Some questions of
conditional Markov processes
Workshop on stochastic and harmonic
analysis of processes with jumps
Angers, France, 2-9 May 2006
Mátyás Barczy and Gyula Pap
University of Debrecen, Hungary
1
(E, %) : complete separable metric space
E : σ-algebra of Borel subsets of E, T > 0
(Zt)0≤t≤T : time-homogeneous Markov process
with state space (E, E)
λ : σ-finite measure on E
∃ transition densities (pZ
t )0<t≤T w.r.t. λ :
Z
P(Zt ∈ A | Zs) =
A
pZ
t−s (Zs , y) λ(dy)
P-a.s.
for all A ∈ E, 0 ≤ s < t ≤ T
2
a, b ∈ E
Bridge from a to b over [0, T ] derived from Z :
a process obtained by conditioning Z to start in
a at time 0 and arrive at b at time T
Problem 1: construct a bridge from
Z
using
only transition densities, question of uniqueness
Problem 2: ”deriving bridge” and ”taking norm”
commute in case of Rd ?
3
Yor, Zambotti (2004):
Z = standard Wiener process in Rd.
If a = 0 or b = 0, the answer is YES.
If a 6= 0, b 6= 0, the answer is NO.
(Statistics & Probability Letters, 68,
(2004), 297-304)
Barczy, Pap (2005):
Z = a special Ornstein-Uhlenbeck process
in Rd
If a = b = 0, the answer is YES.
(Periodica Mathematica Hungarica, 50 (1-2),
(2005), 47-60)
4
Bb,ε := {x ∈ E : %(b, x) < ε} for b ∈ E, ε > 0
(Xtε)0≤t≤T := (Zt)0≤t≤T conditioned that ZT ∈ Bb,ε
Transition densities of X ε : 0 ≤ s < t < T, x, y ∈ E
R
Z (y, z) λ(dz)
p
ε
B
Z (x, y) b,ε T −t
pX
,
(x,
y)
=
p
R
s,t
t−s
Z
Bb,ε pT −s(x, z) λ(dz)
R
provided Bb,ε pZ
T −s (x, z) λ(dz) 6= 0. Indeed,
P(Xtε ∈ A | Xsε = x)
= P(Zt ∈ A | Zs = x, ZT ∈ Bb,ε)
=
P(Zt ∈ A, ZT ∈ Bb,ε | Zs = x)
P(ZT ∈ Bb,ε | Zs = x)
=
R
Z (y, z) λ(dz)
p
B
b,ε T −t
pZ
(x,
y)
λ(dy)
R
t−s
Z
A
Bb,ε pT −s (x, z) λ(dz)
Z
5
We can think of the desired bridge as lim X ε
ε↓0
Definition:
For x, y ∈ E and 0 ≤ s < t < T , let
R
ps,t(x, y) := pZ
t−s(x, y) lim R
Z (y, z) λ(dz)
p
Bb,ε T −t
ε↓0 B pZ (x, z) λ(dz)
b,ε T −s
if the right hand side exists, and ps,t(x, y) := 0
otherwise.
By a bridge from a to b over [0, T ] derived
from Z we mean a Markov process (Xt)0≤t≤T
with P(X0 = a) = 1, P(XT = b) = 1 and with
transition densities (ps,t)0≤s<t<T
provided that
such a process exists.
The Markov process (Xt)0≤t≤T (if it exists) is
in general not time-homogeneous.
6
Lemma 1
Suppose that
(i) (x, y) 7→ ps,t(x, y) measurable ∀ 0 ≤ s < t < T ,
(ii) y 7→ ps,t(x, y) density ∀ x ∈ E, 0 ≤ s < t < T ,
R
(iii) ps,u(x, z) = E ps,t(x, y)pt,u(y, z) λ(dy) holds
∀ x, z ∈ E, 0 ≤ s < t < u < T .
Then ∃1 prob. measure
PZ
a,b,T
on
³
E [0,T ], E [0,T ]
´
such that the coordinate process (Yt)0≤t≤T on
´
[0,T
]
[0,T
]
E
,E
under PZ
a,b,T is a bridge from a
³
to b over [0, T ] derived from Z.
Consequently, if (Xt)0≤t≤T is a bridge from a
to b over [0, T ] derived from Z then its
law on
³
E [0,T ], E [0,T ]
´
is PZ
a,b,T , hence, uniquely
determined.
7
Proposition 1
Suppose that
(i) (x, y) 7→ pZ
t (x, y) continuous ∀ 0 < t ≤ T ,
(ii)
pZ
s+t (x, z)
R
Z
= E pZ
s (x, y)pt (y, z) λ(dy) holds
∀ x, z ∈ E and s, t > 0 with s + t ≤ T ,
(iii) pZ
t (x, b) > 0 holds ∀ x ∈ E, 0 < t ≤ T .
Then
ps,t(x, y) =
Z (y, b)
pZ
(x,
y)p
t−s
T −t
pZ
T −s(x, b)
,
for x, y ∈ E, 0 ≤ s < t < T , and (ps,t)0≤s<t<T
satisfy conditions of Lemma 1.
Example: if
Z = standard Wiener process in
Rd, Proposition 1 is applicable.
8
Proposition 2
Let E = [0, ∞), let λ be the Lebesgue measure
on [0, ∞), and let b = 0. Under mild regularity
conditions on the transition density functions pZ
we have
pZ
T −t (y, ε)
ps,t(x, y) = pZ
,
t−s (x, y) lim Z
ε↓0 p
T −s(x, ε)
for x, y ∈ [0, ∞), 0 ≤ s < t < T , and (ps,t)0≤s<t<T
satisfy conditions of Lemma 1.
Example: If Z = Bessel process with dimension
d
(d ≥ 2), then considering continuous density
functions pZ
t (x, 0) = 0, x ≥ 0, 0 < t ≤ T, and
Proposition 2 is applicable (but Proposition 1 is
not).
9
Standard d-dimensional Wiener process W
(Xt)0≤t≤T :=
W
conditioned to start in 0 at
time 0 and arrive at 0 at time T
(called the d-dimensional Wiener bridge)
Rt := kWtk, t ≥ 0 : the norm of (Wt)t≥0
(called the d-dimensional Bessel process)
(kXtk)0≤t≤T : the norm of the Wiener bridge X
(Yt)0≤t≤T := R conditioned to start in 0 at time
0 and arrive at 0 at time T
(called the d-dimensional Bessel bridge)
By Proposition 2, the transition densities of the
Bessel bridge and the norm of the Wiener bridge
coincide.
10
Special multidimensional OU processes
Let us consider the d-dimensional SDE

dZ = aZ dt + σ dW ,
t
t
t
 Z = 0,
0
t ≥ 0,
where a, σ ∈ R such that σ 6= 0, and (Wt)t≥0
is a standard d-dimensional Wiener process.
∃1 strong solution
Zt = σ
Z t
0
ea(t−s) dWs,
t ≥ 0.
Our result: Transition densities of the bridge
derived from the norm of Z and the norm of the
bridge derived from Z
coincide, if we consider
bridges with endpoints 0.
11
Multidimensional Ornstein-Uhlenbeck
processes
Let us consider the d-dimensional SDE

dZ = AZ dt + Σ dW ,
t
t
t
 Z = 0,
0
t ≥ 0,
where A ∈ Rd×d, Σ ∈ Rd×r
and (Wt)t≥0 is a
standard r-dimensional Wiener process.
We can construct a bridge starting from Z.
Conjecture: ”deriving bridges with endpoints 0”
and ”taking norm” do not commute in general
starting from Z.
Reason: the norm of
Z
is in general not a
Markov process.
12
Lemma 2
If for each 0 < t ≤ T ,
(i) (x, y) 7→ pZ
t (x, y) continuous,
(ii) ∀ x0 ∈ E ∃ δ > 0 such that
sup pZ
t (x, y) < ∞,
sup
x∈Bx0 , δ y∈E
(iii) ∀ y0 ∈ E ∃ δ > 0 such that
sup
(iv) ∀ y ∈ E
sup
pZ
t (x, y) < ∞,
x∈E y∈By0 , δ
R
we have E pZ
t (x, y) λ(dx)
<∞
then the Kolmogorov-Chapman equation
pZ
s+t (x, z) =
Z
E
Z
pZ
s (x, y)pt (y, z) λ(dy)
holds ∀ x, z ∈ E and s, t > 0 with s + t ≤ T .
13
Proposition 2
Let E = [0, ∞), let λ be the Lebesgue measure
on [0, ∞), and let b = 0. Suppose that
(i) (x, y) 7→ pZ
t (x, y) continuous ∀ 0 < t ≤ T ,
R
Z
Z
(ii) pZ
s+t (x, z) = E ps (x, y)pt (y, z) λ(dy) holds
∀ x, z ∈ E and s, t > 0 with s + t ≤ T ,
(iii) ∃ lim
pZ
T −t (y,ε)
Z
ε↓0 pT −s (x,ε)
∀ x, y ∈ [0, ∞), 0 ≤ s < t < T ,
(iv) ∀ 0 ≤ s < t < T , x ∈ [0, ∞) ∃ δ > 0 such that
sup
sup
pZ
T −t(y, ε)
Z
y∈[0,∞) 0<ε<δ pT −s (x, ε)
< ∞.
Then
pZ
T −t (y, ε)
,
ps,t(x, y) = pZ
t−s (x, y) lim Z
ε↓0 p
T −s(x, ε)
for x, y ∈ [0, ∞), 0 ≤ s < t < T , and (ps,t)0≤s<t<T
satisfy conditions of Lemma 1.
14
(Wt)t≥0 : standard d-dimensional Wiener process
pW
t (x, y) =
1
(2πt)d/2
(
exp −
kx − yk2
)
2t
for t > 0, x, y ∈ Rd
(Xt)0≤t≤T : Wiener bridge := W conditioned to
start in 0 at time 0 and arrive at 0 at time T
By Proposition 1,
Ã
pX
s,t (x, y) =
T −s
2π(t − s)(T − t)
(
× exp −
!d/2
kx − yk2
2(t − s)
−
kyk2
2(T − t)
+
kxk2
2(T − s)
for all x, y ∈ Rd and all 0 ≤ s < t < T
15
)
Rt := kWtk, t ≥ 0 : radial part of (Wt)t≥0
(called the d-dimensional Bessel process)
pR
t (x, y) =
y ν+1
txν
(
exp −
x2
+ y2
2t
)
µ
xy
Iν
t
¶
if x, y > 0,
y 2ν+1
(
y2
)
pR
exp −
t (0, y) = ν ν+1
2 t
Γ(ν + 1)
2t
if y > 0, and



0



R
(
)
pt (x, 0) = s
2

2
x


exp
−

 πt
2t
if d > 1,
if d = 1,
where ν := 2d − 1, and Iν is the modified Bessel
function with index ν.
16
(kXtk)0≤t≤T : radial part of the Wiener bridge
Markov process with transition densities:
kXk
ps,t (x, y)
y ν+1
=
(t − s)xν
µ
(
× exp −
¶
µ
¶
T − s ν+1
xy
Iν
T −t
t−s
x2
+ y2
2(t − s)
−
y2
2(T − t)
+
x2
2(T − s)
for all 0 ≤ s < t < T and all x, y > 0, and
µ
¶ν+1
2ν+1
y
T
−
s
kXk
ps,t (0, y) = ν
2 (t − s)ν+1Γ(ν + 1) T − t
(
× exp −
y2
2(t − s)
−
y2
)
2(T − t)
for all 0 ≤ s < t < T and all y > 0.
(Yt)0≤t≤T : Bessel bridge := R conditioned to
start in 0 at time 0 and arrive at 0 at time T
By Proposition 2, the transition densities of the
processes (kXtk)0≤t≤T and (Yt)0≤t≤T coincide.
17
)
Multidimensional Ornstein-Uhlenbeck
processes
Let us consider the d-dimensional SDE

dZ = AZ dt + Σ dW ,
t
t
t
 Z = 0,
0
t ≥ 0,
where A ∈ Rd×d, Σ ∈ Rd×r
and (Wt)t≥0 is a
standard r-dimensional Wiener process.
∃1 strong solution
Zt =
Z t
0
e(t−s)AΣ dWs,
t ≥ 0.
18
If ΣΣ> > 0 then
1
pZ
t (x, y) = q
(2π)d det(Vt)
½
1
× exp − (y − etAx)>Vt−1(y − etAx)
2
¾
for all x, y ∈ Rd and all t > 0, where
Vt :=
Z t
0
>
e(t−v)AΣΣ>e(t−v)A dv,
If all the eigenvalues of
parts then
V
A
t > 0.
have negative real
>
Vt = V − etAV etA ,
t > 0,
where
is the unique solution of the algebraic matrix
equation
AV + V A> = −ΣΣ>
R ∞ uA
> euA> du.
e
ΣΣ
0
given by
V =
19
(Xt)0≤t≤T : OU bridge := Z conditioned to start
in 0 at time 0 and arrive at 0 at time T
By Proposition 1,
pX
s,t (x, y)
v
u
u
=t
det(VeT −s)
(2π)d det(Vet−sVeT −t)
(
× exp
1
−1
− (x − e−(t−s)Ay)>Vet−s
(x − e−(t−s)Ay)
2
)
1
1 > e −1
− y >VeT−1
y
+
x VT −sx
−t
2
2
for all x, y ∈ Rd and all 0 ≤ s < t < T , where
Vet :=
Z t
0
>
e−vAΣΣ>e−vA dv,
t > 0.
20
Special multidimensional OU processes
Let us consider the d-dimensional SDE

dZ = aZ dt + σ dW ,
t
t
t
 Z = 0,
0
t ≥ 0,
where a, σ ∈ R such that σ 6= 0, and (Wt)t≥0
is a standard d-dimensional Wiener process.
∃1 strong solution
Zt = σ
pZ
t (x, y)
Z t
0
ea(t−s) dWs,
t ≥ 0.
(
− eatxk2
ky
exp −
=
d/2
2
2σ 2κa,t
(2πσ κa,t)
where
1

2at − 1


e
κa,t :=


t
2a
)
,
for a 6= 0,
for a = 0.
21
(Xt)0≤t≤T : OU bridge := Z conditioned to start
in 0 at time 0 and arrive at 0 at time T
Ã
pX
s,t (x, y) =
κa,T −s
!d/2
2πσ 2κa,t−s κa,T −t
(
× exp
ky − ea(t−s)xk2 e2a(T −t)kyk2
−
−
2σ 2κa,t−s
2σ 2κa,T −t
+
e2a(T −s)kxk2
)
2σ 2κa,T −s
22
(kXtk)0≤t≤T : radial part of the OU bridge
Markov process with transition densities:
kXk
ps,t (x, y)
e−aν(t−s)y ν+1
=
σ 2κa,t−sxν
½
× exp
Ã

!ν+1 
a(t−s) xy
κa,T −s
e

Iν  2
κa,T −t
σ κa,t−s
e2a(t−s)x2 + y 2 e2a(T −t)y 2
−
−
2
2σ κa,t−s
2σ 2κa,T −t
e2a(T −s)x2
+
2σ 2κa,T −s
¾
for all 0 ≤ s < t < T and all x, y > 0, and
Ã
y 2ν+1
kXk
ps,t (0, y) = ν 2
2 (σ κa,t−s)ν+1Γ(ν + 1)
(
× exp −
y2
2σ 2κa,t−s
−
κa,T −s
!ν+1
κa,T −t
y2
)
2σ 2κa,T −t
for all 0 ≤ s < t < T and all y > 0.
23
(Rt)t≥0 := (kZtk)0≤t≤T : called radial OU process
Time-homogeneous Markov process with transition densities:
pR
t (x, y) =
e−aνty ν+1
σ 2κa,txν
(
exp −
e2atx2
+ y2
)
Ã
Iν
2σ 2κa,t
eatxy
σ 2κa,t
if x, y > 0
(
y 2ν+1
y2
)
pR
exp − 2
t (0, y) = ν 2
ν+1
2 (σ κa,t)
Γ(ν + 1)
2σ κa,t
if y > 0, and



0


pR
t (x, 0) = r



if d > 1,
½
2
πσ 2 κ
a,t
exp
2at x2
e
− 2σ 2κ
a,t
¾
if d = 1,
where ν := 2d − 1.
(Yt)0≤t≤T := R conditioned to start in 0 at time
0 and arrive at 0 at time T
By Proposition 2, the transition densities of the
processes (kXtk)0≤t≤T and (Yt)0≤t≤T coincide.
24
!