
where the process in the first brackets is F̃t-measurable, and the process in the second
brackets is Fw
t -measurable. Then ζ(t) is Mt-measurable as the limit of measurable
functions.
The second moment of the process ζ(t) is finite for all t ≥ 0 because
Ew̃(ηt)2 = EE
(
w̃(ηt)2/Fw
∞
)
= E
(
Ew̃(t̂)2
)
t=ηt
= E
(
t̂
)
t=ηt
= Eηt < ∞.
In the second equality, we used Lemma 1 in [6, p.67].
Let us prove that, for an arbitrary bounded Ms-measurable random value ξ, the
relation E (ζ(t) − ζ(s)) ξ = 0 holds for all t ≥ s. It is enough to check this relation for the
indicators of sets generating the σ-algebra Ms. For all k ≥ 1, ui ∈ [0, +∞), Γi ∈ BS , 1 ≤
i ≤ k, Γ ∈ BC[0,+∞) (the Borel σ-algebra on C([0, +∞)))
E [w̃(ηt) − w̃(ηs)] 1I{w(·)|s0∈Γ}
k∏
i=0
1I{w(ui)∈Γi}1I{ui≤ηs} =
= E
[
1I{w(·)|s0∈Γ}E
(
[w̃(ηt) − w̃(ηs)]
k∏
i=0
1I{w(ui)∈Γi}1I{ui≤ηs}/Fw
∞
)]
=
= E
[
1I{w(·)|s0∈Γ}E
(
[w̃(T2) − w̃(T1)]
k∏
i=0
1I{w(ui)∈Γi}1I{ui≤T1}
)]∣∣∣∣∣
T2=ηt,T1=ηs
= 0,
here we denote the trajectory of the Wiener process {w(t)} on [0, s] by w(·)|s0. Here we
use the independence of the processes {w̃(t)}, {ηt} and Lemma 1 in [6, p.67] in the second
equality.
In the same way, one can observe that the process {w̃(ηt)2 − ηt} is martingale w.r.t.
{Mt}. This means that the characteristics of the martingale {w̃(ηt)} is equal to {ηt}.
Lemma is proved.
Lemma 3. {w(t)} is a Wiener process w.r.t. {Mt}.
Remark 1. This result is not obvious because the σ-algebra {Mt} is larger than {Fw
t }
and σ-algebras {Fw
t } and {F̃t} are not independent.
Proof. It is enough to prove that the process {w(t)} is a martingale with characteristic
t w.r.t. {Mt}. Firstly we show that, for an arbitrary bounded Ms-measurable random
value ξ, the relation E (w(t) − w(s)) ξ = 0 holds for all t ≥ s. Again we check this relation
for the indicators of sets generating the σ-algebra Ms. For all k ≥ 1, ui ∈ [0, +∞), Γi ∈
BS , 1 ≤ i ≤ k, Γ ∈ BC[0,+∞)
E [w(t) − w(s)] 1I{w(·)|s0∈Γ}
k∏
i=0
1I{w(ui)∈Γi}1I{ui≤ηs} =
= E
[
1I{w(·)|s0∈Γ}
k∏
i=0
1I{w(ui)∈Γi}1I{ui≤ηs}E
(
w(t) − w(s)/Fw
s ∨ Fw
∞
)]
= 0.
In the same way one can show that the process {w(t)2 − t} is martingale w.r.t. {Mt}.
This means that the characteristic of the martingale {w(t)} is t.
Lemma is proved.
ON THE MARKOV PROPERTY OF STRONG SOLUTIONS TO SDE 143
For a given x0 ∈ �d,-measurable function α : S → S, and operator β : S → L+(S)
(L+(S) denotes the space of all linear symmetric nonnegative operators on S), we consider
the stochastic equation
(1) x(t) = x0 +
∫ t
0
(
qν + α(xS(τ))
)
dητ +
∫ t
0
β̃(xS(τ))dw̃(ητ ) + w(t)
in �d, where β̃(·) = β1/2(·). We call the strong solution to Eq. (1) to be the {Mt}-
adapted process {x(t)} which satisfies equality (1).
Theorem 1. Assume that there exists K > 0 such that
1. supx∈S
(∣∣∣α(x)
∣∣∣ +
∥∥∥β̃(x)
∥∥∥) ≤ K,
2.
∣∣∣α(x) − α(y)
∣∣∣2 +
∥∥∥β̃(x) − β̃(y)
∥∥∥2
≤ K
∣∣∣x − y
∣∣∣2 , for all x, y ∈ S.
Then, for all T > 0, the solution to Eq. (1) exists and is unique when t ∈ [0, T ].
It is enough to prove the existence and uniqueness for the process {xS(t)}. This,
in turn, is a consequence of the existence and uniqueness theorems for the stochastic
equations with arbitrary martingales and stochastic measures (see [7], p. 278-296).
Remark 2. It was proved in [8] that the solution of Eq. (1) has a measurable modification
as the function of the starting point.
3. Markov property of the constructed process.
By {x(t, x)}, we denote the solution to Eq. (1) started from x ∈ �d.
Theorem 2. {x(t, x)} has Markov property.
Proof. We prove the theorem if we show that the relation
(2) E1I{x(t,x)∈Γ}ξ = EΦt−s(x(s, x), Γ)ξ,
holds for an arbitrary bounded Ms-measurable random value ξ, where Φt−s(·, Γ) : �d →
� is a measurable function for all 0 ≤ s ≤ t, Γ ∈ B�d .
Firstly, we note that the process {x(t, x)} has the property
(3) θsx(t − s, z)|z=x(s,x) = x(t, x), 0 ≤ s ≤ t,
where θs is the shift operator (see, [10, p.121]). We can deal with the process
{x(t, x(s, x))}
because the process {x(t, x)} is measurable as a function of the initial point. Equality
(3) holds true, because each of the processes on both sides of (3) satisfies the equation
(4) x(t, x) = x(s, x)+
∫ t
s
(
qν + α(xS(u, x))
)
dηu +
∫ t
s
β̃(xS(u, x))dw̃(ηu)+w(t)−w(s),
that has the unique solution.
Let us denote (γ(·)−γ(s))|ts = (γ(·)−γ(s))1I[s,t](·) for an arbitrary process {γ(t)}, s ≤
t. Equalities (3) and (4) mean that we can express the process {x(t, x)} in the form
(5) x(t, x) = F (x(s, x), (w(·) − w(s))|ts, (η· − ηs)|ts, (w̃(η·) − w̃(ηs))|ts), 0 ≤ s ≤ t,
where F : �d × C[0, +∞) × C[0, +∞) → �d is a measurable functional. Therefore, we
will prove (2), if we show that, for all 0 ≤ s ≤ t, Γ ∈ B(C[0,+∞))3, the relation
(6) E1I{((w(·)−w(s))|ts,(η·−ηs)|ts,(w(η·)−w(ηs))|ts)∈Γ}ξ = EΦ1
t−s(x(s, x), Γ)ξ,
holds true, where Φ1
t−s(·, Γ) : �d → � is a measurable function for all t, s, Γ.
144 LUDMILA L. ZAITSEVA
We prove (6) in the next way. We construct Φ1
t−s(·, ·) for the indicators of sets gen-
erating the σ-algebra Ms. We show that, for all k ≥ 1, ui ∈ [0, +∞), Γ̃i ∈ BS , 1 ≤ i ≤
k, Γ0, Γ1, Γ2, Γ3 ∈ BC[0,+∞), the relation
E1I{(w(·)−w(s))|ts∈Γ1}1I{(η·−ηs)|ts∈Γ2}1I{(w(η·)−w(ηs))|ts∈Γ3}1I{w(·)|s0∈Γ0}
k∏
i=1
(
1I{w(ui)∈Γi} ×
(7) × 1I{ui≤ηs}
)
= EΦ2
t−s(x(s, x), Γ1, Γ2, Γ3)1I{w(·)|s0∈Γ0}
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs}
holds, where Φ2
t−s(·, Γ1, Γ2, Γ3) : �d → � is a measurable function for all t, s, Γ1, Γ2, Γ3.
Let us take the conditional expectation w.r.t. Fw
s on the left-hand side of relation (7).
Then we obtain
E1I{(w(·)−w(s))|ts∈Γ1}1I{(η·−ηs)|ts∈Γ2}1I{(w(η·)−w(ηs))|ts∈Γ3}1I{w(·)|s0∈Γ0}×
×
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs} = E
(
1I{w(·)|s0∈Γ0}E
[
1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2} ×
(8) × 1I{(w(η·)−w(r))|ts∈Γ3}
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}/Fw
s
]∣∣∣∣∣
z=w(s),r=ηs
⎞⎠ .
Let us show that the sets {(w(·) − z)|ts ∈ Γ1}
⋂{(η· − r)|ts ∈ Γ2}
⋂{(w̃(η·) − w̃(r))|ts ∈
Γ3} and
⋂k
i=1 {w̃(ui) ∈ Γ̃i}
⋂ {ui ≤ r} are independent when the σ-algebra Fw
s is fixed.
Really, for an arbitrary bounded Fw
s -measurable random value ζ, we have
Eζ1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2}1I{(w(η·)−w(r))|ts∈Γ3}
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r} =
= E
(
ζ1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2}E
[
1I{(w(η·)−w(r))|ts∈Γ3} ×
×
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}/Fw
∞
])
= E
(
ζ1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2} ×
× E
[
1I{(w(r(·))−w(r))|ts∈Γ3}
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}
]∣∣∣∣∣
r(·)=η·
⎞⎠ = E
(
ζ1I{(w(·)−z)|ts∈Γ1} ×
× 1I{(η·−r)|ts∈Γ2}E
[
1I{(w(r(·))−w(r))|ts∈Γ3}
∣∣
r(·)=η·
]
E
[
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}
])
=
= E
(
ζ1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2}E
[
1I{(w(η·)−w(r))|ts∈Γ3}/Fw
∞
])×
×E
(
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}
)
= E
(
ζ1I{(w(·)−z)|ts∈Γ1}1I{(η·−r)|ts∈Γ2} ×
× 1I{(w(η·)−w(r))|ts∈Γ3}
)
E
(
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤r}
)
.
In the third equality, we take into account the independence of the sets {(w̃(r(·)) −
w̃(r))|ts ∈ Γ3} and
⋂k
i=1 {w̃(ui) ∈ Γ̃i}
⋂ {ui ≤ r}.
Therefore, we obtain that the expression on the right-hand side of (8) equals
E
(
1I{w(·)|s0∈Γ0}E
[
1I{(w(·)−w(s))|ts∈Γ1}1I{(η·−ηs)|ts∈Γ2}1I{(w(η·)−w(ηs))|ts∈Γ3}/Fw
s
] ×
ON THE MARKOV PROPERTY OF STRONG SOLUTIONS TO SDE 145
(9) × E
[
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs}/Fw
s
])
.
Let us denote the conditional joint distribution of the processes {(w(·)−w(s))|ts} and
{(η· − ηs)|ts} in the following way:
P
{
(w(·) − w(s))|ts ∈ Γ1, (η· − ηs)|ts ∈ Γ2/Fw
s
}
= μt−s(xν(s), Γ1, Γ2).
We recall that the process {ηt} is an additive functional of {xν(t)}, and thus the distri-
bution of {(η· − ηs)|ts} depends only on the distribution of xν(s). Then we have
E
(
1I{(w(·)−w(s))|ts∈Γ1}1I{(η·−ηs)|ts∈Γ2}1I{(w(η·)−w(ηs))|ts∈Γ3}/Fw
s
)
=
=
∫
P {y(·) ∈ Γ1, θ(·) ∈ Γ2, ŵ(θ(·)) ∈ Γ3}μt−s(xν(s), dy(·), dθ(·)) =
= Φ2
t−s(x
ν(s), Γ1, Γ2, Γ3),
where ŵ(·) is a Wiener process in S.
Therefore, (9) equals
E
(
1I{w(·)|s0∈Γ0}E
[
1I{(w(·)−w(s))|ts∈Γ1}1I{(η·−ηs)|ts∈Γ2}1I{(w(η·)−w(ηs))|ts∈Γ3}/Fw
s
] ×
× E
[
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs}/Fw
s
])
= E
(
1I{w(·)|s0∈Γ0}Φ
2
t−s(x
ν(s), Γ1, Γ2, Γ3) ×
× E
[
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs}/Fw
s
])
= E
(
1I{w(·)|s0∈Γ0}Φ
2
t−s(x
ν(s), Γ1, Γ2, Γ3) ×
×
k∏
i=1
1I{w(ui)∈Γi}1I{ui≤ηs}
)
.
This proves equality (7) and then, after the standard limiting procedure, equality (6).
The theorem is proved.
In the next theorem, we characterize the process constructed in Theorem 1 as a gen-
eralized diffusion process in the sense of Portenko (see [1]).
Theorem 3. For all θ ∈ �d, ϕ ∈ C0(�d) (the space of all real-valued continuous functions
in �d with compact support), the relations
lim
t↓0
1
t
∫
�d
ϕ(z)E|x(t, z) − x(0, z)|4dz = 0,
lim
t↓0
1
t
∫
�d
ϕ(z)E(x(t, z) − x(0, z), θ)dz =
∫
S
ϕ(z)(qν + α(z), θ)dσz
lim
t↓0
1
t
∫
�d
ϕ(z)E(x(t, z) − x(0, z), θ)2dz = (θ, θ)
∫
�d
ϕ(z)dz +
∫
S
(β(z)θ, θ)ϕ(z)dσz
hold, where dσ is the Lebesgue measure on S.
This result means that the process {x(t, x)} is a generalized diffusion process with the
generalized drift vector being equal to (qν + α(xS))δS(x) and the generalized diffusion
matrix being equal to I + β(xS)δS(x), here I is the identity matrix in �d, and δS(·) is
the delta-function concentrated on S.
The proof of this theorem is similar to that of the corresponding theorem in [11].
146 LUDMILA L. ZAITSEVA
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E-mail : zaitseva@univ.kiev.ua
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