A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL
PROCESSES
Dominique Lépingle1
Abstract. We consider the problem of strong existence and uniqueness of a Brownian
motion forced to stay in the nonnegative quadrant by an electrostatic repulsion from the sides
that works obliquely. We obtain a kind of two-dimensional Bessel process with oblique drift.
Because of the obliqueness the components are not independent each other. To construct the
solution, we consider the case of a normal drift and use as main tool a comparison lemma.
We tackle the question of hitting the corner or the sides of the quadrant. A few of the
properties of one-dimensional Bessel processes are preserved in this setting. The results are
reminiscent of the study of a Brownian motion with oblique reflection in a wedge. Actually,
the same skew-symmetry condition is involved when looking for a stationary distribution in
product form. The terms of the product are now gamma distributions in place of exponential
ones.
1. Introduction
In the late seventies the study of heavy traffic limits in open multi-station queueing networks has put the question of existence and properties of the Brownian motion obliquely
reflected on the sides of a wedge and more generally on the faces of a polyhedron. In the following decade there was an extensive literary output on that topic, among which we mention
the works of Harrison, Reiman, Williams and co-authtors ([10],[12],[33],[36],[4], to cite a few
of them). But there is another way than normal or oblique reflection to prevent Brownian
motion from overstepping a linear barrier. We may add as drift term the gradient of a concave function that explodes in the neighborhood of the faces of the polyhedron. To be more
specific, let n1 , . . . , nk be unit vectors in Rd and b1 , . . . , bk be real numbers. The state space
S is defined by
S := {x ∈ Rd : nr .x ≥ br , r = 1, . . . , k} .
Let φ1 , . . . , φk be k convex C 1 functions on (0, ∞) with φr (0+) = +∞ for any r = 1, . . . , k.
The potential function Φ on S is defined by
Φ(x) :=
k
X
φr (nr .x − br ) .
r=1
From the general existence and uniqueness theorem on multivalued stochastic differential
systems established in [2], completed with identification of the drift term as in Lemma 3.4 in
[3], we know there exists a unique strong solution living in S to the equation
(1)
dXt = dBt − ∇Φ(Xt )dt
X0 ∈ S
where B is a Brownian motion in Rd . As was proved in Proposition 4.1 of [22], the hypotheses
φr (0+) = +∞ for any r = 1, . . . , k entail that there is no additional boundary process of
Key words and phrases. Singular stochastic differential equation; electrostatic repulsion; Bessel process;
product form stationary distribution; skew-symmetry condition; obliquely reflected Brownian motion.
AMS classification: 60H10; 60J50.
1
Université d’Orléans, MAPMO-FDP, F-45067 Orléans (dominique.lepingle@univ-orleans.fr)
1
2
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
local time type in the r.h.s. of (1) since the repulsion forces are sufficiently strong. As
−∇Φ(x) = −
k
X
φ0r (nr .x − br )nr ,
r=1
the repulsion from the faces Fr = {x ∈ S : nr .x = br } points in a normal direction into
the interior of S. This setting was applied in [3] to a finite model of interacting Brownian
particles given by
k
= d−1
√−er
nr
= er+1
2
br
= 0 √
φr (u) = −β 2 ln u with β > 0.
When β = 1 (resp. 2, resp. 4), this system is associated with the Dyson Brownian motion [7] and describes the motion of the eigenvalues of a symmetric (resp. Hermitian, resp.
symplectic) Brownian motion.
We now introduce vectors q1 , . . . , qk with qr .nr = 0 for r = 1, . . . , k and consider a new
equation
P
dXt = dBt − kr=1 φ0r (nr .x − br )(nr + qr )dt
(2)
.
X0 ∈ S
This is a singular drift and because of the obliqueness induced by q1 , . . . , qk convex analysis
cannot be used as in [2] to get strong existence and uniqueness of the associated stochastic
differential system. A similar problem has been recently tackled by Gassous, Răşcanu and
Rotenstein in [9]. They assume the domain of the subdifferential operator to be closed, which
is not compatible with our hypotheses φr (0+) = +∞.
In this paper we shall concentrate on the nonnegative quadrant in R2 as state space S
and a drift term that derives from electrostatic repulsive forces. It means that the potential
functions φr are logarithmic. In dimension one the associated process is a Bessel process.
Thus we may call the solution to (2) a two-dimensional Bessel process with oblique drift, or
more briefly an oblique two-dimensional Bessel process (O2BP for short). When the drift acts
normally (q1 = q2 = 0) and the two components of the Brownian motion are independent,
we recognize the usual two-dimensional Bessel process which is made of two independent
Bessel processes. A few from all the nice properties of one-dimensional Bessel processes
are preserved in our setting: same scaling property, abolute continuity between the laws of
processes with different parameters, a martingale property.
We obtain strong existence and uniqueness for a large set of parameters and initial conditions, but not for all possible values. In the proofs we naturally fall into the crucial question
of hitting the corner, that is the non-smooth part of the boundary. Using McKean’s argument
on the asymptotic behavior of continuous local martingales obtained by time change from
the real Brownian martingale, we are able to state several sufficient conditions to prevent
the processes from hitting the corner. Actually we do not have any convenient function such
as in [33] where some explicit harmonic function provided a full answer to the question of
hitting the corner for an obliquely reflected Brownian motion in a wedge. Though partial,
our results are reminiscent of those in [33], but our methods are not powerful enough to
allow for necessary conditions. However we shall not restrict to processes avoiding the corner
and, depending on the parameters, we will get existence and uniqueness (in a strong sense)
sometimes in the whole quadrant, sometimes in the punctured quadrant.
For theoretical as well as practical reasons, a great deal of interest was taken in the question
of recurrence of the Brownian motion with a constant drift vector and oblique reflection
and in the computation of the invariant measure ([35],[38],[11],[30]). Under the assumption
that the directions of reflection satisfy a skew-symmetry condition, it was proved that the
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
3
invariant measure has exponential product form density ([13],[14],[36],[37]). Motivated by the
so-called Atlas model of equity markets presented in [8], some authors ([16],[17],[19]) have
recently studied Brownian motions on the line with rank dependent local characteristics. This
model is strongly related to Brownian motion reflected in polyhedral domains. The invariant
probability density has an explicit exponential product form when the volatility is constant
[25] and a sum of products of exponentials form when the volatility coefficients depend on
the rank ([20],[18]). This last kind of density was previously obtained in [5] for a Brownian
motion in a wedge with oblique reflection.
A neighboring way has been recently explored in [24]. Here the process is a Brownian
motion with a drift term that is continuous and depends obliquely, via a regular potential
function, on the position of the process relative to a polyhedral domain. Under the same
skew-symmetry condition as in [13], the invariant density has an explicit product form again.
Following the way in [14], [6] and [11], a more general question is the recurrence or transience of an O2BP. It seems not easy to answer this question. The method in [15], which
provides a full answer for the obliquely reflected Brownian motion in the quadrant, appears
to break down here. Mimicking the computation in [24], we just calculate under a skewsymmetry condition an invariant measure in product form. Now the terms of the product
are gamma distributions with explicit parameters.
Another topic of interest in [24] lies in its Remark 4.12: when a scale parameter goes to
zero, the exponentially reflected Brownian motion should converge to the obliquely reflected
Brownian motion. It is typically a penalty method. This kind of approximation by a sequence
of diffusions with regular drifts living on the whole Euclidean space is also the subject of [29].
With regard to our framework, it is easily checked that a sequence of real Bessel processes
with dimension decreasing to 1 pathwise decreases to a real reflected Brownian motion. It
could be interesting to consider a sequence of oblique two-dimensional Bessel processes with
parameters going together to zero. It should converge to a two-dimensional obliquely reflected
Brownian motion. This time, this would be an interior approximation, as it was done in [38]
and [26] in the particular setting of stationary Markov processes associated with Dirichlet
forms. We leave aside this point for further investigation.
The rest of the paper is organized as follows. In Section 2 we recall some trajectorial
properties of usual Bessel processes. In Section 3 we give the definition of an O2BP and
state two main lemmas of repeated use in the sequel. Section 4 is devoted to the proof of
sufficient conditions to avoid the corner of the quadrant. The main theorems of existence
and uniqueness of an O2BP are given in Section 5. In Section 6 we discuss the question of
hitting the sides of the quadrant. In Section 7 we consider two particular cases where there
exist simple functions of O2BPs that are local martingales and obtain some information on
the asymptotic behavior of the paths. We leave the trajectorial point of view in Section 8
to tackle questions of absolute continuity of the law of an O2BP with respect to the product
of laws of real Bessel processes. The final Section 9 introduces a skew-symmetry condition
that allows us to obtain existence of a stationary probability in form of the product of two
gamma distributions.
2. Some properties of Bessel processes
A Bessel process of dimension d > 1, starting at r ≥ 0, is the unique solution of the
stochastic differential equation
Z
d−1 t 1
ds.
(3)
Rt = r + Wt +
2
0 Rs
where W is a standard driftless real-valued Brownian motion starting at 0.
4
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
We know that:
• d ≥ 2 : the point 0 is a polar set ([27], Proposition V.2.7);
• d = 2 : lim supt→∞ Rt = +∞, lim inf t→∞ Rt = 0 ([27], Theorem V.2.8);
• 1 < d < 2 : the point 0 is instantaneously reflecting ([27], Proposition XI.1.5).
More precisely ([21], p.337 and p.339),
• Dimension d > 2:
P(Rt > 0, ∀t > 0) = 1,
P(Rt → ∞, t → ∞) = 1,
R2−d is a local martingale.
• Dimension d = 2:
P(Rt > 0, ∀t > 0) = 1,
P(supt>a Rt = ∞, inf t>a Rt = 0) = 1 for any a > 0,
ln R is a local martingale.
• Dimension 1 < d < 2:
P (Rt > 0, ∀t > a) = 0 for any a > 0.
Corollary 1. When d ≥ 2, for any a > 0,
Z ∞
ds
= ∞ a.s.
Rs2
a
Proof. From Itô’s formula
R t s d−2 R t ds
ln Rt = ln Ra + a dW
Rs + 2
a Rs2 .
R ∞ ds
From ([27], Proposition IV.1.26) we know that on the set { a R2 < ∞}, the continuous local
s
Rt s
martingale a dW
converges
as
t
→
∞.
But
we
have
seen
that
ln Rt does not converges a.s.
Rs
and the conclusion follows.
We shall also use an absolute continuity result ([27], Exercise XI.1.22 or [21], Proposition
6.1.5.1). On the canonical space Ω = C(R+ , R+ ), we denote by R the canonical map Rt (ω) =
ω(t), by Rt = σ(Rs , s ≤ t) the canonical filtration and by Pdr the law of the Bessel process of
dimension d ≥ 2 starting at r > 0. Then,
d−2
Z
Rt 2
(d − 2)2 t ds
d
. P2r |Rt .
(4)
Pr |Rt =
exp −
2
r
8
0 Rs
3. Oblique Bessel process in the quadrant
The general state space is the nonnegative quadrant S = R+ × R+ . The corner 0 = (0, 0)
will play a crucial role and in some cases it will be necessary to restrict the state space to the
punctured nonnegative quadrant S 0 = S \ {0}.
Definition 2. Let (Bt , Ct ) be a driftless Brownian motion in the plane starting from 0, with
covariance matrix
1 ρ
.
ρ 1
It is adapted to a filtration F = (Ft ) with usual conditions. Let α, β, γ, δ be four real constants
with α > 0, δ > 0. We say that an F-adapted continuous process (X, Y ) with values in S is
an Oblique Two-dimensional Bessel Process (O2BP) if for any t ≥ 0
R t ds
Rt
Xt = X0 + Bt + α 0 X
+ β 0 Ydss ≥ 0
s
Rt
R t ds
(5)
Yt = Y0 + Ct + γ 0 X
+ δ 0 Ydss
≥ 0
s
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
5
where X0 and Y0 are non-negative F0 -measurable random variables, and
Rt
Rt
1
ds
=
0
1
ds = 0
{X
=0}
s
R0t {Ys =0} ds
R0t
ds
<
∞
1
1
0 {Ys >0} Ys < ∞ .
0 {Xs >0} Xs
The drift in (5) is of type (2) with d = k = 2 and
φ1 (x) = −α ln x
b1 = 0
n1 = (1, 0)
q1 = (0, αγ )
φ2 (y) = −δ ln y
b2 = 0
n2 = (0, 1)
q2 = ( βδ , 0)
The case with β = γ = 0 is a particular case of (1). In the sequel, we will note (U, V ) the
solution of the system
Rt
Ut = X0 + Bt + α 0 Udss ≥ 0
Rt
(6)
Vt = Y0 + Ct + δ 0 Vdss
≥ 0.
The processes U and V are Bessel processes. Actually, U is a Bessel process of dimension
2α + 1, and the point 0 is intanstaneously reflecting for U if α < 21 and polar if α ≥ 12 . If
ρ = 0, U 2 + V 2 is the square of a Bessel process of dimension 2α + 2δ + 2 [31], and so the
corner 0 is polar for (U, V ) in this case.
Comparison between X and U , Y and V will play a key role in the construction of the
solution (X, Y ) and the study of its behavior close to the sides of the quadrant. The following
simple lemma will be of constant use.
Lemma 3. For T > 0, α > 0, let x1 and x2 be nonnegative continuous solutions on [0, T ] to
the equations
Rt
x1 (t) = v1 (t) + α 0 x1ds(s)
Rt
x2 (t) = v2 (t) + α 0 x2ds(s)
where v1 , v2 are continuous functions such that 0 ≤ v1 (0) ≤ v2 (0), and v2 − v1 is nondecreasing. Then x1 (t) ≤ x2 (t) on [0, T ].
Proof. Assume there exists t ∈ (0, T ] such that x2 (t) < x1 (t). Set
τ := max{s ≤ t : x1 (s) ≤ x2 (s)} .
Then,
x2 (t) − x1 (t) = x2 (τ ) − x1 (τ ) + (v2 (t) − v1 (t)) − (v2 (τ ) − v1 (τ )) + α
≥ 0,
a contradiction.
Rt
1
τ ( x2 (s)
−
1
x1 (s) )ds
Another main tool will be the following convergence result, whose argument goes back to
McKean ([23], p.31 and p.47). The statement and the proof below are borrowed from ([27],
Theorem V.1.7 and Proposition V.1.8). The only change lies in the introduction of a stopping
time τ up to which the local martingale M is now defined.
Lemma 4. Let (τn )n≥1 be a nondecreasing sequence of stopping times with limit τ . let M be
a continuous process with M0 = 0 defined on the time set ∪n≥1 [0, τn ] such that the stopped
process (Mτn ∧t )t≥0 is a uniformly integrable martingale for any n ≥ 1. Let (hM it )0≤t≤τ be
its quadratic variation.
(1) On {hM iτ < ∞}, limt→τ Mt exists a.s. in R.
(2) On {hM iτ = ∞}, lim supt→τ Mt = − lim inf t→τ Mt = +∞ a.s.
6
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
Proof. (1) Existence and uniqueness of the quadratic variation hM i on ∪n≥1 [0, τn ] has
been obtained in ([27], Exercise IV.1.48). So we may define hM iτ := limn→∞ hM iτn . For any
p ≥ 1 let
σp = inf{t ≥ 0 : hM it ≥ p}.
The stopped process (Mt∧τn ∧σp )t≥0 is a L2 -bounded martingale and as m, n → ∞
E[(Mτn ∧σp − Mτm ∧σp )2 ] = E[|hM iτn ∧σp − hM iτm ∧σp |] → 0.
We set
M p := lim Mτn ∧σp = lim Mt∧σp .
n→∞
t→τ
On {hM iτ < ∞}, the stopping times σp are a.s. infinite from some p on and we can set
Mτ := limp→∞ M p . Thus on this set
Mt → Mτ
as
t → τ.
(2) Let for any t ≥ 0
Tt = inf{0 ≤ s ≤ τ : hM is > t}.
There exist an enlargement (Ω̃, (F̃t ), P̃) of (Ω, (FTt ), P) and a Brownian motion β̃ on Ω̃ independent of M such that the process
MTt
if t < hM iτ
Bt =
Mτ + β̃t−hM iτ if t ≥ hM iτ
is a standard linear Brownian motion. As
lim sup Bt = − lim inf Bt = +∞
t→∞
t→∞
a.s.
we obtain on {hM iτ = ∞}
lim sup MTt = − lim inf MTt = +∞
t→∞
t→∞
and therefore
lim sup Mt = − lim inf Mt = +∞
t→τ
t→τ
.
We will also need the following consequence of the results in [2] on multivalued stochastic
differential systems, completed with the method used in [3] and developed in [22] to check
the lack of additional boundary process.
Proposition 5. Let α > 0, δ ≥ 0, σ = (σji ; i, j = 1, 2) a 2 × 2-matrix, (B, C) a Brownian
motion in the plane, b1 and b2 two Lipschitz functions on R2 , Z01 and Z02 two F0 -measurable
nonnegative random variables. There exists a unique strong solution (Z 1 , Z 2 ) to the system
Rt
Rt
Zt1 = Z01 + σ11 Bt + σ21 Ct + α 0 Zds1 + 0 b1 (Zs1 , Zs2 )ds
s
Rt
Rt
(7)
Zt2 = Z02 + σ12 Bt + σ22 Ct + δ 0 Zds2 + 0 b2 (Zs1 , Zs2 )ds
s
with the conditions
Zt1
≥ 0 if δ = 0 and
Zt1
≥
0, Zt2
≥ 0 if δ > 0.
It is worth noticing that the solutions to (5) enjoy the Brownian scaling property. It
means that if (X, Y ) is a solution to (5) starting from (X0 , Y0 ) with driving Brownian motion
(Bt , Ct ), then for any c > 0 the process (Xt0 := c−1 Xc2 t , Yt0 := c−1 Yc2 t ; t ≥ 0) is a solution to
(5) starting from (c−1 X0 , c−1 Y0 ) with driving Brownian motion (c−1 Bc2 t , c−1 Cc2 t ).
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
7
4. Avoiding the corner
We shall see in Section 5 that existence and uniqueness of the solution to (5) are easily
obtained as soon as the solution process keeps away from the corner. Thus the question of
attaining the corner in finite time is of great interest. For some class of reflection matrix, a
necessary and sufficient condition is given in [28]. Unfortunately we are not able to provide
such a complete answer and we have to be content with a set of sufficient conditions ensuring
nonattainability. We will just see at the end of Section 5 a degenerated case where the corner
is reached with full probability in finite time.
Our sufficient conditions are stated in the following theorem. In some cases (conditions
C2a and C2b below), the comparison with Bessel processes suffices to conclude. In other cases
(conditions C1 and C3 below), we are first looking for a C 2 -function f on the punctured
quadrant S 0 with limit −∞ at the corner. Then we use Lemma 4 to show that f (Xt , Yt )
cannot go to −∞ in finite time.
Theorem 6. Let (X, Y ) be a solution to (5). We set
τ 0 := inf{t > 0 : (Xt , Yt ) = 0}
with the usual convention inf ∅ = ∞. Then P(τ 0 < ∞) = 0 if one of the following conditions
is satisfied:
(1) C1 : β ≥ 0, γ ≥ 0 and −1 ≤ ρ ≤ α + δ.
(2) C2a : α ≥ 21 and β ≥ 0
(3) C2b : δ ≥ 21 and γ ≥ 0
(4) C3 : There exist λ > 0 and µ > 0 such that
• λα + µγ ≥ 0
• λβ
p + µδ ≥ 0
p
• ( λ(λα + µγ) + µ(λβ + µδ))2 ≥ 12 (λ2 + µ2 + 2ρλµ).
Proof.
Condition C1 . For > 0 let
σ
= 1{(X0 ,Y0 )=0} inf{t > 0 : Xt + Yt ≥ }
0,
τ
= inf{t > σ : (Xt , Yt ) = 0} .
As ↓ 0, σ ↓ 0 and τ 0, ↓ τ 0 . We set Rt = Xt2 + Yt2 . From Itô’s formula we get for
t ∈ [σ , τ 0, )
ln Rt
Rt
= ln Rσ + 2 σ
≥ ln Rσ +
Rt
Xs dBs +Ys dCs
+ 2 σ (α+δ)ds
Rs
Rs
Rt
Mt + 2 σ P (XRs2,Ys ) ds
s
+2
Rt
Xs
σ (β Ys
Ys ds
+ γX
) − 4ρ
s Rs
Rt
σ
Xs Ys
ds
Rs2
where M is a continuous local martingale on [σ , τ 0, ) and P (x, y) is the second degree
homogeneous polynomial
P (x, y) = (α + δ)(x2 + y 2 ) − 2ρxy.
This polynomial is nonnegative on S if ρ ≤ α + δ. Therefore
Z τ 0,
P (Xs , Ys )
0≤
ds ≤ ∞
Rs2
σ
From Lemma 4 we know that depending on wether hM iτ 0, is finite or not, the local martingale
Mt either converges in R as t → τ 0, or oscillates between +∞ and −∞. It cannot converge
to −∞. Thus Rτ 0, > 0 on {τ 0, < ∞}, proving that P(τ 0, < ∞) = 0 and finally that
P(τ 0 < ∞) = 0.
Condition C2a (resp. C2b ). Remember the Bessel processes U and V in (6). From Lemma 1
8
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
we get Xt ≥ Ut (resp. Yt ≥ Vt ) and in this case 0 is polar for U (resp. V), so Xt > 0 (resp.
Yt > 0) for t > 0.
Condition C3 . We use again > 0, σ and τ 0, . We set St = λXt + µYt for t ≥ 0, λ > 0 and
µ > 0. From Itô’s formula we get for t ∈ [σ , τ 0, )
ln St
Rt
Rt
s
+
(λα
+
µγ)
= ln Sσ + σ λdBsS+µdC
σ
s
Rt
− 21 (λ2 + µ2 + 2ρλµ) σ Sds2
s
Rt
s ,Ys )
= ln Sσ + Mt + σ PX(X
ds
2
s Ys S
ds
Xs Ss
+ (λβ + µδ)
Rt
ds
σ Ys Ss
s
where M is a continuous local martingale on [σ , τ 0, ) and P (x, y) is the second degree
homogeneous polynomial
P (x, y)
= λ(λβ + µδ)x2 + µ(λα + µγ)y 2 + [λ(λα + µγ) + µ(λβ + µδ) − 21 (λ2 + µ2 + 2ρλµ)]xy .
This polynomial is nonnegative on S if
p
1 2
(λ + µ2 + 2ρλµ) − [λ(λα + µγ) + µ(λβ + µδ)] ≤ 2 λ(λβ + µδ)µ(λα + µγ).
2
This is exactly the condition C3 . Therefore
Z t
P (Xs , Ys )
0≤
ds < ∞
2
σ Xs Ys Ss
and so
Z
τ 0,
P (Xs , Ys )
ds ≤ ∞ .
Xs Ys Ss2
σ
Using again Lemma 4, we see that the continuous local martingale M either converges to a
finite limit or oscillates between +∞ and −∞ when t → τ 0, . It cannot converge to −∞ and
thus Sτ 0, > 0 on {τ 0, < ∞}, proving P(τ 0, < ∞) = 0. Letting finally → 0 we obtain
P(τ 0 < ∞) = 0.
0≤
Condition C3 is not explicit. We give two concrete examples of application.
Corollary 7. When ρ = 0, α = δ and |β| = |γ|, the condition C3 is satisfied if
(8)
• β 2 ≤ α − 41
• −β ≤ α − 14
when β = −γ
when β = γ < 0 .
Proof. In both cases we take λ = µ > 0.
When β = −γ ≥ 0, the condition C3 writes
• α − β√≥ 0
p
√
• 1 ≤ ( α + β + α − β)2 = 2α + 2 α2 − β 2 .
If β 2 ≤ α − 14 , then
α2 − β 2 ≥ α2 − α +
1
≥0
4
and
(1 − 2α)2 = 1 − 4α + 4α2 ≤ 4(α2 − β 2 ).
When β = γ < 0, the condition C3 writes
• α+β √
≥0
• 1 ≤ (2 α + β)2 = 4(α + β)
and this is −β ≤ α − 41 .
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
9
Corollary 8. Assume
• max{α, δ} ≥ 12
• 2ρ ≤ βδ + αγ .
Then P(τ 0 < ∞) = 0 .
Proof. We may assume α ≥ 12 . If β ≥ 0, then condition C2a holds true and the conclusion
follows. If β < 0, we will use condition C3 . We take λ = δ and µ = −β. Then
λα + µγ = δα − βγ
≥ δα − βα(2ρ − βδ )
= αδ (δ 2 + β 2 − 2ρβδ)
≥ αδ (δ − ρβ)2
≥ 0
and
=
≥
≥
≥
p
p
( λ(λα + µγ) + µ(λβ + µδ))2 − 21 (λ2 + µ2 + 2ρλµ)
λ(λα + µγ) − 12 (λ2 + µ2 + 2ρλµ)
α(δ 2 + β 2 − 2ρβδ) − 12 (δ 2 + β 2 − 2ρβδ)
(α − 12 )(δ − ρβ)2
0.
5. Existence and uniqueness
We now proceed to the question of existence and uniqueness of a global solution to (5).
We consider separately the three cases: first β ≥ 0 and γ ≥ 0, second βγ < 0, third β ≤ 0
and γ ≤ 0. In the first and second cases, we construct the solution by switching from one
side to the other and patching the paths together. Thus it is essential to avoid the corner,
as it was supposed in [29] in order to weakly approximate an obliquely reflected Brownian
motion. The third case uses a different proof and does not requires avoiding the corner. All
three proofs heavily use the comparison method of Lemma 3.
5.1. Case β ≥ 0 and γ ≥ 0.
Theorem 9. Assume β ≥ 0, γ ≥ 0 and one of the conditions C1 . C2a , C2b , C3 is satisfied.
(1) There is a unique solution to (5) in S 0 .
(2) There is a solution to (5) in S starting from 0.
(3) If αδ ≥ βγ, there is a unique solution to (5) in S.
Proof. 1. Let a > 0, > 0 and define for (x, z) ∈ R+ × R
ψ (x, z) :=
1
.
max(γx + z, α)
This is a Lipschitz function. From Proposition 5 we know that the system
R t ds
Rt
Xt = X0 + Bt + α 0 X
+ αβ 0 ψ (Xs , Zs )ds ≥ 0
s
Rt
(9)
Zt = −γX0 + α(Y0 + 1{Y0 =0} a) − γBt + αCt + α(αδ − βγ) 0 ψ (Xs , Zs )ds
has a unique solution. Let
τY := inf{t > 0 : γXt + Zt < α} .
10
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
If 0 < η < < a we deduce from the uniqueness that (X , Z ) and (X η , Z η ) are identical on
[0, τY ]. Patching together we can set
Xt := lim→0 Xt
Yt := lim→0 α1 (γXt + Zt )
on {(ω, t) ∈ Ω × [0, ∞) : Y0 (ω) > 0 and 0 ≤ t < τY0 (ω)}, where
τY0 := lim τY .
→0
On this set, (X, Y ) is the unique solution to (5). As we already noted, we have Xt ≥ Ut and
Yt ≥ Vt . Therefore, on {Y0 > 0} ∩ {τY0 < ∞},
R τY0 ds
R τY0 ds
R τY0 ds R τY0 ds
and
0 Xs ≤ 0 Us < ∞
0 Ys ≤ 0 V s < ∞
and we can define
(10)
Xτ 0
Y
Yτ 0
Y
R τ 0 ds
R τY0 ds
:= limt→τ 0 Xt = X0 + Bτ 0 + α 0 Y X
+
β
0 Y
Y
Y
R τY0 ds s
R τY0 ds s
:= limt→τ 0 Yt = Y0 + Cτ 0 + γ 0 Xs + δ 0 Ys .
Y
Y
We have Yτ 0 = 0. From Theorem 6 we know that Xτ 0 > 0. In exactly the same way we can
Y
Y
construct a solution on {Y0 > 0} in the interval [T1 , T2 ], where T1 = τY0 , T2 = inf{t > T1 :
Xt = 0}. Iterating, we get a solution on {Y0 > 0} × [0, limn→∞ Tn ) where
T2p
:= inf{t > T2p−1 : Xt = 0}
T2p+1 := inf{t > T2p : Yt = 0} .
On {Y0 > 0} ∩ {limn→∞ Tn < ∞} we set Xlimn→∞ Tn := limp→∞ XT2p = 0 and Ylimn→∞ Tn :=
limp→∞ YT2p+1 = 0. The polarity of 0 entails this is not possible in finite time and thus
limn→∞ Tn = ∞. So we have obtained a unique global solution on {Y0 > 0}. In the same
way we obtain a unique global solution on {X0 > 0} and as P((X0 , Y0 ) = 0) = 0 the proof is
complete.
2. Assume now X0 = Y0 = 0. Let (yn )n≥1 be a sequence of real numbers (strictly)
decreasing to 0. From the above paragraph it follows there exists for any n ≥ 1 a unique
solution (X n , Y n ) with values in S 0 to the system
Rt
Rt
Xtn = Bt + α 0 Xdsn + β 0 Ydsn
sR
sR
t
t
Ytn = yn + Ct + γ 0 Xdsn + δ 0 Ydsn .
s
s
Let
τ := inf{t > 0 : Xtn+1 < Xtn } .
Using Lemma 3 we obtain Ytn+1 ≤ Ytn on [0, τ ]. We note that (Xτn , Yτn ) ∈ S 0 on {τ < ∞}.
On {Yτn+1 = Yτn } ∩ {τ < ∞}, since Xτn+1 = Xτn and the solution starting at time τ is unique,
it follows that Xtn+1 = Xtn and Ytn+1 = Ytn on [τ, ∞). On {Yτn+1 < Yτn } ∩ {τ < ∞}, the
continuity of solutions at time τ entails there exists ρ > 0 such that Ytn+1 ≤ Ytn on [τ, τ + ρ].
A second application of Lemma 3 proves that Xtn+1 ≥ Xtn on [τ, τ + ρ], a contradiction to
the definition of τ . Therefore P(τ = ∞) = 1. It follows that Xtn+1 ≥ Xtn and Ytn+1 ≤ Ytn for
any t ∈ [0, ∞), and we may define
Xt := lim ↑ Xtn
n→∞
As
Ytn
Yt := lim ↓ Ytn .
n→∞
≥ Vt where (U, V ) is the solution to (6) with X0 = Y0 = 0, we have
Rt
Rt
Xt = Bt + α limn→∞ 0 Xdsn + β limn→∞ 0 Ydsn
s
R t ds
R t sds
= Bt + α 0 X
+
β
0 Ys
s
< ∞
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
11
and also
Rt
Rt
Yt = limn→∞ yn + Ct + γ limn→∞ 0 Xdsn + δ limn→∞ 0 Ydsn
s
s
R t ds
R t ds
= Ct + γ 0 X
+
δ
0 Ys
s
< ∞.
3. Assume finally αδ − βγ ≥ 0. As the conclusion holds true if β = γ = 0, we may also
assume β > 0. Let (X, Y ) be the solution to (5) with X0 = Y0 = 0 obtained in the previous
paragraph and let (X 0 , Y 0 ) be another solution. Considering (X n , Y n ) again and replacing
(X n+1 , Y n+1 ) with (X 0 , Y 0 ), the previous proof works and we finally obtain Xt0 ≥ Xt and
Yt0 ≤ Yt . Then,
Z t
1
1
(αδ − βγ)( 0 −
)ds ≤ 0.
(11)
0 ≤ δ(Xt0 − Xt ) − β(Yt0 − Yt ) =
X
X
s
0
s
Thus Xt0 = Xt and Yt0 = Yt , proving uniqueness. Replacing (δ, β) with (γ, α) in equation (11)
we obtain the same conclusion if γ > 0.
Remark 10. The statement in Theorem 9 is not complete since the problem of uniqueness
when starting at the corner and αδ < βγ is not solved. When considering the solution (X, Y )
in the above proof of existence, we have noted that X 0 ≥ X and Y 0 ≤ Y for any other
solution (X 0 , Y 0 ). Thus uniqueness in law would be sufficient to obtain path uniqueness. A
possible way to prove weak uniqueness could be the method in [1]. This would be far from
our trajectorial methods and we don’t go further in that direction.
5.2. Case βγ < 0.
Theorem 11. Assume β > 0, γ < 0 and one of the conditions C2a or C3 is satisfied. Then,
there exists a unique solution to (5) in S 0 .
Proof. The proof is similar to the proof of 1 in Theorem 9. The only change is that now
Yt ≤ Vt . Therefore, on {Y0 > 0} ∩ {τY0 < ∞},
Z τ0
Z τ0
Y ds
Y ds
δ
≤ Vτ 0 − Y0 − Cτ 0 − γ
<∞
Y
Y
Ys
Us
0
0
and we can define Xτ 0 and Yτ 0 as previously done. The application of Theorem 6 to the
Y
Y
process on the time interval [0, τY0 ] shows that Xτ 0 > 0.
Y
There is obviously an analogous statement if β < 0, γ > 0.
5.3. Case β ≤ 0 and γ ≤ 0. In this case we can give a full answer to the question of existence
and uniqueness. Our condition of existence is exactly the condition found in [34] for the
reflected Brownian in a wedge being a semimartingale, i.e. there is a convex combination of
the directions of reflection that points into the wedge from the corner. It amounts to saying
that the matrix
α β
γ δ
is completely-S in the terminology of ([32],[6],[37],[4],[28],[29],[30]).
Theorem 12. Assume β ≤ 0 and γ ≤ 0.
(1) If αδ > βγ, there exists a unique solution to (5) in S.
(2) If αδ ≤ βγ and 1 + ρ + |α + γ| + |β + δ| > 0, there is no solution.
(3) If 1 + ρ = α + γ = β + δ = 0 and (X0 , Y0 ) 6= 0 there exists a unique solution.
12
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
(4) If 1 + ρ = α + γ = β + δ = 0 and (X0 , Y0 ) = 0 there is no solution.
Proof. 1. Assume first αδ > βγ.
a) Existence. Let (hn , n ≥ 1) be a (strictly) increasing sequence of bounded positive nonincreasing Lipschitz functions converging to 1/x on (0, ∞) and to +∞ on (−∞, 0]. For instance
we can take
hn (x) = (1 − n1 ) x1
on [ n1 , ∞)
= n−1
on (−∞, n1 ] .
We consider for each n ≥ 1 the system
(12)
Rt
Rt
Xtn = X0 + Bt + α 0 Xdsn + β 0 hn (Ysn )ds
s
Rt
Rt
Ytn = Y0 + Ct + γ 0 hn (Xsn )ds + δ 0 Ydsn .
s
From Proposition 5 it follows there exists a unique solution to this system. We set
τ := inf{s > 0 : Xsn+1 > Xsn } .
We have hn+1 (Xtn+1 ) ≥ hn (Xtn ) on [0, τ ]. A first application of Lemma 3 shows that Ytn+1 ≤
Ytn on [0, τ ]. Since hn+1 (Yτn+1 ) > hn (Yτn ) on {τ < ∞}, we deduce from the continuity of
solutions that there exists σ > 0 such that hn+1 (Ytn+1 ) ≥ hn (Ytn ) on [τ, τ + σ]. A second
application of Lemma 3 shows that Xtn+1 ≤ Xtn on [τ, τ + σ], a contradiction to the definition
of τ . Thus P(τ = ∞) = 1 proving that on the whole [0, ∞) we have Xtn+1 ≤ Xtn and
Ytn+1 ≤ Ytn . Then we can set for any t ∈ [0, ∞)
Xt := lim Xtn
n→∞
and
Yt := lim Ytn .
n→∞
If αδ > βγ, there is a convex combination of the directions of repulsion pointing into the
positive quadrant, i.e. there exist λ > 0 and µ > 0 such that λα + µγ > 0 and µδ + λβ > 0.
For n ≥ 1 and t ≥ 0,
(13)
λUt + µVt ≥ λXtn + µYtn
Rt
≥ λX0 + µY0 + λBt + µCt + (λα + µγ) 0
Letting n → ∞ in (13) we obtain
Z t
0
ds
< ∞ and
Xs
Z
0
t
ds
Xsn
+ (µδ + λβ)
Rt
ds
0 Ysn
.
ds
< ∞.
Ys
Then we may let n go to ∞ in (12) proving that (X, Y ) is a solution to (5).
b) Uniqueness. Let (X 0 , Y 0 ) be another solution to (5). Replacing (X n+1 , Y n+1 ) with (X 0 , Y 0 )
we follow the above proof to obtain for t ∈ [0, ∞) and n ≥ 1
Xt0 ≤ Xtn
and Yt0 ≤ Ytn
Xt0 ≤ Xt
and Yt0 ≤ Yt .
Letting n → ∞ we conclude
With the same λ > 0 and µ > 0 as above,
Z t
1
1
1
1
0
0
− 0 ) + (µδ + λβ)( − 0 )]ds ≤ 0
0 ≤ λ(Xt − Xt ) + µ(Yt − Yt ) =
[(λα + µγ)(
Xs Xs
Ys Ys
0
and therefore Xt0 = Xt , Yt0 = Yt .
2. If αδ ≤ βγ there exist λ > 0 and µ > 0 such that λα + µγ ≤ 0 and µδ + λβ ≤ 0. For that,
just take
(14)
α
µ
−β
≤ ≤
.
−γ
λ
δ
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
13
Let us consider the nonnegative quadratic (λ2 + µ2 + 2ρλµ). It is positive if ρ ≥ 0. It is larger
than (1 − ρ2 )λ2 > 0 if −1 < ρ < 1. It may be positive if ρ = −1 and |α + γ| + |β + δ| > 0
because from (14) we may take λ 6= µ. Thus, if (X, Y ) is a solution to (5), for any t ≥ 0,
0 ≤ λXt + µYt ≤ λX0 + µY0 + λBt + µCt .
This is not possible since the paths of the Brownian martingale (λ2 +µ2 +2ρλµ)−1/2 (λBt +µCt )
are not bounded below. So there is no global solution.
3. Assume now 1 + ρ = |α + γ| + |β + δ| = 0 and X0 + Y0 > 0. The system becomes
Rt
R t ds
− δ 0 Ydss ≥ 0
Xt = X0 + Bt + α 0 X
s
R t ds
Rt
(15)
Yt = Y0 − B t − α 0 X
+ δ 0 Ydss ≥ 0.
s
This entails for any t ≥ 0
Xt + Yt = X0 + Y0
and the first equation in (15) reduces to
Z t
Z t
ds
ds
(16)
0 ≤ Xt = X0 + Bt + α
−δ
≤ X0 + Y0 ..
X
X
+
Y0 − Xs
s
0
0
0
Clearly pathwise uniqueness holds for equation (16) since if there are two solutions X and
X 0 , for any t ≥ 0
Rt
Rt
(Xt − Xt0 )2 = 2α 0 (Xs − Xs0 )( X1s − X10 )ds − 2δ 0 (Xs − Xs0 )( X0 +Y10 −Xs − X0 +Y10 −X 0 )ds
s
s
≤ 0.
Consider now for any c > 0 the equation
Z
0 ≤ Xt = X0 + Bt + α
0
t
ds
−δ
Xs
Z
0
t
ds
≤ c.
c − Xs
This is a one-dimensional equation of type (1) associated with the convex function
φ(x) = −α ln x − δ ln(c − x)
for
0 < x < c.
It has a unique solution living on the interval [0, c]. Thus there is a weak and then a unique
strong solution to (16). Setting
Yt = X0 + Y0 − Xt
we obtain a unique strong solution to (15).
4. We must have
Xt + Yt = X0 + Y0 = 0
and thus Xt = Yt = 0. But this is not possible for a solution to (5).
We end this section by considering a degenerate case where the solution hits the corner
with probability one.
Proposition 13. Assume ρ = 1, αδ > βγ, max{α, δ} <
P(τ 0 < ∞) = 1.
1
2
and max{β, γ} ≤ 0. Then,
Proof. We set
0
τX
0
τY
:= inf{t > 0 : Xt = 0}
:= inf{t > 0 : Yt = 0} .
The dimension of each Bessel process U and V in (6) is less than 2, and X ≤ U , Y ≤ V .
0 < ∞ = 1) and P(τ 0 < ∞ = 1). Assume first α + β ≤ γ + δ. Using Lemma 3 we
Then P(τX
Y
0 , ∞). Therefore P(τ 0 < ∞) = 1. Same result
obtain Xt ≤ Yt ≤ Vt on the time interval [τX
when α + β ≥ γ + δ.
14
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
In the following pictures, we display the x-repulsion direction vector rx = (α, γ) and the
y-repulsion direction vector ry = (β, δ) in three illustrative instances.
y
6
β ry
y
y
6
6
ry
δ @
I
@
ry
1rx
γ
MB
B
B
1
rx
@
@H
α
-x
•
β > 0, γ > 0, αδ > βγ
-x
•
β < 0, γ > 0, αβ + γδ = 0
HH
H
jrx
H
-x
•
β < 0, γ < 0, αδ > βγ
6. Avoiding the sides
We now consider the question of hitting a side of the quadrant. Remember the definitions
(17)
0
τX
0
τY
:= inf{t > 0 : Xt = 0}
:= inf{t > 0 : Yt = 0} .
0 < ∞) = 0 if α ≥ 1 and β ≥ 0. Conversely, a comparison with
We already know that P(τX
2
0 < ∞) = 1 if α < 1 and β ≤ 0. If we know that the
the Bessel process U shows that P(τX
2
corner is not hit and α ≥ 12 , we can get rid of the nonnegativity assumption on β. Since we
are only interested in one coordinate, we are looking for a function that is C 2 on (0, ∞) and
goes to −∞ when approaching 0. A natural candidate is the logarithmic function.
0 < ∞) = 0.
Proposition 14. Assume P(τ 0 < ∞) = 0. If α ≥ 12 , then P(τX
Proof. For η > 0 let
η
θX
= 1{X0 =0} inf{t > 0 : Xt ≥ η}
η
0,η
: Xt = 0} .
τX = inf{t > θX
η
0,η
0 . For t ∈ [θ η , τ 0,η ), from Itô’s formula
As η ↓ 0, θX
↓ 0 and τX
↓ τX
X X
R t dBs
Rt
Rt
ln Xt = ln Xθη + θη Xs + (α − 12 ) θη Xds2 + β θη
(18)
X
s
X
X
X
ds
Xs Ys
.
0,η
Since P(τ 0 < ∞) = 0, on the set {τX
< ∞} we have Yτ 0,η > 0 because Xτ 0,η = 0. On this
X
X
set, from the definition of an O2BP,
Z τ 0,η
Z τ 0,η
X
X ds
ds
<∞,
< ∞.
η
η
Xs
Ys
θX
θX
0,η
η
As Ys > 0 on some interval [χ, τX
] with positive measure and Xs > 0 on [θX
, χ] we see that
Z τ 0,η
X
ds
|β|
< ∞.
η
Xs Ys
θX
0,η
As t → τX
, the local martingale in the r.h.s. of (18) cannot converge to −∞. This entails
0,η
0 < ∞) = 0.
that P(τX < ∞) = 0 and therefore P(τX
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
15
We may also be interested in hitting either side. This time we are looking for a function
that is C 2 in the interior of S and goes to −∞ when approaching either side.
0 < ∞) = P(τ 0 < ∞) = 0 if one of
Proposition 15. Assume α ≥ 21 and δ ≥ 12 . Then P(τX
Y
the following conditions is satisfied:
(1) β ≥ 0
(2) γ ≥ 0
(3) 0 < βγ ≤ (α − 12 )(δ − 12 ).
Proof. The proofs in the cases (1) and (2) ) are direct consequences of Theorem 6 (under
conditions C2a or C2b ) and Proposition 14. Assume now the conditions in (3) hold true. For
> 0 let
σ = 1{X0 Y0 =0} inf{t > 0 : Xt Yt ≥ }
τ = inf{t > σ : Xt Yt = 0} .
For λ > 0 and µ > 0 we set
Rt = λ ln Xt + µ ln Yt .
From Itô’s formula we get for t ∈ [σ , τ )
R t λ(α− 1 ) µ(δ− 1 )
Rt
Rt = Rσ + σ ( Xλs dBs + Yµs dCs ) + σ [ X 2 2 + Y 2 2 + (λβ+µγ)
Xs Ys ]ds
s
s
R t Q(Xs ,Ys )
= Rσ + Nt + σ X 2 Y 2 ds
s
s
where N is a continuous local martingale and Q(x, y) is the second degree homogeneous
polynomial
1
1
Q(x, y) = µ(δ − )x2 + λ(α − )y 2 + (λβ + µγ)xy.
2
2
If the conditions (3) are satisfied, we may take
λ = 2(α − 12 )(δ − 12 ) − βγ > 0
µ = β2
> 0.
Then, using again conditions (3), we can check that
1
1
1
1
1
1
2
2
2
x + 2 α−
δ−
− βγ α −
y +2 α −
δ−
βxy
Q(x, y) = β δ −
2
2
2
2
2
2
is nonnegative on S. The proof terminates as previously in Theorem 6 .
7. Associated local martingales
Scale functions play a major role in the study of one-dimensional stochastic differential
equations ([27] Section VII.3). There is no equivalent functions on the plane. However, in
some particular cases we can find simple functions of O2BPs that are supermartingales or
local martingales.
Proposition 16. Assume the following set of conditions:
(19)
α > 1/2
δ > 1/2
β
γ
2δ−1 + 2α−1 ≥ ρ
ρ > −1.
Then
Mt := Xt1−2α Yt1−2δ
is a positive supermartingale on (0, ∞) which tends to 0 as t → ∞. It is a local martingale
if the third inequality in (19) is an equality.
16
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
Proof. For > 0 let
σ = 1{X0 Y0 =0} inf{t > 0 : Xt Yt ≥ }
τ = inf{t > σ : Xt Yt = 0} .
As ↓ 0, σ ↓ 0. Up to τ , we get from Itô’s formula on {σ < ∞}
Rt
s +(1−2δ)Xs dCs
Mt = Mσ + σ (1−2α)Ys dB
X 2α Y 2δ
s
s
[(1 − 2α)β + (1 − 2δ)γ + ρ(2α − 1)(2δ − 1)]
Rt
ds
σ Xs2α Ys2δ .
The function f (x, y) = x1−2α y 1−2δ is C 2 in the interior of S and goes to +∞ when approaching
the sides of the quadrant, whereas the finite variation part in the semimartingale decomposition of Mt is nonincreasing. We apply again Lemma 4 and obtain that P(τ < ∞) = 0.
Then, letting → 0, we see that M is a positive supermartingale on (0, ∞). As such it tends
to a limit H ≥ 0 when t → ∞ ([27], Corollary II.2.11 ). If β ≥ 0 and γ ≥ 0, then X ≥ U
and Y ≥ V where U and V are the Bessel processes in (6). We have seen in Section 2 that
Ut → ∞ and Vt → ∞ as t → ∞; so Mt → 0 as t → ∞. If now min{β, γ} < 0, say γ < 0,
then we have Y ≤ V . From Corollary 1 we deduce that for any > 0
Z ∞
Z ∞
ds
ds
(20)
≥
= ∞.
2
2
Y
V
σ
σ
s
s
We consider the quadratic variation hM i given by
Rt
hM it = σ Xs−4α Ys−4δ [(1 − 2α)2 Ys2 + (1 − 2δ)2 Xs2 + 2ρ(1 − 2α)(1 − 2δ)Xs Ys ]ds
Rt
2
2
+ (1−2δ)
+ 2ρ (1−2α)(1−2δ)
]ds.
= σ Ms2 [ (1−2α)
Xs Ys
X2
Y2
s
s
If ρ ≥ 0,
hM it ≥
and if
ρ2
Z
t
σ
Ms2
(1 − 2δ)2
ds
Ys2
< 1,
t
(1 − 2δ)2
ds.
Ys2
σ
In both cases, using (20) we should have hM i∞ = ∞ on the set {H > 0} ∩ {σ < ∞}, and
then lim supt Mt = − lim inf t Mt = ∞, a contradiction with Mt → H. Thus H = 0 a.s.
hM it
2
Z
≥ (1 − ρ )
Ms2
0 < ∞) = P(τ 0 < ∞) = 0. In fact this is not a
During the proof we have seen that P(τX
Y
new result since here the conditions in Proposition 15 are in force. There is another case of
interest.
Proposition 17. Assume α = δ = 1/2, βγ 6= 0, ρβγ < |βγ|, and there exists a solution to
(5) satisfying P(τ 0 < ∞) = 0. Then
Mt := γ ln Xt − β ln Yt
is a continuous local martingale on (0, ∞) and
lim sup Mt = − lim inf Mt = ∞
.
t→∞
t→∞
Proof. We deduce from Proposition 15 that Xt > 0 and Yt > 0 for any t > 0. Let again
for > 0
σ = 1{X0 Y0 =0} inf{t > 0 : Xt Yt ≥ }.
Now Itô’s formula gives
Z t
dBs
dCs
Mt = M σ +
γ
−β
Xs
Ys
σ
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
17
and the quadratic variation is
hM it
Z
t
=
σ
γ2
β2
βγ
+
− 2ρ
2
2
Xs
Ys
Xs Ys
ds.
Then, if ρβγ ≤ 0,
hM it
Z
t
≥
σ
γ2
β2
+
Xs2 Ys2
ds,
and if ρ2 < 1,
Z t 2 γ2
β
≥ (1 − ρ ) max
ds,
ds .
2
2
σ Xs
σ Ys
If β > 0, then X ≥ U where U is the Bessel process of dimension two in (6) and
hM it
Z
2
t
lim sup (ln Xt ) ≥ lim sup (ln Ut ) = ∞
t→∞
t→∞
If β < 0, then X ≤ U and
lim inf (ln Xt ) ≤ lim inf (ln Ut ) = −∞
t→∞
t→∞
But
Z
t
ln Xt = ln Xσ +
σ
and
dBs
+β
Xs
Z
t
σ
ds
.
Xs Ys
Z 1
ds
1
1 t
+
ds.
≤
2 σ Xs2 Ys2
σ Xs Ys
Thus ln Xt would converge a.s. as t → ∞ if
Z ∞
Z ∞
ds
ds
< ∞ and
< ∞.
2
2
σ Ys
σ Xs
Z
t
Therefore hM i∞ = ∞ on {σ < ∞} and the assertion is proved.
8. Absolute continuity properties
In this section we suppose ρ = 0. In some cases we easily obtain an absolute continuity
property between the laws of O2BPs with various parameters. When there exists a unique
solution to (5), we denote by Pα,β,γ,δ
the law on C(R+ , S) = C(R+ , R+ ) × C(R+ , R+ ) of the
x,y
solution starting at (x, y) ∈ S. We denote by (U, V ) the canonical map (Ut (u, v), Vt (u, v)) =
(u(t), v(t)) and by Ut = σ((Us , Vs ), s ≤ t) the canonical filtration. Recall that we denote by
Pdr the law of one-dimensional Bessel process of dimension d starting at r ≥ 0.
Proposition 18. Assume ρ = 0, δ ≥ 12 and y > 0. Then
Z t
Z t
Z
dUs
ds
β 2 t ds
α,β,0,δ
− αβ
−
. P2α+1
⊗ P2δ+1
|Ut .
(21)
Px,y
|Ut = exp β
x
y
2
V
U
V
2
V
s
s
s
0
0
0
s
Proof. Under P2α+1
⊗ P2δ+1
the process
x
y
Z
Bt := Ut − x − α
0
t
ds
Us
is a one-dimensional Brownian motion. The assumptions on δ and on y imply that Vt > 0
for any t ≥ 0 and thus
Z t
ds
<∞
P2δ+1
− a.s.
y
2 (s)
v
0
18
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
R t dBs
Therefore, for P2δ+1
-almost every v, 0 v(s)
is a P2α+1
-centered Gaussian variable with variy
x
R t ds
ance 0 v2 (s) and
Z t
Z Z
Z
dBs
β 2 t ds
2α+1
exp β
dPx (u) dP2δ+1
(v) = 1.
−
y
2
v(s)
2
v(s)
0
0
We see that
Z
Zt := exp β
0
is a
P2α+1
x
⊗
P2δ+1
-positive
y
t
dBs β 2
−
Vs
2
Z
0
t
ds
Vs2
martingale with expectation 1. Setting for any T > 0
QT := ZT . P2α+1
⊗ P2δ+1
|UT
x
y
we check that under QT
Z
Ut − x − α
0
t
ds
−β
Us
t
Z
0
ds
,
Vs
0≤t≤T
is a real Brownian motion independent of the Brownian motion {Vt − y − δ
Thus,
α,β,0,δ
QT = Px,y
|UT .
Rt
ds
0 Vs ,
0 ≤ t ≤ T }.
A second set of conditions is obtained using Novikov’s criterion.
Proposition 19. Assume ρ = 0 and the following set of conditions is satisfied:
|β| ≤ δ − 1/2
x>0
(22)
|γ| ≤ α − 1/2
y > 0.
The process
Z
Zt := exp β
0
is a
(23)
P2α+1
x
⊗
t
dUs
−γ
Vs
P2δ+1
-positive
y
Z
0
t
dVs
− (αβ + γδ)
Us
t
Z
ds
β2
−
Us V s
2
0
Z
0
t
ds
γ2
−
Vs2
2
Z
0
t
ds
,
Us2
martingale with expectation 1 and
Pα,β,γ,δ
|Ut = Zt . P2α+1
⊗ P2δ+1
|Ut .
x,y
x
y
Proof. Under P2α+1
⊗ P2δ+1
the processes
x
y
t
Z
Bt := Ut − x − α
0
and
Z
ds
Us
t
ds
0 Vs
are independent Brownian motions. Then Zt may be written
Z t
Z t
Z
Z
dCs β 2 t ds
γ 2 t ds
dBs
Zt = exp β
+γ
−
−
,
2 0 Vs2
2 0 Us2
0 Us
0 Vs
Ct := Vt − y − δ
which shows that Zt is a positive local martingale. Using (4) we compute
RR
R
2 Rt
2 Rt
exp { γ2 0 Uds2 } dP2α+1
⊗ dPy2δ+1 =
exp { γ2 0 Rds2 } dP2α+1
x
x
s
R Rt α−1/2 s
2 −γ 2 R t
=
(x)
exp {− (α−1/2)
2
0
< ∞
ds
}
Rs2
dP2x
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
19
since a Bessel process of dimension two has finite moments of any order. Finally,
RR
2 Rt
2 Rt
⊗ dP2δ+1
exp { β2 0 Vds2 } exp { γ2 0 Uds2 } dP2α+1
x
y
s
s
R
2δ+1 R
2α+1
γ 2 R t ds
β 2 R t ds
= ( exp { 2 0 R2 } dPx )( exp { 2 0 R2 } dPx )
s
s
< ∞
and Novikov’s criterion ([27], Proposition VIII.1.15) proves that Zt has expectation 1 with
Rt
Rt
respect to P2α+1
⊗ P2δ+1
. We easily see that {Bt − β 0 Vdss , 0 ≤ t ≤ T } and {Ct − γ 0 Udss , 0 ≤
x
y
t ≤ T } are independent Brownian motions under the probability with density ZT and this
proves that the new probability is Pα,β,γ,δ
|UT .
x,y
9. Product form stationary distribution
For a > 0 and c > 0, let Γ(a, c) be the probability measure on [0, ∞) with density
ca a−1 −cx
γ(x; a, c) :=
x e .
Γ(a)
For a > 0, b > 0, let B(a, b) be the probability measure on [0, 1] with density
β(x; a, b) :=
Γ(a + b) a−1
x (1 − x)b−1 .
Γ(a)Γ(b)
It is well-known that the function g(x) = xd−1 is an invariant measure density on [0, ∞)
for the Bessel process of dimension d. To get a stationary probability we have to introduce
a negative drift that entails positive recurrence. Then, the process
Z
d − 1 t ds
− θt,
Xt = X0 + Bt +
2
0 Xs
with θ > 0, has γ(x; d, 2θ) as stationary density. As for the Brownian motion reflected at
0 with constant drift −θ, it has the stationary exponential density 2θ exp{−2θx}. In the
bidimensional case, the drifted obliquely reflected Brownian motion has a stationary density
in the form of product of two exponential densities if and only if it satisfies at once ([13],[37]):
• an inversibility condition on the reflection matrix;
• a positivity condition on the exponential coefficients;
• a skew-symmetry condition.
Therefore it is natural to ask wether one can find similar conditions for drifted O2BPs ensuring
existence of a stationary distribution in the form of product of two gamma distributions.
The answer is positive and given in the next theorem. In fact, this may be considered as a
consequence of the study in [24] which introduced a generalised reflected Brownian motion
associated with a potential U regular on the whole real line. The difference is that here
the logarithmic potential is defined only on the positive axis. The proof below is just an
adaptation of the proof in [24] to this special case.
We introduce an additional constant drift (−θ, −η) in order to make the solution a recurrent
process in the nonnegative quadrant. We consider the system
R t ds
Rt
+ β 0 Ydss − θt
Xt = X0 + Bt + α 0 X
s
R
R
(24)
t ds
t
Yt = Y0 + Ct + γ 0 X
+ δ 0 Ydss − ηt
s
with the conditions Xt ≥ 0, Yt ≥ 0. Changing probability through a Girsanov transformation,
we easily check that Theorem 6 in Section 4 is still valid for this drifted system. Moreover,
the proofs in Section 5 do not bother whether the Brownian motions B and C are drifted or
not drifted. Therefore existence and uniqueness results in Section 5 hold true for the solution
to (24).
20
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
Theorem 20. Assume there exists a unique solution to (24) in S 0 or S. This process has
an invariant distribution in the form Γ(a, c) ⊗ Γ(b, d) if and only if at once
• αδ − βγ > 0
• αη − γθ > 0
• 2ρ = βδ + αγ
and
(inversibility)
δθ − βη > 0 (positivity)
(skew-symmetry)
Under these conditions, the unique solution is given by
•
•
•
•
a
b
c
d
=
=
=
=
1 + 2α
1 + 2δ
δθ−βη
2α αδ−βγ
αη−γθ
2δ αδ−βγ .
Proof. Let
for x ≥ 0, y ≥ 0 .
q(x, y) = γ(x; a, c)γ(y; b, d)
The infinitesimal generator of the diffusion (24) is given by
1 ∂2
∂2
∂2
α β
∂
γ
δ
∂
L = ( 2 + 2 + 2ρ
) + ( + − µ)
+ ( + − ν) .
2 ∂x
∂y
∂x∂y
x
y
∂x
x y
∂y
By a density argument, to prove that q is an invariant density, it is enough to check that
Z ∞Z ∞
Lf (x, y)q(x, y) dxdy = 0
0
0
for any f (x, y) = g(x)h(y) with g, h ∈ Cc2 ((0, ∞)) (compactly supported twice continuously
differentiable functions on (0, ∞)). Integrating by parts, we get
Z ∞Z ∞
Z ∞Z ∞
L(gh)(x, y)q(x, y) dxdy =
g(x)h(y)J(x, y) dxdy
0
0
0
0
where
J(x, y) = ρ(x, y) [A + Bx−1 + Cx−2 + Dy −1 + Ey −2 + F x−1 y −1 ]
with
A
B
C
D
E
F
= 12 c2 + 21 d2 − θc − ηd + ρcd
= −(a − 1)c + θ(a − 1) + αc + γd − ρ(a − 1)d
= 12 (a − 1)(a − 2) − α(a − 2)
= −(b − 1)d + η(b − 1) + βc + δd − ρ(b − 1)c
= 12 (b − 1)(b − 2) − δ(b − 2)
= −[β(a − 1) + γ(b − 1)] + ρ(a − 1)(b − 1) .
Letting A = B = C = D = E = F = 0 we obtain the necessary and sufficient conditions and
the specified values for a, b, c, d.
It was already proved in [36] (see also [28]) that under a skew-symmetry condition the
obliquely reflected Brownian motion does not reach the non-smooth part of the boundary.
In the same way the above skew-symmetry equality is reminiscent of the second condition in
Corollary 8, which is an inequality. We therefore get a partial but handy statement.
Corollary 21. Assume
•
•
•
•
max{α, δ} ≥
αδ − βγ > 0
αη − γθ > 0
2ρ = βδ + αγ
1
2
and
(inversibility)
δθ − βη > 0 (positivity)
(skew-symmetry).
A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES
21
Then the unique solution to (24) has an invariant distribution given by
δθ − βη
αη − γθ
Γ 1 + 2α, 2α
⊗ Γ 1 + 2δ, 2δ
.
αδ − βγ
αδ − βγ
Proof. This is a straightforward consequence of Corollary 8, Theorems 9, 11, 12 and 20.
Remark 22. When ρ = 0, the skew-symmetry condition αβ + γδ = 0 means that rx and ry
are orthogonal, and the positivity condition means that (θ, η) points into the interior of the
quadrant designed by rx and ry .
Remark 23. With the same proof, we may check that under the skew-symmetry condition,
when θ = η = 0, the function q(x, y) = x2α y 2δ is a non-integrable invariant density that does
not depend on the other parameters.
A simple change of variables (beta-gamma algebra) provides the following result.
Corollary 24. With the conditions and notations of Theorem 20, the two-dimensional process
(25)
t
Wt := cXcX
t +dYt
Zt := cXt + dYt
has B(a, b) ⊗ Γ(a + b, 1) for invariant distribution on [0, 1] × [0, ∞).
Acknowledgments. The author is deeply indebted to the anonymous referee for careful
reading, encouragements and sound advices. They have given rise to fundamental improvements in the content of this paper.
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