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J. Differential Equations ••• (••••) •••–•••
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www.elsevier.com/locate/jde
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Stochastic variational inequalities on non-convex
domains
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12Q2
Rainer Buckdahn , Lucian Maticiuc , Étienne Pardoux ,
Aurel Răşcanu b,d
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a Laboratoire de Mathématiques, CNRS-UMR 6205, Université de Bretagne Occidentale, 6, avenue Victor Le Gorgeu,
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29238 Brest Cedex 3, France
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b Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd., no. 9, Iasi, 700506, Romania
c LATP, Université de Provence, CMI, 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France
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d “Octav Mayer” Mathematics Institute of the Romanian Academy, Iasi branch, Carol I Blvd., no. 8, Iasi, 700506,
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Romania
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Received 5 December 2013; revised 2 May 2015
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Abstract
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The objective of this work is to prove in a first step the existence and the uniqueness of a solution of the
following multivalued deterministic differential equation:
dx(t) + ∂ − ϕ(x(t))(dt) dm(t), t > 0,
x(0) = x0 ,
where m : R+ → Rd is a continuous function and ∂ − ϕ is the Fréchet subdifferential of a (ρ, γ )-semiconvex
function ϕ; the domain of ϕ can be non-convex, but some regularities of the boundary are required.
The continuity of the map m → x : C([0, T ]; Rd ) → C([0, T ]; Rd ) associating to the input function m
the solution x of the above equation, as well as tightness criteria allows to pass from the above deterministic
case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion:
⎧
t
t
⎪
⎪
⎪
⎨
Xt + Kt = ξ + F (s, Xs )ds + G(s, Xs )dBs , t ≥ 0,
⎪
0
0
⎪
⎪
⎩
dKt (ω) ∈ ∂ − ϕ(Xt (ω))(dt).
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E-mail addresses: [email protected] (R. Buckdahn), [email protected] (L. Maticiuc),
[email protected] (É. Pardoux), [email protected] (A. Răşcanu).
http://dx.doi.org/10.1016/j.jde.2015.08.023
0022-0396/© 2015 Published by Elsevier Inc.
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© 2015 Published by Elsevier Inc.
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MSC: 60H10; 60J60
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Keywords: Skorohod problem; Stochastic variational inequalities; Fréchet subdifferential
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1. Introduction
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Given a multi-dimensional Brownian motion B = (Bt ) and a proper (ρ, γ )-semiconvex function ϕ (for the definition the reader is referred to Definition 11 from the next section) defined over
a possibly non-convex domain Dom(ϕ) and a random variable ξ independent of B, which takes
its values in the closure of Dom(ϕ), we are interested in the following multi-valued stochastic
differential equations (also called stochastic variational inequality) driven by the Fréchet subdifferential operator ∂ − ϕ:
⎧
t
t
⎪
⎪
⎨ X + K = ξ + F (s, X ) ds + G (s, X ) dB , t ≥ 0,
t
t
s
s
s
⎪
0
0
⎪
⎩
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt) .
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(1)
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⎧
⎪
⎪
⎨
⎪
⎪
⎩
x (t) + k (t) = x0 +
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t
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f (s, x (s)) ds + m (t) ,
t ≥ 0,
(2)
0
dk (t) ∈ ∂ − ϕ (x (t)) (dt)
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dk (t) ∈ ∂ − ϕ (x (t)) (dt)
(for the notation
see Definition 11 and Remark 15).
With the particular choice of f ≡ 0 and ϕ as convexity indicator function of a closed domain
E ⊂ Rd ,
ϕ (x) = IE (x) :=
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0,
+ ∞,
if x ∈ E,
if x ∈
/ E,
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However, in order to study the above system, we shall first solve the following deterministic
counterpart of the above equation. Given a continuous function m : R+ → Rd and an initial value
x0 ∈ Dom(ϕ) we look for a pair of continuous functions (x, k) : R+ → R2d such that
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equation (2) turns out to be just the Skorohod problem, i.e., a reflection problem, associated
to the data x0 , m and E. For this reason we will refer to equation (1) as Skorohod equation.
The existence of solutions for both the Skorohod equation (2) and for the stochastic equation (1), has been well studied by different authors for the case, where ϕ is a convex function. In
this case ∂ − ϕ becomes a maximal monotone operator and the domain Dom(ϕ) in which the solution is kept is convex. By replacing ∂ − ϕ by a general maximal monotone operator A, E. Cépa
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generalized in [14] the above equation in the finite dimensional case, while A. Răşcanu [32]
investigated the infinite dimensional case.
Deterministic variational inequalities with regular inputs, i.e., deterministic equations of
type (2), with convex ϕ and m = 0 have been well studied and the corresponding results have
by now become classical. As concerns the non-convex framework, the reader is referred to
A. Marino, M. Tosques [26], M. Degiovanni, A. Marino, M. Tosques [19], A. Marino, C. Saccon, M. Tosques [25] or R. Rossi, G. Savaré [36,37]. They used the concept of φ-convexity
(see, e.g., [26, Definition 4.1]) and they provide the existence, uniqueness and continuous dependence on the initial data of the solutions for the evolution equation of type (2) in the case
m = 0, f = 0 (or f (t, x (t)) = f (t) in [37]). We specify that our notion of (ρ, γ )-semiconvex
function corresponds to the particular case of a φ-convex function of order r = 1 with
φ (x, y, ϕ (x) , ϕ (y) , α) := ρ + γ |α|. This particular form was required by the presence of the
singular input dm/dt .
Related to our problem is the research on non-convex differential inclusions, see, e.g., F. Papalini [30], T. Cardinali, G. Colombo, F. Papalini, M. Tosques [10] and A. Cernea, V. Staicu [15].
In [15] it is proved the existence of a solution for the Cauchy problem
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x (0) = x0 ,
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where ϕ is a φ-convex function of order r = 2, F is an upper semicontinuous and cyclically
monotone multifunction and f is a Carathéodory function.
The particular case of a reflection problem, i.e., with ϕ = IE , was extended to that of moving
domains E (t), t ≥ 0, by considering the following problem (which solution is called sweeping
process):
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x (t) + NE(t) (x (t)) f (t, x (t)) ,
t ≥ 0,
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where NE(t) (x (t)) = ∂ − IE(t) (x (t)) is the external normal cone to E (t) in x (t).
This problem was introduced by J.J. Moreau (see [29]) for the case f ≡ 0 and convex sets
E (t), t ≥ 0, and it has been intensively studied since then by several authors; see, e.g., C. Castaing [11], M.D.P. Monteiro Marques [28]. For the case of a sweeping process without the
assumption of convexity on the sets E (t), t ≥ 0, we refer to [3,4,16,18]. The extension to the case
f ≡ 0 was considered, e.g., in C. Castaing, M.D.P. Monteiro Marques [12] and J.F. Edmond, and
L. Thibault [21] (see also the references therein). Another extension was made in [27] and [34]
by considering the quasivariational sweeping process
x (t) + NE(t,x(t)) (x (t)) 0,
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x (0) = x0 ,
t ≥ 0,
x (0) = x0 .
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x ∈ −∂ − ϕ (x) + F (x) + f (t, x) ,
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The works mentioned above concern the case with vanishing driving force m = 0. Let us discuss
now the case of equation (2) with singular input dm/dt . The associated reflection problem with
singular input dm/dt has been investigated by A.V. Skorohod in [39] and [40] (for the particular
case of E = [0, ∞)), by H. Tanaka in [41] (for a general convex domain E), and by P.-L. Lions
and A.S. Sznitman in [24] and Y. Saisho in [38] for a non-convex domains. Generalizations from
the reflection problem (with ∂ϕ = ∂IE ) to the case of ∂ϕ for a general convex function ϕ, and
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even to the case of a maximal monotone operator A, were discussed by A. Răşcanu in [32],
V. Barbu and A. Răşcanu in [2] and by E. Cépa in [13] and [14].
Concerning the stochastic equations, we shall mention the papers [24] by P.-L. Lions and
A.S. Sznitman and [38] by Y. Saisho, but also [20] by P. Dupuis and H. Ishii for stochastic
differential equations with reflecting boundary conditions. On the other hand, A. Răşcanu [32],
I. Asiminoaei and A. Răşcanu [1] as well as A. Bensoussan and A. Răşcanu [5] studied stochastic
variational inequalities (1) in the convex case.
More recently, A.M. Gassous, A. Răşcanu and E. Rotenstein obtained in [22] existence and
uniqueness results for stochastic variational inequalities with oblique subgradients. More precisely, in their equation the direction of reflection at the boundary of the convex domain differs
from the normal direction, an effect which is caused by the presence of a multiplicative Lipschitz
matrix acting on the subdifferential operator. In the authors’ approach it turned out to be crucial
to pass first by a study of the Skorohod problem with generalized reflection.
The objective of the present work is twofold: We generalize both the (non-)convex reflection problem as well as convex variational inequalities to non-convex variational inequalities.
Some studies in this direction have been made already by A. Răşcanu and E. Rotenstein in [33]:
a non-convex setup for multivalued (deterministic) differential equations driven by oblique subgradients has been established and the uniqueness and the local existence of the solution has been
proven.
Our approach here in the present manuscript is heavily based on an a priori discussion of
the generalized Skorohod problem (2) with f ≡ 0 and a (ρ, γ )-semiconvex ϕ. We give useful a
priori estimates and prove the existence and the uniqueness of a solution (x, k) for the generalized
Skorohod problem:
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x (t) + k (t) = x0 + m (t) ,
dk (t) ∈ ∂ − ϕ (x (t)) (dt) ,
t ≥ 0,
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(3)
where x0 ∈ Dom (ϕ), the input m is a continuous function starting from zero, and ∂ − ϕ is the
Fréchet subdifferential of a proper, lower semicontinuous and (ρ, γ )-semiconvex function ϕ.
Here the set Dom (ϕ) is not necessarily convex, but however two assumptions are required:
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1. Dom (ϕ) satisfies the uniform exterior ball condition (see Definition 1);
2. Dom (ϕ) satisfies the (γ , δ, σ )-shifted uniform interior ball condition, i.e.
there are some suitable constants γ ≥ 0 and δ, σ > 0 such that, for all y ∈ Dom (ϕ), there are
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some λy ∈ (0, 1] and vy ∈ Rd , vy ≤ 1 with λy − vy + λy γ ≥ σ and
B x + vy , λy ⊂ Dom (ϕ) ,
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for all x ∈ Dom (ϕ) ∩ B (y, δ) .
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We observe that this condition is fulfilled if, in particular, the domain Dom (ϕ) satisfies the uniform interior drop condition (see Definition 5). It is worth pointing out that the shifted uniform
interior ball condition is comparable with assumption (5) of P.-L. Lions, A.S. Sznitman [24]
(or Condition (B) from [38]) (see
20–21).
the Remarks
The application m → x : C [0, T ] ; Rd → C [0, T ] ; Rd , which associates to the input function m the solution x of (3), will be proven to be continuous. This allows to derive the existence
of a solution to the associated stochastic equation with additive noise M = (Mt ):
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Xt (ω) + Kt (ω) = ξ (ω) + Mt (ω) , t ≥ 0, ω ∈ ,
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt) .
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(4)
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After having the existence, the uniqueness and properties for the equations (3) and (4), we will
be able to extend the study to the more general equations (2) and (1), where f (respectively F )
is continuous, satisfies a one-sided Lipschitz condition with respect to the second variable and a
boundedness assumption.
The article is organized as it follows: The next section is devoted to a recall of such basic notions as those of as a semiconvex set, a (ρ, γ )-semiconvex function or a Fréchet subdifferential.
Some notions, like for instance that of a (ρ, γ )-semiconvex function, are illustrated by an example. In Section 3 we give the definition of the solution to the generalized Skorohod problem (3),
we prove the existence and the uniqueness of a solution (x, k) and we give some useful a priori
estimations. Moreover, we extend equation (3) to the stochastic case (4). Section 4 is devoted to
the proof of the both main results of Section 3. Finally, the Sections 5 and 6 study the extension
of the results established in Section 3 to the equations (2) and (1). Appendix A is devoted to
important auxiliary results
such as applications of Fatou’s Lemma, some complements concerning tightness in C R+ ; Rd or a very useful forward stochastic inequality, which are used in our
approach.
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We introduce first some definitions and results concerning the notions of normal cone, uniform
exterior ball conditions, semiconvex sets, (ρ, γ )-semiconvex functions and Fréchet subdifferential of a function.
Here and everywhere below E will be a nonempty closed subset of Rd . Let NE (x) be the
closed external normal cone of E at x ∈ Bd (E) i.e.
dE (x + δu)
d
NE (x) := u ∈ R : lim
= |u| ,
δ0
δ
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where dE (z) := inf {|z − x| : x ∈ E} is the distance of a point z ∈ Rd to E.
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where B (x, r) denotes the ball from Rd of center x and radius r
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for all x ∈ Bd (E)
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and
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for all x ∈ Bd (E) and u ∈ NE (x) such that |u| = r0 it holds that dE (x + u) = r0 .
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NE (x) = {0}
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or equivalently if
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B (x + u, r0 ) ∩ E = ∅,
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Definition 1. Let r0 > 0. We say that E satisfies the r0 -uniform exterior ball condition (we write
it r0 -UEBC for brevity) if
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2. Preliminaries
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We remark that for all v ∈ NE (x) with |v| ≤ r0 we also have dE (x + v) = |v|.
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Definition 2. Let γ ≥ 0. A set E is γ -semiconvex if for all x ∈ Bd (E) there exists x̂ ∈ Rd {0}
such that
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x̂, y − x ≤ γ x̂ |y − x|2 ,
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We have the following equivalence:
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Lemma 3. (See [31, Lemma 6.47].) Let r0 > 0. The set E satisfies the r0 -UEBC if and only if E
is 2r10 -semiconvex.
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For a given z ∈ Rd we denote by E (z) the set of elements x ∈ E such that dE (z) = |z − x|.
Obviously, E (z) is nonempty since E is nonempty and closed. Moreover, under the r0 -uniform
exterior ball condition, it follows that the set E (z) is a singleton for all z such that dE (z) < r0 .
In this case πE (z) will denote the unique element of E (z) and it is called the projection of z
on E. We recall the following well-known property of the projection.
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Lemma 4. Let the r0 -UEBC be satisfied, ε ∈ (0, r0 ) and U ε (E) := y ∈ Rd : dE (y) ≤ ε denoting the closed ε-neighborhood of E.
Then:
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1 NE (x) = x̂ : x̂, y − x ≤
x̂ |y − x|2 , ∀y ∈ E ;
2r0
• the projection πE restricted to U ε (E) is Lipschitz with Lipschitz constant Lε = r0 / (r0 − ε);
and
• the function d2E (·) is of class C 1 on U ε (E) with
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and z − πE (z) ∈ NE (πE (z)) , ∀z ∈ U ε (E) .
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Dx (v, r) := conv x, B (x + v, r) = x + t (u − x) : u ∈ B (x + v, r) , t ∈ [0, 1]
is called (v, r)-drop of vertex x and running direction v. Remark that if |v| ≤ r, then Dx (v, r) =
B (x + v, r).
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Definition 5. The set E ⊂ Rd satisfies the uniform interior drop condition if there exist r0 , h0 > 0
such that for all x ∈ E there exists vx ∈ Rd with |vx | ≤ h0 and
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Dx (vx , r0 ) ⊂ E
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Let us introduce now the notion of drop of vertex x and running direction v.
Let x, v ∈ Rd , r > 0. The set
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1 2
∇d (z) = z − πE (z)
2 E
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• the closed external normal cone of E at x is given by
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∀y ∈ E.
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(we also say that E satisfies the uniform interior (h0 , r0 )-drop condition).
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Remark 6. It is easy to see that if there exists r0 > 0 such that E c satisfies the r0 -UEBC, then E
satisfies the uniform interior (h0 , r0 )-drop condition.
Indeed let x ∈ Bd (E c ) = Bd (E) and ux ∈ NE c (x) with |ux | = r0 .
Then
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Dx (ux , r0 /2) ⊂ Dx (ux , r0 ) = B (x + ux , r0 ) ⊂ E.
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It is easy to see that, for any x ∈ int (E), there exists a direction vx
Dx (vx , r0 /2) ⊂ E.
(i)
(ii)
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such that |vx | ≤ r0 and
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Example 8. Let E be a set for which there exists a function φ ∈ Cb R
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such that
E = x ∈ Rd : φ (x) ≤ 0 ,
(ii) Int (E) = x ∈ Rd : φ (x) < 0 ,
(iii) Bd (E) = x ∈ Rd : φ (x) = 0 and |∇φ (x)| = 1, ∀ x ∈ Bd (E) .
(i)
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Then the set E satisfies the uniform exterior ball condition and the uniform interior drop condition.
Indeed, using the definition of E we see that, for x ∈ Bd (E), the gradient ∇φ (x) is a
unit normal vector to the boundary, pointing towards the exterior of E. Therefore, for any
x ∈ Bd (E), the normal cone is given by NE (x) = {c∇φ (x) : c ≥ 0} and NE c (x) = {−c∇φ (x) :
c ≥ 0}.
Since φ (y) ≤ 0 = φ (x), for all y ∈ E, x ∈ Bd (E),
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∇φ (x) , y − x = φ (y) − φ (x) −
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∇φ (x + λ (y − x)) − ∇φ (x) , y − x dλ
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≤ M |y − x|2 = M |∇φ (x)| |y − x|2 ,
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(this condition will be called (γ , δ, σ )-SUIBC).
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(5)
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λy − vy + λy γ ≥ σ,
B x + vy , λy ⊂ E, ∀ x ∈ E ∩ B (y, δ)
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Proposition 7. Let the set E be as above with Int (E) = ∅. If set E satisfies the uniform interior
(h0 , r0 )-drop condition then E satisfies the shifted uniform interior ball condition, which means
d
that there exist γ ≥ 0 and δ, σ > 0, and for every y ∈ E there exist λy ∈ (0, 1] and vy ∈ R ,
vy ≤ 1 such that
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∈ Rd
We state below that the drop condition implies a weaker condition, but is not equivalent with
this (for the proof see Proposition 4.35 in [31]).
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9
2
5
6
8
1
46
which means, using Definition 2 and Lemma 3, that E satisfies
1
2M -UEBC.
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Since φ (y) ≥ 0 = φ (x), for all y ∈ E c , x ∈ Bd (E),
2
4
−∇φ (x) , y − x = −φ (y) +
5
≤ M |y − x|2 = M |−∇φ (x)| |y − x|2 ,
7
9
10
11
12
which yields that E c satisfies
form interior drop condition.
17
18
19
20
21
22
23
24
25
28
29
32
37
38
39
40
41
42
43
44
45
46
47
9
11
12
Example 9. Let E ⊂ Rd be a closed convex set with nonempty interior. If there exists r0 > 0 such
that (the r0 -interior of E) Er0 = ∅ and h0 = supz∈E dEr0 (z) < ∞ (in particular if E is a bounded
closed convex set with nonempty interior), then E satisfies the uniform interior (h0 , r0 )-drop
condition.
r0
Moreover for every 0 < δ ≤ 2(1+h
∧ 1, E satisfies (γ , δ, σ )-SUIBC with λy = σ = δ.
0)
1
ŷ
−
y
. Hence
For the proof, let y ∈ E, ŷ the projection of y on the set Er0 and vy = 1+h
0
ŷ − y ≤ h0 , vy ≤ 1 and for all x ∈ E ∩ B (y, δ)
15
r0 B x + v y , δ ⊂ B y + vy ,
⊂ conv y, B ŷ, r0 = Dy ŷ − y, r0 ⊂ E.
1 + h0
14
16
17
18
19
20
21
22
23
24
25
26
Let ϕ: Rd → (−∞, +∞] be a function with domain defined by
Dom (ϕ) := v ∈ Rd : ϕ (v) < +∞ .
27
28
29
30
We recall now the definition of the Fréchet subdifferential (for this kind of subdifferential operator see, e.g., [26] and the monograph [35, cap. VIII]):
31
32
33
Definition 10. The Fréchet subdifferential ∂ − ϕ is defined by:
35
36
7
13
33
34
6
10
30
31
5
8
and consequently (see Remark 6) E satisfies the uni-
26
27
4
Eε := {x ∈ E : dE c (x) ≥ ε} .
14
16
1
2M -UEBC
If E denotes a closed subset of Rd let Eε be the ε-interior of E, i.e.
13
15
3
∇φ (x + λ (y − x)) − ∇φ (x) , y − x dλ
0
6
8
2
1
3
1
34
35
/ Dom (ϕ) and
a1 ) ∂ − ϕ (x) = ∅, if x ∈
a2 ) for x ∈ Dom (ϕ),
ϕ (y) − ϕ (x) − x̂, y − x
−
d
∂ ϕ (x) = {x̂ ∈ R : lim inf
≥ 0}.
y→x
|y − x|
Taking into account this definition we will say that ϕ is proper if the domain Dom (ϕ) = ∅
and has no isolated points.
We set
Dom ∂ − ϕ = x ∈ Rd : ∂ − ϕ (x) = ∅ ,
∂ − ϕ = x, x̂ : x ∈ Dom ∂ − ϕ , x̂ ∈ ∂ − ϕ (x) .
36
37
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1
In the particular case of the indicator function of the closed set E,
2
3
4
ϕ (x) = IE (x) :=
5
6
7
8
9
10
11
12
17
18
19
20
21
22
23
24
25
2
0,
+ ∞,
if x ∈ E,
if x ∈
/ E,
3
4
5
6
7
x̂, y − x
−
d
∂ IE (x) = x̂ ∈ R : lim sup
≤0 .
y→x, y∈E |y − x|
8
9
10
11
Moreover, in this particular case we deduce that, for any closed subset E of a Hilbert space,
∂ − IE (x) = NE (x)
14
16
1
the function ϕ is lower semicontinuous and the Fréchet subdifferential becomes
13
15
9
12
(6)
Definition 11. Let ρ, γ ≥ 0. The function ϕ
function if
16
→ (−∞, +∞] is said to be (ρ, γ )-semiconvex
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
18
20
a1 ) Dom (ϕ)
is
γ -semiconvex,
a2 ) Dom ∂ − ϕ = ∅,
a3 ) For all x, x̂ ∈ ∂ − ϕ and y ∈ Rd :
21
22
23
x̂, y − x + ϕ (x) ≤ ϕ (y) + ρ + γ x̂ |y − x|2 .
24
25
26
Remark 12. Let E be a nonempty closed subset of Rd . We have:
28
29
17
19
26
27
14
15
(for the proof see Colombo and Goncharov [17]).
: Rd
13
27
28
1. IE is (0, γ )-semiconvex iff E is γ -semiconvex (see (6) and Definitions 2 and 11; we also
mention that in the definition of γ -semiconvexity we can take x ∈ E, but in this case x̂ should
be taken 0).
2. IE is (0, γ )-semiconvex (or, see [26, Definition 4.1], φ-convex of order r = 1 with
φ(x, y, ϕ (x) , ϕ (y) , α) = γ |α|) iff there exist δ, μ > 0 such that x → dE (x) + μ |x|2 is
convex on B (y, δ), for any y ∈ E (see [31, Lemma 6.47]).
3. A convex function is also (ρ, γ )-semiconvex function, for any ρ, γ ≥ 0 (see Definition 11
and the definition of the subdifferential of a convex function).
4. If E is convex, then E is γ -semiconvex for any γ ≥ 0 (see the supporting hyperplane Theorem 4.1.6 from [7]).
5. If E has nonempty interior and is 0-semiconvex, then E is convex (see Definition 2 and
[8, Exercise 2.27]).
6. If ϕ : Rd → (−∞, +∞] is a (ρ, γ )-semiconvex function, then there exist a, b ∈ R+ and
c ∈ R such that
ϕ (y) + a |y|2 + b |y|2 + c ≥ 0,
for all y ∈ Rd .
Indeed, by Definition 11, we have, for a fixed x0 , x̂0 ∈ ∂ − ϕ : a = ρ + γ x̂0 , b = 2a |x0 | +
x̂0 and c = a |x0 |2 + x̂0 , x0 − ϕ (x0 ).
29
30
31
32
33
34
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Example 13. If the bounded set E satisfies the r0 -UEBC and g ∈ C 1 Rd (or g : Rd → R is a
convex function), then ϕ : Rd → (−∞, +∞] given by
3
ϕ (x) := IE (x) + g (x)
4
is a lower semicontinuous and 2rL0 , 2r10 -semiconvex function, where |∇g (x)| ≤ L, for any
x ∈ E (or |∂g (x)| ≤ L, for any x ∈ E). Moreover
6
5
7
8
5
9
|ϕ (x) − ϕ (y)| ≤ L |x − y| , ∀ x, y ∈ Dom (ϕ) = E.
11
13
14
15
16
11
In order to define the solution for the deterministic problem envisaged by our work it is necessary to introduce the bounded variation function space.
Let T > 0, k : [0, T ] → Rd and D be the set of the partitions of the interval [0, T ].
Set
S (k) =
19
n−1
22
23
kT := sup S (k),
∈D
25
(7)
26
27
28
BV([0, T ] ; R ) = {k : [0, T ] → R : kT < ∞}.
d
31
d
32
38
∗
C([0, T ] ; R ) = {k ∈ BV([0, T ] ; Rd ) : k(0) = 0}
d
45
34
35
36
37
38
40
42
T
y (t) , dk (t) .
(y, k) →
0
46
47
33
41
42
44
31
39
given by the Riemann–Stieltjes integral
41
43
30
32
The space BV([0, T ] ; Rd ) equipped with the norm ||k||BV([0,T ];Rd ) := |k (0)| + kT is a Banach
space.
Moreover, we have the duality
39
40
25
29
30
37
24
26
where : 0 = t0 < t1 < · · · < tn = T .
Write
29
36
16
21
and
24
35
15
20
23
34
14
19
21
33
13
18
|k(ti+1 ) − k(ti )|
i=0
20
28
12
17
18
27
8
10
17
22
7
9
10
12
2
3
4
6
1
43
44
45
46
We will say that a function k ∈ BV loc (R+ ; Rd ) if, for every T > 0, k ∈ BV([0, T ] ; Rd ).
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1
11
3. Generalized Skorohod problem
1
2
3
4
2
The aim of this section is to prove the existence and uniqueness result for the following deterministic Cauchy type differential equation:
5
6
7
8
9
t > 0,
6
(8)
9
10
x0 ∈ Dom (ϕ),
(ii) m ∈ C R+ ; Rd , m (0) = 0,
(i)
12
13
11
(9)
14
and
15
ϕ : Rd → (−∞, +∞] is a proper lower semicontinuous
and (ρ, γ ) -semiconvex function.
16
(10)
18
19
20
23
24
25
26
27
28
29
(jv) ∀ 0 ≤ s ≤ t, ∀y : R+ → Rd continuous:
t
t
t
y (r) − x (r) , dk (r) + ϕ (x (r)) dr ≤ ϕ (y (r)) dr
s
s
38
39
40
41
24
25
26
(11)
s
of the generalized Skorohod problem
−In this case
the pair (x, k) is said to
be the solution
∂ ϕ; x0 , m (denoted by (x, k) = SP ∂ − ϕ; x0 , m ).
If ϕ = IE then ∂ − ϕ = NE and we say that (x, k) is solution of the Skorohod problem
(E; x0 , m) and we write (x, k) = SP (E; x0 , m).
46
47
29
31
32
34
35
36
37
38
39
Remark 15. The notation
40
−
dk (t) ∈ ∂ ϕ (x (t)) (dt)
41
42
means that x, k : R+ → Rd are continuous functions satisfying conditions (11-j, jj, jv).
44
45
28
33
42
43
27
30
2
s
33
37
23
|y (r) − x (r)| (ρdr + γ d kr ) .
+
32
36
20
22
t
31
35
19
21
(j ) x (t) ∈ Dom (ϕ), ∀ t ≥ 0 and ϕ (x (·)) ∈ L1loc (R+ ) ,
(jj) k ∈ BV loc R+ ; Rd , k (0) = 0,
(jjj) x (t) + k (t) = x0 + m (t) , ∀ t ≥ 0,
30
34
17
18
Definition 14 (Generalized Skorohod problem). A pair (x, k) of continuous functions x, k :
R+ → Rd , is a solution of equation (8) if
21
22
12
13
15
17
7
8
where
11
16
4
5
dx (t) + ∂ − ϕ (x (t)) (dt) dm (t) ,
x (0) = x0 ,
10
14
3
43
44
and will
The next result provides an equivalent condition for (11-jv)
be used later in the proof
of the continuity of the mapping (x0 , m) → (x, k) = SP ∂ − ϕ; x0 , m and for the main existence
result in the stochastic case.
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12
Lemma 16. We suppose that ϕ satisfies assumption (10) and let x, k : R+ → Rd be two continuous functions satisfying (11-j, jj). Then the pair (x, k) satisfies (11-jv) if and only if there exists
a continuous increasing function A : R+ → R+ , such that
4
jv
5
6
7
8
s
s
12
13
14
15
16
and let λ, η : R+ → [0, 1] and θ : R+ → Rd with |θ (r)| ≤ 1, for any r ∈ R+ , be some measurable
functions (given by the Radon–Nikodym’s theorem) such that
18
dk (r) = θ (r) dQr ,
22
24
dr = λ (r) dQr
17
21
and dAr = η (r) dQr .
22
t+ε
t+ε
t+ε
z − x (r) , θ (r) dQr +
ϕ (x (r)) λ (r) dQr ≤ ϕ (z)
λ (r) dQr
25
27
t−ε
28
t−ε
27
28
t+ε
|z − x (r)|2 η (r) dQr
+
30
29
30
t−ε
31
31
32
32
and therefore
34
37
38
39
40
33
t+ε
z,
t+ε
θ (r) dQr −
t−ε
t+ε
x (r) , θ (r) dQr +
t−ε
ϕ (x (r)) λ (r) dQr ≤ ϕ (z)
t−ε
t−ε
35
λ (r) dQr
36
t−ε
t+ε
t+ε
t+ε
2
|x (r)|2 η (r) dQr .
+ |z|
η (r) dQr − 2z,
x (r) η (r) dQr +
t−ε
34
t+ε
37
38
(13)
t−ε
41
42
43
44
45
39
40
41
1
Multiplying by
and using the Lebesgue–Besicovitch differentiation theorem,
Q ([t − ε, t + ε])
we
deduce, passing to the limit in the seven above integrals, that there exists 1 ⊂ R+ with
1 dQr = 0, such that for all z ∈ Dom (ϕ) and r ∈ R+ 1
46
47
24
26
t−ε
29
36
20
23
26
35
19
Clearly d kr = |θ (r)| dQr . From (12) we deduce that, for all t ∈ R+ , ε > 0 and z ∈ Dom (ϕ)
25
33
11
Qr := r + kr + Ar
21
23
10
(12)
Proof. We only need to prove that (12) ⇒ (11-jv).
Denote
17
20
8
s
16
19
7
9
|y (r) − x (r)|2 dAr .
+
12
18
6
t
11
15
3
5
s
10
14
2
4
∀ 0 ≤ s ≤ t, ∀y : R+ → Rd continuous:
t
t
t
y (r) − x (r) , dk (r) + ϕ (x (r)) dr ≤ ϕ (y (r)) dr
9
13
1
42
43
44
45
46
z − x (r) , θ (r) + ϕ (x (r)) λ (r) ≤ ϕ (z) λ (r) + |z − x (r)|2 η (r) .
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1
13
Hence, from the definition of the Fréchet subdifferential we obtain
1
2
2
−
θ (r) ∈ ∂ IDom(ϕ) (x (r)) ,
3
∀ r ∈ 2 \ 1
3
4
5
4
and
5
6
7
8
9
10
11
6
θ (r)
∈ ∂ − ϕ (x (r)) , ∀ r ∈ (R+ 2 ) \ 1 ,
λ (r)
where 2 = {r ≥ 0 : λ (r) = 0} with 2 dr = 2 λ (r) dQr = 0.
Since IDom(ϕ) is (0, γ )-semiconvex,
7
8
9
10
11
12
12
13
y (r) − x (r) , θ (r) ≤ γ |θ (r)| |y (r) − x (r)| ,
2
14
15
16
17
18
19
13
∀ r ∈ 2 \ 1 .
14
On the other hand, since ϕ is a (ρ, γ )-semiconvex function, we have for any continuous y :
R+ → Rd ,
y (r) − x (r) ,
20
θ (r) + ϕ (x (r)) ≤ ϕ (y (r)) + |y (r) − x (r)|2
λ (r)
θ (r) ρ +γ
,
λ (r)
17
19
20
21
22
23
16
18
∀ r ∈ (R+ 2 ) \ 1 .
21
15
22
Therefore (with the convention 0 · (+∞) = 0) we deduce that, for all r ∈ R+ 1 ,
23
24
24
25
y (r) − x (r) , θ (r) + ϕ (x (r)) λ (r) ≤ ϕ (y (r)) λ (r)
25
26
+ |y (r) − x (r)| (ρλ (r) + γ |θ (r)|) .
26
2
27
28
27
Integrating on [s, t] with respect to the measure dQr we infer that (11-jv) holds.
29
30
Lemma 17. If
dk (t) ∈ ∂ − ϕ (x (t)) (dt)
and
d k̂ (t) ∈ ∂ − ϕ
2
28
29
x̂ (t) (dt), then for all 0 ≤ s ≤ t :
30
31
32
33
31
t
34
35
32
x (r) − x̂ (r)2 2ρdr + γ d kr + γ d k̂ r
33
34
s
t
36
+
37
38
35
36
x (r) − x̂ (r) , dk (r) − d k̂ (r) ≥ 0.
(14)
39
40
41
39
Proof. The conclusion follows from (11-jv) written for (x, k) with y = x̂ and for (x̂, k̂) with
y = x. 2
42
43
44
45
46
47
37
38
s
40
41
42
Notation 18. Let x[s,t] := sup |xr | and xt := x[0,t] .
43
r∈[s,t]
44
Theorem 19 (Uniqueness). Let assumptions (9) and (10) be satisfied. If (x, k) = SP ∂ − ϕ; x0 , m
and (x̂, k̂) = SP(∂ − ϕ; x̂0 , m̂) then for all t ≥ 0:
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14
x − x̂ 2 ≤ 2 x0 − x̂0 2 + m − m̂2 + 2 m − m̂ k − k̂t
t
t
t
1
2
· exp(8ρt + 4γ kt + 4γ k̂ t ) .
3
4
5
1
(15)
In particular the uniqueness of the problem SP ∂ − ϕ; x0 , m follows.
5
6
Proof. We clearly have
8
7
8
x (t) − m (t) − x̂ (t) + m̂ (t)2 = x0 − x̂0 2
9
10
t
11
+2
12
9
10
t
m (r) − m̂ (r) , dk (r) − d k̂ (r) − 2
0
13
11
x (r) − x̂ (r) , dk (r) − d k̂ (r).
12
0
13
14
15
14
Using (14) it follows that
16
15
16
x (t) − x̂ (t)2 − m (t) − m̂ (t)2 ≤ x (t) − m (t) − x̂ (t) + m̂ (t)2
2
≤ x0 − x̂0 + 2 m − m̂t k − k̂ t
t
2 + 2 x (r) − x̂ (r) 2ρdr + γ d kr + γ d k̂ r ,
1
2
17
18
19
20
21
22
17
18
19
20
21
22
0
23
24
23
which implies, via Gronwall’s inequality, the desired conclusion. 2
24
25
26
27
25
To derive the uniform boundedness and the continuity of the solution of the generalized Skorohod problem we need to introduce some additional assumptions:
28
|ϕ (x) − ϕ (y)| ≤ L + L |x − y| , ∀ x, y ∈ Dom (ϕ)
30
(16)
34
35
36
37
38
39
32
Dom (ϕ) satisfies the (γ , δ, σ ) -shifted uniform interior ball condition
(17)
(for the definition of (γ , δ, σ )-SUIBC, see definition (5)).
We mention that assumption (16) is obviously satisfied by the function ϕ given in the Example 13.
Using Proposition 7 we see that assumption (17) is fulfilled if we impose that
44
45
(18)
34
35
36
37
38
39
41
42
condition which can be more easily visualized.
Note that the lower semicontinuity of ϕ and the assumption (16) clearly yield that the Dom (ϕ)
is a closed set, and, from the assumption (17) it can be derived that
46
47
33
40
Dom (ϕ) satisfies the (h0 , r0 ) -drop condition,
42
43
29
31
and
40
41
27
30
32
33
26
28
29
31
3
4
6
7
2
43
44
45
46
Int (Dom (ϕ)) = ∅.
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5
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15
Remark 20. Technical condition (5) from assumption (17) will provide an estimate for the total variation of k (see Lemma 29). On the other hand, assumption (5) from P.-L. Lions and
A.S. Sznitman [24] (or [38, Condition (B)]) has the same role (see [24, Lemma 1.2]).
In the particular case ϕ = IE , as in [24], it is essentially used in the representation of the
bounded variation process k and in our case it is used in the subdifferential inequality (11-jv).
Hence assumption (17) is required by the transition from the particular case of the indicator
function to the case of a general convex l.s.c. function ϕ.
8
9
10
11
14
19
22
23
24
25
26
27
28
37
38
M := sup m < ∞ and CT ,M := sup CT ,m < ∞.
m∈M
m∈M
41
(c)
22
23
24
25
26
30
31
32
33
If moreover m̂ ∈ C
[0, T ] ; Rd
36
37
39
|x (t) − x (s)| + kt − ks ≤ CT ,m · μm (t − s),
35
38
xT ≤ |x0 | + CT ,m ,
(21)
∀ 0 ≤ s ≤ t ≤ T.
40
41
42
(x̂, k̂) = SP(∂ − ϕ; x̂
, x̂0 ∈ Dom (ϕ) and
0 , m̂), then
x − x̂T + k − k̂T ≤ A CT ,m , CT ,m̂ · x0 − x̂0 + m − m̂T ,
46
47
21
34
(b)
45
14
29
(20)
Theorem 23. Let assumptions (9), (10), (16) and (17) be satisfied. Then there exists a constant C,
depending only on the constants from the assumptions, such that if (x, k) = SP ∂ − ϕ; x0 , m
then
40
44
13
28
The main results of this section are the following two theorems whose proofs will be given in
the next section:
(a) kBV [0,T ];Rd = kT ≤ CT ,m ,
43
11
27
Remark 22. It is easy to prove that, for any compact subset M ⊂ C [0, T ] ; Rd ,
39
42
10
19
34
36
9
20
31
35
7
18
The function μ y : [0, T ] → [0, μ y (T )] is a strictly increasing continuous function and therefore
the inverse function μ −1
y : [0, μ y (T )] → [0, T ] is well defined and is also a strictly increasing
continuous function. Using this inverse function let C be a positive constant and
2
μ−1
m := 1/μ
m δ exp[−C (1 + T + mT )] ,
(19)
CT ,m := exp C (1 + T + mT + m ) .
30
33
6
17
where my (ε) is the modulus of continuity, given by
my (ε) := sup |y (t) − y (s)| : |t − s| ≤ ε, t, s ∈ [0, T ] .
29
32
5
16
20
21
4
15
μ y (ε) = ε + my (ε) ,
16
18
3
12
In order to
prove some a priori estimates, let us introduce the following notation: for y ∈
C [0, T ] ; Rd and ε > 0 write
15
17
2
8
Remark 21. We notice that assumption (18) is similar to Condition (B’) from [38] (the uniform
interior cone condition). But the running direction from the drop condition (18) is not required
to be uniform with respect to the vertex, like in [38, Condition (B’)].
12
13
1
43
44
(22)
45
46
where A is a continuous function.
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We
can now
derive the following continuity result of the mapping (x0 , m) → (x, k) =
SP ∂ − ϕ; x0 , m .
3
4
5
Corollary 24. Let assumptions (9), (10), (16) and (17) be satisfied. If x0n , x0 ∈ Dom (ϕ),
mn , m ∈ C R+ ; Rd , mn (0) = 0 and
7
(xn , kn ) = SP
x0n → x0 ,
i)
ii)
8
9
and there exist x, k ∈ C R+ ; R
d
CT ,M := sup CT ,ν < ∞.
23
ν∈M
ν∈M
43
44
45
|xn (t) − xn (s)| + kn t − kn s ≤ CT ,M · μ M (t − s)
and
24
27
28
30
31
33
34
35
36
xn − xl T + kn − kl T ≤ CT ,M · |x0n − x0l | + ||mn − ml ||T .
Therefore there exist x, k, A ∈ C R+ ; R
xn → x,
d
37
38
39
such that
40
kn → k, in C([0, T ] ; R ), as n → ∞,
d
and, by Arzelà–Ascoli’s theorem, on a subsequence denoted also with kn ,
kn → A, in C([0, T ] ; Rd ), as n → ∞,
46
47
22
32
xn T + kn T ≤ a + CT ,M ,
35
42
21
29
Let a > 0 be such that |x0n | ≤ a. By the estimates established in Theorem 23 we obtain: for all
n, l ∈ N∗ and for all s, t ∈ [0, T ], s ≤ t ,
34
41
19
26
μ M (ε) := sup μ ν (ε) 0, as ε 0.
33
40
18
25
Also
32
39
16
20
29
38
15
Proof.
Let us
fix arbitrary T > 0. The set M = {m, mn : n ∈ N∗ } is a compact subset of
C [0, T ] ; Rd . If CT ,m is the constant defined by (19), then, using (20), it follows that
28
37
13
17
27
36
10
14
25
31
9
(a) xn − xT + kn − kT → 0,
(b) (x, k) = SP ∂ − ϕ; x0 , m .
24
30
8
such that, for any T ≥ 0,
23
26
7
12
19
22
,
sup kn T < ∞, ∀T ≥ 0,
18
21
5
6
n∈N∗
17
20
0n , mn
4
11
14
16
∂ − ϕ; x
then
13
15
mn → m in C [0, T ] ; Rd , ∀T ≥ 0,
iii)
10
12
2
3
6
11
1
41
42
43
44
45
46
where A is an increasing function starting from zero.
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1
2
17
Clearly, the
pair (x, k)
satisfies (11-j, jj, jjj) and (12), which means, using Lemma 16, that
(x, k) = SP ∂ − ϕ; x0 , m . 2
3
4
5
7
8
9
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
2
3
Theorem 25. Let assumptions (9), (10), (16) and (17) be satisfied.
Then the generalized Skorohod problem
6
10
1
4
5
6
x (t) + k (t) = x0 + m (t) ,
dk (t) ∈ ∂ − ϕ (x (t)) (dt)
t ≥ 0,
7
8
9
has a
unique solution (x, k), in the sense of Definition 14 (and we write (x, k) = SP ∂ − ϕ;
x0 , m ).
Before giving the proof of the main results, Theorems 23 and 25, let us examine the particular
case of the indicator function of the closed set E (which yields the classical Skorohod problem).
If E satisfies the r0 -UEBC, then, by Lemmas 3 and 4 and Definition 11, the set E is
1
1
2r0 -semiconvex and the indicator function IE (x) is a (0, 2r0 )-semiconvex function. Hence assumptions (10) and (16) are satisfied.
We write the definition of the solution of the Skorohod problem in the case of the indicator
function.
Definition 26 (Skorohod problem). Let E ⊂ Rd be a set satisfying the r0 -UEBC. A pair (x, k) is
a solution of the Skorohod problem if x, k : R+ → Rd are continuous functions and
(j )
x (t) ∈ E,
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
d
(jj) k ∈ BV loc R+ ; R , k (0) = 0,
(jjj) x (t) + k (t) = x0 + m (t) ,
(jv) ∀ 0 ≤ s ≤ t ≤ T , ∀y ∈ C (R+ ; E)
t
t
1
y (r) − x (r) , dk (r) ≤
|y (r) − x (r)|2 d kr .
2r0
s
25
26
27
(23)
29
30
31
s
32
33
The following theorem is a consequence of the main existence Theorem 25.
34
be a continuous function such that m (0) = 0. If E
Theorem 27. Let x0 ∈ E and m : R+ →
satisfies the r0 -UEBC, for some r0 > 0, and the shifted uniform interior ball condition, then there
exists a unique solution of the Skorohod problem, in the sense of Definition 26.
Moreover, each one of the following conditions is equivalent with (23-jv):
Rd
⎧
t
⎪
⎪
⎪
⎪
⎪
⎪ kt = 1x(s)∈Bd(E) d ks ,
⎪
⎪
⎪
0
⎨
t
jv
⎪
⎪
⎪
k (t) = nx(s) d ks ,
⎪
⎪
⎪
⎪
⎪
0
⎪
⎩
where nx(s) ∈ NE (x (s)) and |nx(s) | = 1, d ks -a.e.,
28
35
36
37
38
39
40
41
42
43
(24)
44
45
46
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18
and
1
jv
2
∃ β > 0 such that ∀ y : R+ → E continuous,
t
t
y (r) − x (r) , dk (r) ≤ β |y (r) − x (r)|2 d kr .
4
5
6
s
7
8
9
10
3
4
(25)
s
Proof. The uniqueness is ensured by Theorem 25. For a direct proof it is sufficient
to use inequality (15) and the next one: if (x, k) = SP (E; x0 , m) and (x̂, k̂) = SP E; x̂0 , m̂ , then from (23)
we get (see also Lemma 17)
x (t) − x̂ (t) , dk (t) − d k̂ (t) +
14
16
17
18
19
20
23
24
25
t
t
s
28
31
32
33
34
35
36
t
y (r) − x (r) , nx(r) 1x(r)∈Bd(E) d kr
=
41
s
≤
1
2r0
1
≤
2r0
nx(r) |y (r) − x (r)|2 1x(s)∈Bd(E) d kr
t
14
15
16
17
18
19
20
21
23
24
26
27
30
|y (r) − x (r)| d kr .
31
s
32
Clearly (23-jv) =⇒ (25).
Proof of (25) =⇒ (24): let [s, t] be an interval such that x (r) ∈ Int (E) for all r ∈ [s, t]. Then
there exists δ = δs,t > 0 such that
inf dBd(E) x (r) ≥ δ.
r∈[s,t]
Let λ ∈ [0, δ] and α ∈ C [0, T ] ; Rd such that αT ≤ 1. Setting y (r) = x (r) + λα (r) in (25)
we obtain
33
34
35
36
37
38
39
40
41
42
t
t
α (r) , dk (r) ≤ βλ
s
43
d kr .
s
46
47
13
29
2
44
45
12
28
s
42
43
10
25
t
38
40
9
22
37
39
s
29
30
y (r) − x (r) , nx(r) d kr
y (r) − x (r) , dk (r) =
26
27
2
1 x (t) − x̂ (t) (d kt + d k̂ t ) ≥ 0.
2r0
The existence is due to Theorem 25 (but, for a direct proof of the existence, with condition
(23-jv) replaced by (24), we refer the reader to [24]).
Proof of (24) =⇒ (23-jv): using Lemma 3 we see
21
22
8
11
12
15
6
7
11
13
5
44
45
46
Hence, passing to the limit, for λ → 0, and taking supαT ≤1 , we deduce the implication:
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x (r) ∈ Int (E) , ∀r ∈ [s, t] =⇒ kt − ks = 0.
1
19
(26)
2
3
2
Let (r) be a measurable function such that | (r)| = 1, d kr -a.e. and
3
4
4
t
5
7
5
(r) d kr .
k (t) =
6
6
7
0
8
9
8
Since (25) holds for all 0 ≤ s ≤ t , we deduce, using the Lebesgue–Besicovitch theorem, that
10
d kr -a.e.,
11
12
14
12
for any y ∈ C ([0, T ] ; E).
Therefore, from Lemma 4, we infer that
15
13
14
15
(r) ∈ NE (x (r)) , d kr -a.e.
16
(27)
17
18
18
19
4. Generalized Skorohod problem: proofs
20
21
22
23
24
25
26
21
In order to prove Theorem
25 let us first prove some auxiliary results.
Let (x, k) = SP ∂ − ϕ; x0 , m and y ∈ C (R+ ; E), where E = Dom (ϕ). From (16) and
(11–jv) we have, for all 0 ≤ s ≤ t,
t
s
28
t
29
33
29
|y (r) − x (r)| (ρdr + γ d kr ) .
+
2
(28)
s
32
Suppose that x (r) ∈ Int (Dom (ϕ)) for all r ∈ [s, t], and let
34
r∈[s,t]
36
39
t
41
43
44
45
46
47
λb
s
35
36
d
Write y (r) = x (r) + λbα (r) with α ∈ C R+ ; R , α[s,t] ≤ 1 and 0 < λ < 1. Hence the above
−1
inequality becomes, for λ = (1 + γ ) (1 + b)2
40
42
31
33
0 < b ≤ inf dBd(E) (x (r)) .
35
38
30
32
34
37
24
27
s
31
23
26
|y (r) − x (r)| dr
30
22
25
t
y (r) − x (r) , dk (r) ≤ L (t − s) + L
27
28
16
17
2
We have thus proved inequality (24), since we have (26) and (27).
19
20
9
10
β |y (r) − x (r)|2 − (r) , y (r) − x (r) ≥ 0,
11
13
1
37
38
39
40
2
α (r) , dk (r) ≤ (L + Lb) (t − s) + λ2 b ρ (t − s) + γ (kt − ks )
41
42
43
≤ L + Lb + λ2 b2 ρ (t − s) +
λb
(kt − ks ) .
1+b
Taking the supremum over all α such that α[s,t] ≤ 1 we see that
44
45
46
47
JID:YJDEQ AID:7984 /FLA
λb2
(kt − ks ) ≤ L + Lb + λ2 b2 ρ (t − s) .
1+b
1
2
3
4
5
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20
Lemma 28. Let ϕ such that assumption (16) is satisfied and (x, k) = SP ∂ − ϕ; x0 , m . If x (r) ∈
Int (Dom (ϕ)), for all r ∈ [s, t], then there exists a positive constant C = C (L, ρ, γ , b) such that
kt − ks ≤ C (t − s)
9
10
r∈[s,t]
13
16
17
18
Lemma 29. Let ϕ be a (ρ, γ )-semiconvex function and (x, k) = SP ∂ − ϕ; x0 , m . Assume that
ϕ satisfies assumption (16) and set Dom (ϕ) satisfies assumption (17). If 0 ≤ s ≤ t and
sup |x (r) − x (s)| ≤ δ,
r∈[s,t]
20
24
25
26
27
1
3L + 4ρ
|k (t) − k (s)| +
(29)
(t − s) .
σ
σ
Proof. Let us fix arbitrarily α ∈ C R+ ; Rd such that α[s,t] ≤ 1. From assumptions (16)–(17),
if
|ϕ (y (r)) − ϕ (x (r))| ≤ 3L.
23
24
25
26
27
29
33
34
t
α (r) , dk (r) ≤ −
λx(s)
37
38
40
t
s
vx(s) , dk (r) + (3L + 4ρ) (t − s)
t
+γ
s
41
42
43
s
44
47
19
39
From (28) we deduce that
41
46
18
36
38
45
17
35
and
37
43
16
32
|y (r) − x (r)| ≤ vx(s) + λx(s) ≤ 2
35
42
15
31
34
40
13
30
then y (r) ∈ E.
Moreover
33
39
12
28
y (r) := x (r) + vx(s) + λx(s) α (r) , r ∈ [s, t] ,
30
36
9
22
kt − ks ≤
29
32
8
21
then there exists σ > 0 such that
28
31
7
20
21
23
6
14
More generally we have
19
22
5
11
0 < b ≤ inf dBd(E) (x (r)) .
12
15
4
10
where
11
14
2
3
Consequently, the following result is proved:
8
1
44
vx(s) + λx(s) 2 d kr .
45
46
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1
21
Taking the supremum over all α such that α[s,t] ≤ 1 we see, using also (5), that
2
2
σ (kt − ks ) ≤ |k (t) − k (s)| + (3L + 4ρ) (t − s)
3
4
5
6
7
8
11
12
13
2
5
Proof of Theorem 23. We denote by C, C , C generic constants independent of x0 , x̂0 , m, m̂
and T , but possibly depending on constants L, δ, σ , ρ, γ provided by the assumptions.
|x (t) − x (s)| ≤ |m (t) − m (s)| + kt − ks .
|x (t) − x (s) − m (t) + m (s)| = 2
22
m (r) − m (s) , dk (r)
2
25
t
26
x (s) − x (r) , dk (r) .
+2
27
s
28
t
30
x (s) − x (r) , dk (r) ≤ L (t − s) + L
31
|x (s) − x (r)| dr
s
t
35
37
43
44
1
2
|α|2 ≤ |α − β|2 + |β|2 , we get
36
39
40
1
|x (t) − x (s)|2 ≤ |m (t) − m (s)|2 + 2mm (t − s) (kt − ks ) + C (t − s)
2
t
+ C |x (r) − x (s)|2 (dr + d kr ) ,
45
s
46
47
34
38
Thus, using also the inequality
40
42
33
37
s
38
41
32
35
|x (s) − x (r)|2 (ρdr + γ d kr ) .
+
36
27
29
t
s
26
28
From (11-jv) written for y (r) ≡ x (s) and (16) we have
33
23
24
s
25
39
19
21
t
24
34
18
20
We clearly have
23
32
15
17
22
31
14
16
21
30
10
13
it follows that
19
29
8
12
|x (t) − x (s) − m (t) + m (s)| = |k (t) − k (s)| ≤ kt − ks ,
18
20
7
11
Let 0 ≤ s ≤ t ≤ T .
Since
15
17
6
9
Step 1. Some estimates of the modulus of continuity of the function x.
14
16
3
4
and the lemma follows.
9
10
1
41
42
43
44
45
46
and, by Gronwall’s inequality,
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22
|x (t) − x (s)|2 ≤ m2m (t − s) + mm (t − s) (kt − ks ) + (t − s)
· exp C (1 + t − s + kt − ks ) ,
1
2
1
(30)
3
4
3
for all 0 ≤ s ≤ t ≤ T .
4
5
6
7
5
Step 2. Estimates of the differences |x (t) − x (s)| and kt − ks under the assumption
|x (t) − x (s)| ≤ δ.
8
9
10
12
13
14
21
16
Step 3. Adapted time partition and local estimates.
26
27
t0 = T0 = 0,
T1 = inf t ∈ [t0 , T ] : dBd(E) (x (t)) ≤ δ/4 ,
29
30
31
t1 = inf {t ∈ [T1 , T ] : |x (t) − x (T1 )| > δ/2} ,
T2 = inf t ∈ [t1 , T ] : dBd(E) (x (t)) ≤ δ/4 ,
32
33
34
·········
35
ti = inf {t ∈ [Ti , T ] : |x (t) − x (Ti )| > δ/2}
Ti+1 = inf t ∈ [ti , T ] : dBd(E) (x (t)) ≤ δ/4
36
37
38
·········
39
40
20
21
(31)
22
23
24
25
27
28
29
30
31
32
33
34
35
36
37
38
39
42
0 = T0 = t0 ≤ T1 < t1 ≤ T2 < · · · < ti ≤ Ti+1 < ti+1 ≤ · · · ≤ T .
43
44
Let ti ≤ r ≤ Ti+1 . Then x (r) ∈ Int (E) and dBd(E) (x (r)) ≥ δ/4. By Lemma 28 we get
46
47
19
41
44
45
18
40
Clearly
42
43
17
26
Let the sequence given by (the definition is suggested by [24])
28
41
14
15
|x (t) − x (s)| + kt − ks ≤ μ m (t − s) · exp C (1 + T + mT ) .
23
25
13
since (t − s) + mm (t − s) = μm (t − s) ≤ T + 2 mT .
Hence if 0 ≤ s ≤ t ≤ T and |x (t) − x (s)| ≤ δ then
22
24
12
|x (t) − x (s)|2 ≤ μ m (t − s) exp C (1 + T + mT ) , for all 0 ≤ s ≤ t ≤ T ,
18
20
11
Using the estimate (30), it clearly follows that
17
19
7
9
kt − ks ≤ C |k (t) − k (s)| + C (t − s)
= C |x (t) − x (s) − m (t) + m (s)| + C (t − s)
μm (t − s) .
≤ C |x (t) − x (s)| + Cmm (t − s) + C (t − s) ≤ Cδ + Cμ
11
16
6
8
Let 0 ≤ s ≤ r ≤ t ≤ T such that |x (t) − x (s)| ≤ δ. From (29) we have
10
15
2
45
46
|k (t) − k (s)| ≤ kt − ks ≤ C (t − s) for ti ≤ s ≤ t ≤ Ti+1 .
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1
23
Also for ti ≤ s ≤ t ≤ Ti+1 :
1
2
2
|x (t) − x (s)| ≤ |k (t) − k (s)| + |m (t) − m (s)| ≤ C (t − s) + |m (t) − m (s)|
3
4
6
7
6
7
μm (t − s) .
|x (t) − x (s)| + kt − ks ≤ Cμ
9
11
15
16
17
18
19
20
21
for all Ti ≤ s ≤ t ≤ ti ,
and consequently, for all Ti ≤ s ≤ t ≤ ti , inequality (31) holds.
If Ti ≤ s ≤ ti ≤ t ≤ Ti+1 then
|x (t) − x (s)| + kt − ks
≤ |x (t) − x (ti )| + kt − kti + |x (ti ) − x (s)| + kti − ks
μm (t − ti ) + μ m (ti − s) · exp C (1 + T + mT )
≤ Cμ
≤ μ m (t − s) × exp C (1 + T + mT ) .
22
23
9
11
|x (t) − x (s)| ≤ δ,
13
8
10
On each of the intervals [Ti , ti ], we have
12
14
5
and then
8
10
4
μm (t − s)
≤ Cμ
5
3
12
13
14
15
16
17
18
19
20
21
22
Consequently for all i ∈ N and Ti ≤ s ≤ t ≤ Ti+1 , inequality (31) holds.
24
23
24
25
Step 4. Getting inequalities (21).
25
26
μm (T ) → [0, T ] is well defined and is a strictly increasing continuous funcSince μ −1
m : 0,μ
tion, from
26
≤ |x (ti ) − x (Ti )| ≤ μm (ti − Ti ) × exp C (1 + T + mT )
≤ μ m (Ti+1 − Ti ) · exp C (1 + T + mT ) ,
29
27
28
29
δ
2
30
31
32
33
34
35
36
37
38
41
δ2
m
Ti+1 − Ti ≥ μ −1
exp
−2C
+
T
+
)
(1
T
m
4
2
≥ μ −1
> 0.
m δ exp −2C (1 + T + mT )
46
47
31
32
34
35
36
37
38
40
41
42
43
45
30
39
Hence the bounded increasing sequence (Ti )i≥0 has a finite number of terms, therefore there
exists j ∈ N∗ such that T = Tj . Then
42
44
28
33
we deduce that
39
40
27
T = Tj =
j
43
(Ti − Ti−1 ) ≥ j −1
m ,
i=1
2
μ−1
where m := 1/μ
m δ exp −C (1 + T + mT ) (see definition (19)).
44
45
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24
Let 0 ≤ s ≤ t ≤ T . We have
2
3
kt − ks =
4
j 1
k(t∧Ti )∨s − kt∧Ti−1 ∨s
i=1
5
7
i=1
8
≤ j μ m (t − s) · exp C (1 + T + mT )
≤ T m μ m (t − s) · exp C (1 + T + mT )
9
10
11
12
13
16
19
20
21
22
25
26
27
28
29
32
33
34
35
36
37
38
|x (t)| = |x0 + m (t) − k (t)| ≤ |x0 | + mt + kt ≤ |x0 | + mT + kT .
Hence there exists a positive constant C = C (L, δ, σ, ρ, γ ) such that, under notations (19),
kT ≤ CT ,m ,
and
xT ≤ |x0 | + CT ,m ,
41
≤ CT ,m μ m (t − s) .
44
45
46
47
10
11
14
15
18
19
20
21
22
24
25
26
27
28
29
30
Step 5. Getting inequality (22).
Since kT + k̂ T ≤ CT ,m + CT ,m̂ , from inequality (15) we have
x − x̂ 2 ≤ 2 x0 − x̂0 2 + m − m̂2 + 2 m − m̂ k − k̂T
T
T
T
· exp 4γ (2t + kT + k̂ T )
2 ≤ A2 CT ,m , CT ,m̂ x0 − x̂0 + m − m̂T ,
31
32
33
34
35
36
37
38
39
(where
A is a continuous function) and the conclusion follows since k − k̂ = x0 − x̂0 + m − m̂ −
x − x̂ . 2
42
43
9
23
which is part of conclusion (21).
In order to end the proof of (21) it is sufficient to remark that, for any 0 ≤ s ≤ t ≤ T ,
|x (t) − x (s)|2 ≤ m2m (t − s) + mm (t − s) CT ,m + (t − s) · exp C 1 + CT ,m
39
40
8
17
30
31
7
16
and
23
24
6
13
kT ≤ T m μ m (T ) · exp C (1 + T + mT ) ≤ exp C (1 + T + mT + m )
15
18
4
12
and consequently
14
17
3
5
j ≤
μ m (t ∧ Ti ) ∨ s − (t ∧ Ti−1 ) ∨ s · exp C (1 + T + mT )
6
2
40
41
42
Proof of the Theorem
25. Uniqueness was proved in Theorem 19. To prove existence, let
mn ∈ C 1 R+ ; Rd with mn (0) = 0 be such that mn − mT → 0 for all T ≥ 0. Since mn ∈
C 1 R+ ; Rd , we deduce, using the results from
the papers [19] or [37], that there exists a
unique solution (xn , kn ) of the SP ∂ − ϕ; x0 , mn , and by Corollary 24 we see that there exist
x, k ∈ C R+ ; Rd such that for all T ≥ 0
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44
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25
xn − xT + kn − kT → 0, as n → ∞, and
(x, k) = SP ∂ − ϕ; x0 , m ,
1
2
1
2
3
4
3
which completes the proof.
2
4
5
6
5
5. Generalized Skorohod equations
6
7
8
9
7
Consider the next (non-convex) variational inequality with singular input (which will be called
generalized Skorohod differential equation):
10
12
13
14
11
t ≥ 0,
(32)
17
16
(for the notation
we recall Remark 15).
We introduce the following supplementary assumptions:
17
18
f (·, x) : R+ → R is measurable, ∀x ∈ R ,
d
19
20
and there exists
22
23
μ ∈ L1loc (R+ ),
(i)
24
x → f
d
26
27
28
(33)
21
22
(t, x) : Rd
→ Rd
23
is continuous,
24
25
(34)
28
0
29
where
30
f # (t) := sup |f (t, x)| : x ∈ Dom (ϕ) .
31
32
34
35
36
37
38
Proposition 30 (Generalized Skorohod equation). Let ϕ : Rd → (−∞, ∞] and f : R+ × Rd →
Rd be such that assumptions (9), (10), (16), (17) and (33), (34) are satisfied.
Then the generalized Skorohod equation (32) has a unique solution.
43
|x (t) − x̂ (t) |2 + 2
34
35
36
37
38
41
42
x (r) − x̂ (r) , dk (r) − d k̂ (r)
43
44
0
t
45
47
33
40
t
44
46
32
39
Proof. Let (x, k) and (x̂, k̂) be two solutions. Then
41
42
31
(35)
Clearly, assumption (34-iii) is satisfied if, as example, f (t, x) = f (t) or if Dom (ϕ) is bounded.
39
40
26
27
29
33
19
20
such that a.e. t ≥ 0:
(ii) x − y, f (t, x) − f (t, y) ≤ μ (t) |x − y|2 , ∀x, y ∈ Rd ,
T
f # (s) ds < ∞, ∀T ≥ 0,
(iii)
25
30
13
15
dk (t) ∈ ∂ − ϕ (x (t)) (dt)
18
21
12
14
15
16
9
10
⎧
t
⎪
⎪
⎨ x (t) + k (t) = x + f (s, x (s)) ds + m (t) ,
0
⎪
0
⎪
⎩
dk (t) ∈ ∂ − ϕ (x (t)) (dt)
11
8
=2
0
x (r) − x̂ (r) , f (r, x (r)) − f r, x̂ (r) dr ≤ 2
t
0
45
μ+ (r) |x (r) − x̂ (r) |2 dr,
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26
and using Lemma 17 it follows that
1
2
2
x (t) − x̂ (t)2 ≤ 2
3
4
5
t
x (r) − x̂ (r)2 dAr ,
3
4
5
0
6
7
6
where
7
8
t
9
At = 2ρt + γ kt + γ k̂ t +
10
11
12
13
14
15
16
17
18
19
20
21
24
27
⎧
xn (t) = x0 , for t < 0,
⎪
⎪
⎪
t
⎪
⎨
xn (t) + kn (t) = x0 + f s, xn (s − 1/n) ds + m (t) ,
⎪
⎪
⎪
0
⎪
⎩
−
dkn (t) ∈ ∂ ϕ (xn (t)) (dt) .
For any i ∈ N, for t ∈
i i+1
n, n
32
35
38
39
42
43
18
for t ≥ 0,
(36)
19
20
21
25
f s, xn (s − 1/n) ds + m (t) − m (i/n) ,
therefore by iteration over the intervals
(xn , kn ) = SP ∂ − ϕ; x0 , mn , with
t
mn (t) =
i i+1
n, n
26
27
28
i/n
29
there exists (via Theorem 25) a unique pair
30
31
32
33
f s, xn (s − 1/n) ds + m (t) .
0
34
35
36
37
Let T > 0. If M denotes the set
M := mn : n ∈ N∗ ,
then M is a relatively compact subset of C
subset of C [0, T ] ; Rd .
Indeed
[0, T ] ; Rd
38
39
40
since it is a bounded and equicontinuous
41
42
43
44
T
45
47
17
44
46
15
24
xn (t) + kn (t) − kn (i/n) = xn (i/n) +
40
41
14
23
t
36
37
13
16
, we can write
33
34
12
22
29
31
11
28
30
10
Applying a Gronwall’s type inequality, we see that x = x̂.
Concerning the existence, we shall obtain the solution (x, k) as the limit in C [0, T ] ; Rd ×
C [0, T ] ; Rd of the sequence (xn , kn )n∈N∗ defined by an approximate Skorohod equation
25
26
9
μ+ (r) dr.
0
22
23
8
mn T ≤
45
f # (s) ds + mT
0
46
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1
27
and, for s < t ,
1
2
2
t
3
|mn (t) − mn (s)| ≤
4
5
f (r) dr + |m (t) − m (s)| .
4
5
s
6
7
3
#
6
7
Then by Theorem 23 and Remark 22,
8
8
9
xn T + kn T ≤ |x0 | + CT ,M
10
11
12
16
17
18
19
20
24
: n ∈ N∗ }
is a relatively compact subset of
Hence, by Arzelà–Ascoli’s theorem, the set {xn
C [0, T ] ; Rd .
Let x ∈ C [0, T ] ; Rd be such that, along a sequence still denoted by {xn : n ∈ N∗ },
24
26
f (s, x(s)) ds + m (t) , as n → ∞,
mn (t) →
29
30
and
31
31
t
32
kn (t) → k (t) = x0 +
39
40
41
32
f (s, x(s)) ds + m (t) − x (t) , as n → ∞.
0
35
38
44
47
34
36
Using Corollary 24 we infer that
37
⎛
(x, k) = SP ⎝∂ − ϕ; x0 ,
·
⎞
f (s, x(s)) ds + m⎠
0
38
39
40
41
42
i.e. (x, k) is a solution of problem (32). The uniqueness of the solution implies that the whole
sequence (xn , kn ) is convergent to that solution (x, k). The proof is completed now. 2
45
46
33
35
42
43
27
28
0
29
37
20
25
28
36
19
23
t
27
34
18
22
Then, uniformly with respect to t ∈ [0, T ],
26
33
17
21
xn − xT → 0, as n → ∞.
25
30
14
16
m∈M
22
13
15
where μ M (ε) := ε + sup mm .
21
23
10
12
|xn (t) − xn (s)| + kn t − kn s ≤ CT ,M μ M (t − s),
14
9
11
and for all 0 ≤ s ≤ t:
13
15
(37)
43
44
45
Remark 31. As it can be seen in the above proof, assumption (34-iii) is essential in order to
obtain inequality (37), i.e. the boundedness of the sequence (xn, kn )n defined by (36).
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28
Remark 32. If we replace assumptions (16), (17) and (34-iii) by the hypotheses
2
5
2
Int (Dom (ϕ)) = ∅
3
4
1
3
4
and
5
T
6
6
fR# (s) ds
7
8
< ∞,
∀R, T > 0,
7
8
0
9
10
9
where
10
fR# (t) := sup |f (t, x)| : |x| ≤ R ,
11
12
13
14
15
16
17
18
19
20
21
22
23
24
11
12
then, following the calculus from the convex case (see, e.g., Remark 4.15 and Proposition 4.16
from [31]), we deduce
1
r0
r0
|x (t) − u0 |2 + kt +
2
2
2
t
27
28
29
30
18
1
≤ x − u0 2t + C0 + C0 mT
4
t
t
+
2
+ 2 μ (r) x − u0 r dr + 2 μ+ (r) x − u0 2r (ρdr + γ d kr ),
19
20
21
22
(38)
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
23
24
0
25
where u0 ∈ Rd and r0 ∈ [0, 1) are such that B (u0 , r0 ) ⊂ Int (Dom (ϕ)).
In the convex case, namely γ = 0, this inequality yields the boundedness (21-a, b), but in the
non-convex case (γ = 0) we cannot obtain (21-a, b).
Of course, if there exists R > 0 such that Dom (ϕ) ⊂ B (0, R), then xT ≤ R and from (38)
we obtain kT ≤ C, if γ is such that
31
32
15
17
25
26
14
16
|f (r, x (r))| dr
0
0
13
0≤γ <
27
28
29
30
31
r0
.
2 (r0 + R + |u0 |)
32
33
If in the above proposition we take ϕ = IE we get, via Theorem 27,
34
35
→ Rd
be a continuous function such
Corollary 33 (Skorohod equation). Let x0 ∈ E and m : R+
that m (0) = 0. If f satisfies assumptions (33)–(34) and E satisfies the r0 -UEBC (for some
r0 > 0) and the shifted uniform interior ball condition, then there exists a unique pair (x, k)
such that:
(j ) x, k ∈ C (R+ ; E) , k (0) = 0,
(jj) k ∈ BV loc R+ ; Rd ,
t
(jjj) x (t) + k (t) = x0 + 0 f (s, x(s)) ds + m (t) ,
t
(jv) kt = 0 1x(s)∈Bd(E) d ks ,
t
(v) k (t) = 0 nx(s) d ks , where nx(s) ∈ NE (x (s)) and
nx(s) = 1, d ks -a.e.
26
36
37
38
39
40
41
42
43
(39)
44
45
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29
6. Non-convex stochastic variational inequalities
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
2
In the last section of the paper we will study the following multivalued stochastic differential
equation (also called stochastic variational inequality) considered on a non-convex domain:
⎧
t
t
⎪
⎪
⎨ X + K = ξ + F (s, X ) ds + G (s, X ) dB ,
t
t
s
s
s
⎪
0
0
⎪
⎩
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt) ,
22
23
24
25
26
27
5
6
t ≥ 0,
7
(40)
0
ξ ∈ L , F0 , P; Dom (ϕ)
and M is a progressively measurable and continuous stochastic process (p.m.c.s.p. for short)
with M0 = 0, then there exists a unique pair (X, K) of p.m.c.s.p., solution of the problem
Xt (ω) + Kt (ω) = ξ (ω) + Mt (ω) , t ≥ 0, ω ∈ ,
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt)
(in this case we shall write (X· (ω) , K· (ω)) = SP ∂ − ϕ; ξ (ω) , M· (ω) , P-a.s.).
(X· (ω) , K· (ω)) = SP ∂ − ϕ; ξ (ω) , M· (ω)
32
47
19
20
21
22
23
24
25
26
27
29
30
31
35
37
39
40
is continuous for each 0 ≤ t ≤ T , the stochastic process X is progressively measurable. Hence
the conclusion follows. 2
41
42
43
The next assumptions will be needed throughout this section:
45
46
18
38
(ξ, M) → X : Dom (ϕ) × C([0, t] ; Rd ) → C([0, t] ; Rd )
43
44
17
36
Since (ω, t) → Mt (ω) is progressively measurable and the mapping
40
42
15
34
(X· (ω) , K· (ω)) ∈ C(R+ ; Rd ) × C(R+ ; Rd ).
38
41
14
33
36
39
13
32
has a unique solution
34
37
12
28
Proof. Let ω be arbitrary but fixed. By Theorem 25, the Skorohod problem
31
35
10
16
30
33
9
Corollary 34. Let (, F, P, Ft , Bt )t≥0 be given. If
28
29
8
11
19
21
4
where ϕ is (ρ, γ )-semiconvex function and {Bt : t ≥ 0} is an Rk -valued Brownian motion
with
respect to a
stochastic basis (which is supposed to be complete and right-continuous)
, F, P, {Ft }t≥0 .
First we derive directly from Theorem 25 the existence result in the additive noise case.
18
20
3
44
45
(A1 ) (Carathéodory conditions.)The functions
F (·, ·, ·) : × R+ × Rd → Rd and G (·, ·, ·) :
d
d×k
d
× R+ × R → R
are P, R -Carathéodory functions, i.e.
46
47
JID:YJDEQ AID:7984 /FLA
F (·, ·, x) and G (·, ·, x) are p.m.s.p., ∀ x ∈ Rd ,
F (ω, t, ·) and G (ω, t, ·) are continuous function dP ⊗ dt-a.e.
1
2
3
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30
T
6
10
13
16
(i)
18
(ii)
19
21
24
11
12
13
x − y, F (t, x) − F (t, y) ≤ μ (t) |x − y|2 ,
|G(t, x) − G(t, y)| ≤ (t) |x − y|,
∀ x, y
∀ x, y ∈ Rd ,
∈ Rd .
29
30
0
31
(jv)
32
33
34
35
20
21
∀ 0 ≤ s ≤ t, ∀y : R+ → Rd continuous
t
t
t
y (r) − Xr , dKr + ϕ (Xr ) dr ≤ ϕ (y (r)) dr
s
s
t
38
which means that
41
42
27
28
29
30
31
(44)
32
33
34
36
2
s
39
24
35
s
|y (r) − Xr | (ρdr + γ d Kr ) ,
+
37
23
26
0
36
37
38
39
40
(X· (ω) , K· (ω)) = SP ∂ − ϕ; ξ (ω) , M· (ω) ,
41
P-a.s.,
42
43
where
44
t
45
47
18
25
Xt ∈ Dom (ϕ), ∀ t ≥ 0, ϕ (X· ) ∈ L1loc (R+ ) ,
(jj) K· ∈ BV loc R+ ; Rd , K0 = 0,
t
t
(jjj) Xt + Kt = ξ + F (s, Xs ) ds + G (s, Xs ) dBs , ∀t ≥ 0,
(j )
28
46
16
22
27
44
15
19
Definition 35. Let (, F, P, Ft , Bt )t≥0 be given. A pair (X, K) : × R+ → Rd × Rd of continuous Ft -p.m.c.s.p. is a strong solution of the stochastic Skorohod equation (40) if P-a.s.,
26
43
14
17
(43)
We define now the notions of strong and weak solutions for the stochastic Skorohod equation (40).
25
40
7
10
22
23
6
8
(A3 ) (Monotonicity and Lipschitz conditions.) There exist μ ∈ L1loc (R+ ) and ∈ L2loc (R+ ) with
≥ 0, such that dP ⊗ dt-a.e.
17
20
0
F # (t) := sup |F (t, x)| : x ∈ Dom (ϕ) ,
G# (t) := sup |G(t, x)| : x ∈ Dom (ϕ)
12
15
(42)
9
where
11
14
5
|G# (s) |2 ds < ∞, P -a.s.,
and
0
8
4
T
F # (s) ds < ∞
7
2
3
(A2 ) (Boundedness conditions.) For all T ≥ 0:
5
9
1
(41)
Mt =
t
F (s, Xs ) ds +
0
45
G (s, Xs ) dBs .
0
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2
3
31
Definition 36. Let F (ω, t, x) := f (t, x), G (ω, t, x) := g (t, x) and ξ (ω) := x0 (be independent of ω). If there exist a stochastic basis (, F, P, Ft )t≥0 , an Rk -valued Ft -Brownian motion
{Bt : t ≥ 0} and a pair (X· , K· ) : × R+ → Rd × Rd of p.m.c.s.p. such that
(X· (ω) , K· (ω)) = SP ∂ − ϕ; x0 , M· (ω) ,
4
5
t
Mt =
10
15
16
17
18
19
20
21
22
23
24
25
−
dKt ∈ ∂ ϕ (Xt ) (dt)
−
d K̂t ∈ ∂ ϕ(X̂t ) (dt) ,
by Lemma 17, for p ≥ 1 and λ > 0
Xt − X̂t , (F (t, Xt ) dt − dKt ) − (F (t, X̂t )dt − d K̂t )
1
+ mp + 9pλ |G (t, Xt ) − G(t, X̂t )|2 dt ≤ |Xt − X̂t |2 dVt ,
2
32
33
t
36
41
42
μ (s) ds +
Vt =
1
2
mp + 9pλ 2 (s) ds + 2ρds + γ d Ks + γ d K̂ s .
0
p E 1 ∧ e−V (X − X̂)T ≤ Cp,λ E 1 ∧ |ξ − ξ̂ |p ,
47
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
36
37
38
40
41
42
43
and the uniqueness follows.
2
45
46
14
39
Therefore, by Corollary 51 (from Appendix A), we get
43
44
13
35
39
40
12
34
where
35
38
and
10
11
Proof. Let (X, K) and (X̂, K̂) be two solutions corresponding to ξ and respectively ξ̂ . Since
31
37
0
Proposition 37 (Pathwise uniqueness). Let (, F, P, Ft , Bt )t≥0 be given and assumptions (9),
(10) and (16) be satisfied. The functions F and G are such that assumptions (A1 )–(A3 ) are
satisfied. Then the stochastic Skorohod equation (40) has at most one strong solution.
30
34
9
g (s, Xs ) dBs ,
Since the stochastic process K is uniquely determined from (X, B) through equation (44-jjj),
we can also say that X is a strong solution (and respectively (, F, P, Ft , Bt , Xt )t≥0 is a weak
solution).
We first give a uniqueness result for strong solutions.
27
29
f (s, Xs ) ds +
then the collection (, F, P, Ft , Bt , Xt , Kt )t≥0 is called a weak solution of the stochastic Skorohod equation (40).
26
28
8
t
0
11
14
5
7
9
13
3
6
where
8
12
2
4
P-a.s.,
6
7
1
44
45
Remark also that in the case of additive noise (i.e. G does not depend upon X) we have
existence of a strong solution.
46
47
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32
Lemma 38. Let (, F, P, Ft , Bt )t≥0 be given and assumptions (9), (10), (16) and (17) be satisfied. The function F is such that assumptions (A1 )–(A3 ) are satisfied.
If
4
8
6
and M is a p.m.c.s.p. with M0 = 0, then there exists a unique pair (X, K) of p.m.c.s.p., solution
of the problem
9
(X· (ω) , K· (ω)) = SP ∂ ϕ; ξ (ω) , M· (ω) ,
10
11
12
13
−
16
10
P-a.s.,
11
13
⎪
⎪
⎩
17
18
t
Xt (ω) + Kt (ω) = ξ (ω) +
14
F (ω, s, Xs (ω)) ds + Mt (ω) ,
15
t ≥ 0,
16
0
17
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt) ,
18
19
19
P-a.s.
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
21
Proof. Applying Corollary 34 to the approximating problem
22
⎧
t
⎪
⎪
⎨ X n (ω) + K n (ω) = ξ (ω) + F (ω, s, X n
t
t
s−1/n (ω))ds + Mt (ω) ,
⎪
⎪
0
⎩
dKtn (ω) ∈ ∂ − ϕ Xtn (ω) (dt) ,
23
24
t ≥ 0,
25
26
27
28
(X n , K n )
we conclude that there exists a unique solution
of p.m.c.s.p. The solution (X, K) is
obtained as the limit of the sequence (X n , K n ), exactly as in the proof of Proposition 30. 2
29
In order to study the general stochastic Skorohod equation (40) we shall consider only the case
when F , G and ξ are independent of ω and, to highlight this, the coefficients will be denoted by
f and g respectively.
Let us consider equation
32
36
37
38
39
40
41
44
47
31
33
34
35
37
t ≥ 0,
38
(45)
39
40
41
42
where f : R+ × Rd → Rd and g : R+ × Rd → Rd×k .
We recall the definition of f # , g # given as in (35).
45
46
30
36
⎧
t
t
⎪
⎪
⎨ X + K = x + f (s, X ) ds + g (s, X ) dB ,
t
t
0
s
s
s
⎪
0
0
⎪
⎩
dKt (ω) ∈ ∂ − ϕ (Xt (ω)) (dt) ,
42
43
8
12
⎧
⎪
⎪
⎨
15
7
9
i.e.
14
20
3
5
6
7
2
4
ξ ∈ L , F0 , P; Dom (ϕ)
0
5
1
43
44
45
Theorem 39. Let assumptions
(9), (10), (16) and (17) be satisfied. The functions f and g are
suppose to be B1 , Rd -Carathéodory functions satisfying moreover the boundedness conditions
46
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33
T |f # (s) |2 + |g # (s) |4 ds < ∞, ∀T ≥ 0.
1
2
3
1
2
3
0
4
5
4
If x0 ∈ Dom (ϕ) then equation (45) has a weak solution (, F, P, Ft , Xt , Kt , Bt )t≥0 .
5
6
7
8
9
10
6
Remark 40. Usually, when G is Lipschitz, a fixed point argument is used (based on Banach
contraction theorem). But, in our case this argument doesn’t work even for the drift part F ≡ 0,
as it can be seen from inequality (22), we have different order of the estimates in the left and in
the right side of the inequality.
11
12
7
8
9
10
11
Proof.
12
13
13
14
Step 1. Approximating sequence.
14
15
Let , F, P, FtB , Bt t≥0 be a stochastic basis. Applying Lemma 38, we deduce that there
exists a unique pair (X n , K n ) : × R+ → Rd × Rd of FtB -p.m.c.s.p. such that
15
16
17
18
⎧
⎪
⎪
⎨
19
20
21
⎪
⎪
⎩
22
23
t
Xtn
+ Ktn
= x0 +
t
n
f (s, Xs−1/n
)ds
+
0
dKtn (ω) ∈ ∂ − ϕ Xtn (ω) (dt) .
(46)
0
22
23
25
29
35
36
37
38
41
42
0
0
27
28
29
$
E
n
sup Mt+θ
0≤θ≤ε
31
⎡⎛
⎤
⎞4
%
t+ε
2⎥
4
2
t+ε ⎢
− Mtn ≤ C ⎣⎝ f # (s) ds ⎠ + t g # (s) ds ⎦
⎡
⎢
≤ εC ⎣supt∈[0,T ]
t
32
33
34
35
⎤
⎞2
⎛ t+ε
# 2
⎝ f (s) ds ⎠ + sup
36
t+ε # 4 ⎥
g (s) ds ⎦
37
t∈[0,T ] t
38
t
39
d
using Proposition 46, we deduce that the family of laws of {M n : n ≥ 1} is tight on C R+ ; R .
Therefore by Theorem 45 for all T ≥ 0
%
$
43
44
lim
45
N∞
46
47
26
30
39
40
t n
n
f s, Xs−1/n ds + g s, Xs−1/n
dBs .
Since
32
34
t
Mtn :=
28
33
21
24
27
31
18
20
t ≥ 0,
Denote
26
30
17
19
n
g(s, Xs−1/n
)dBs ,
24
25
16
sup P M n T ≥ N = 0,
n≥1
40
41
42
43
44
45
46
and, for all a > 0 and T > 0,
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34
$
1
%
lim sup P {mM n (ε) ≥ a}
2
ε0
= 0.
(47)
n≥1
3
4
4
Recalling the definition
5
μ M n = ε + mM n (ε) ,
6
6
7
7
where m is the modulus of continuity, we see that we can replace in (47) mM n by μ M n .
8
9
10
9
10
Step 2. Tightness.
11
12
13
14
11
Let T ≥ 0 be arbitrary. We now show that
the family of laws of the random variables
is tight on C [0, T ] ; R2d+1 .
From (21-c) we deduce that
(X n , K n , K n )
15
17
19
where G : C [0, T ] ; R
22
23
26
27
28
32
33
34
43
44
45
22
23
24
25
26
27
28
29
30
35
L(X̄, K̄, V̄ , B̄) = L(X, K, V , B)
32
33
34
36
37
38
39
and
40
P-a.s.
(X̄ n , K̄ n , V̄ n , B̄ n ) −−−→ (X̄, K̄, V̄ , B̄), as n → ∞, in C([0, T ] ; R2d+1+k ).
41
n
n
n n From Proposition 49 we deduce that B̄ n , {FtX̄ ,K̄ ,V̄ ,B̄ } , n ≥ 1, and B̄, {FtX̄,K̄,V̄ ,B̄ } are
Brownian motions.
43
46
47
21
, ,
L(X̄ n , K̄ n , V̄ n , B̄ n ) = L X n , K n , -K n - , B n
38
42
in law, as n → ∞
20
31
37
41
on C [0, T ] ; R2d+1+k and applying the Skorohod theorem, we can choose a probability space
(, F, P) and some random quadruples (X̄n , K̄ n , V̄ n , B̄ n ), (X̄, K̄, V̄ , B̄) defined on (, F, P)
such that
36
40
19
, , X n , K n , -K n - , B → (X, K, V , B)
35
39
18
From (20) we see that G is bounded on compact subset of C [0, T ] ; Rd and therefore by Propo
sition 47, {U n ; n ∈ N∗ } is tight on C [0, T ] ; Rd .
Using the Prohorov theorem we see that there exists a subsequence (still denoted with n) such
that
30
14
17
→ R+ and
13
16
29
31
d
12
15
G (x) := CT ,x = exp C (1 + T + xT + Bx ) ,
2 −C(1+T +xT )
μ−1
Bx := 1/μ
e
δ
.
x
21
25
=
n
20
24
Un
mU n (ε) ≤ G M
μ M n (ε), a.s.,
16
18
2
3
5
8
1
42
44
45
46
Step 3. Passing to the limit.
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35
Since we also have (X n , K n , K n , B) → (X̄, K̄, V̄ , B̄), in law, then by Corollary 43 we
deduce that for all 0 ≤ s ≤ t , P-a.s.
3
5
6
Moreover, since for all 0 ≤ s < t,
4
(48)
7
8
t
9
10
11
ϕ Xrn dr ≤
s
t
t
t
13
+
14
y (r) − Xrn , dKrn
ϕ (y (r)) dr −
s
12
t
19
20
ϕ X̄r dr ≤
t
y (r) − X̄r , d K̄r
ϕ (y (r)) dr −
s
+
26
d K̄r ∈ ∂ − ϕ X̄r (dr) .
28
30
t
St (Y, B) := x0 +
t
31
g (s, Ys ) dBs , t ≥ 0.
f (s, Ys ) ds +
0
34
0
n
n
n
n
n
n
n
Since for every t ≥ 0,
36
37
38
39
40
Xtn + Ktn − St (X n , B n ) = 0, a.s.,
42
33
35
, ,
L(X̄ , K̄ , V̄ , B̄ , St (X̄ , B̄ )) = L X , K , -K n - , B n , St (X n , B n )
n
32
34
By Proposition 48 it follows
41
41
42
43
then by Corollary 43 we have
44
44
X̄tn + K̄tn − St (X̄ n , B̄ n ) = 0, a.s.,
46
47
27
29
Let
33
45
23
25
32
43
22
(49)
24
Hence, based on (48) and (52) and Lemma 16 we have
31
40
20
21
y (r) − X̄r 2 ρdr + γ d V̄r .
s
28
39
19
s
t
27
38
18
25
37
14
17
t
24
36
13
16
23
35
12
15
22
30
11
s
, , y (r) − X n 2 ρdr + γ d -K n - , a.s.,
r
r
s
21
29
10
then by Corollary 44 we infer that
18
26
9
s
15
17
5
6
n ∈ N∗
8
16
2
3
X̄0 = x0 , K̄0 = 0, X̄t ∈ Dom (ϕ),
, ,
, ,
-K̄ - − -K̄ - ≤ V̄t − V̄s and 0 = V̄0 ≤ V̄s ≤ V̄t
t
s
4
7
1
45
46
and consequently, letting n → ∞,
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36
X̄t + K̄t − St (X̄, B̄) = 0, a.s.
1
1
2
3
2
Hence we obtain that, P-a.s.,
3
4
4
t
5
X̄t + K̄t = x0 +
6
7
8
9
10
f s, X̄s ds +
0
t
g s, X̄s d B̄s , ∀ t ∈ [0, T ] ,
8
t≥0
is a weak solution.
2
11
12
13
14
15
18
19
9
10
11
Since the stochastic process K is uniquely determined by (X, B) via equation (45), then a
weak solution for the stochastic differential equation is a sextuplet (, F, P, {Ft }t≥0 , X, B). We
know that weak existence and pathwise uniqueness implies strong existence (see Theorem 3.55
in [31] or Theorem 1.1 in [23]). Hence we deduce from Theorem 39 and Proposition 37:
16
17
6
7
0
¯ F̄, P̄, FtB̄,X̄ , X̄t , K̄t , B̄t
and consequently ,
5
12
13
14
15
16
Theorem 41. Let assumptions
(9), (10), (16) and (17) be satisfied. The functions f and g are
suppose to be B1 , Rd -Carathéodory functions satisfying moreover assumption (A3 ) and boundedness conditions
20
17
18
19
20
22
T |f # (s) |2 + |g # (s) |4 ds < ∞, ∀T ≥ 0.
23
0
23
21
24
25
If x0 ∈ Dom (ϕ) then equation (45) has a unique strong solution (Xt , Kt )t≥0 .
26
27
30
37
40
33
A.1. Applications of Fatou’s Lemma
34
35
The next result is a well known consequence of weak convergence of probability measures
(for its proof see, e.g. [31, Proposition 1.22]).
law
Xn −−→ X, as n → ∞,
47
39
40
42
43
and there exists a continuous function α : X → R such that
45
46
37
41
(i)
43
44
36
38
Proposition 42. Let (X, ρ) be a separable metric space. Let ϕ : X → (−∞, ∞] be a lower
semicontinuous function. If X and Xn are X-valued random variables, for n ∈ N∗ , such that
41
42
30
32
38
39
29
31
Appendix A
35
36
25
28
The work of A.R. and R.B. was supported by grant “Deterministic and stochastic systems with
state constraints”, code 241/05.10.2011 and the work of L.M. by grant POSDRU/89/1.5/S/49944.
33
34
24
27
31
32
22
26
Acknowledgments
28
29
21
44
45
(ii) α (x) ≤ ϕ (x) , ∀ x ∈ X,
(iii) {α (Xn ) : n ∈ N∗ } is a uniformly integrable family,
46
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37
then the expectations Eϕ (X) and Eϕ (Xn ) exist for all n ∈ N, and
2
1
2
−∞ < Eϕ (X) ≤ lim inf Eϕ (Xn ) .
3
3
n→+∞
4
5
6
7
8
9
4
For 0 ≤ s < t ≤ T , we denote by X[s,t] (similar to (7)) the total variation of X· on [s, t],
that is
n−1
Xt (ω) − Xt (ω) : n ∈ N∗ , s = t0 < t1 < · · · < tn = t
X[s,t] (ω) = sup
i
i+1
i=0
10
11
12
We also use XT := X[0,T ] .
Applying successively Proposition 42 for
16
17
14
15
16
17
18
where s = t0 < t1 < . . . < tN = t is an arbitrary partition of [s.t], and respectively
20
19
20
ϕ (x) = (x (s) − x (t))+ ,
21
21
22
26
27
22
we get the next result:
23
d
Corollary 43. Let s, t be arbitrary fixed such that 0 ≤ s ≤ t ≤ T . If g : C [0, T ] ; R → R+
n
n
∗
is a continuous
function and X, V , X , V , n ∈ N , are random variables with values in
d
C [0, T ] ; R , such that
law
X n , V n −−→ (X, V ) , as n → ∞,
28
29
34
35
32
(a) If Xtn ∈ F a.s., then Xt ∈ F , a.s., whenever F is closed subset of Rd ;
(b) If X n [s,t] ≤ g (V n ) a.s., then X[s,t] ≤ g (V ), a.s.;
(c) If d = 1 and Xsn ≤ Xtn a.s., then Xs ≤ Xt , a.s.
44
45
46
47
34
35
37
N : s = r0 < r1 < . . . < rN = t, ri+1 − ri =
40
43
33
36
Now let us consider the partition
39
42
27
31
38
41
26
30
then the following implications hold true:
36
37
25
29
32
33
24
28
30
31
9
12
i=0
18
25
8
13
15
24
7
11
ϕ (x) = dF (x (t)) ,
/+
.N−1
|x (ti+1 ) − x (ti )| − g (y) ,
ϕ (x, y) =
14
23
6
10
13
19
5
and the function g : C
[0, T ] ; Rd
0
N−1
t −s
N
→ [0, 1], defined by
+
i=0
39
40
41
g (x, k, v) :=
∧1
i=0 |k (ri+1 ) − k (ri )| − v (t) + v (s)
⎡ t
⎤+
N−1
x (ri ) , k (ri+1 ) − k (ri ) − mx (1/N ) (v (t) − v (s))⎦ ∧ 1.
+ ⎣ ϕ (x (r)) dr −
s
38
42
43
44
45
46
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38
Applying again the generalization of the Fatou’s Lemma (Proposition 42), it can be proved:
2
3
4
Corollary 44. Let (X, K, V ),
random variables, such that
(X n , K n , V n ),
6
n
n
X ,K ,V
n
7
9
and for all 0 ≤ s < t, and
× C ([0, T ] ; R)-valued
5
6
n→∞
7
−−−−→ (X, K, V )
8
9
, n, , n,
-K - − -K - ≤ V n − V n a.s.
t
s
t
s
11
If ϕ
: Rd
10
11
→ (−∞, +∞] is a lower semicontinuous function and
13
t
14
15
16
ϕ Xrn dr ≤
s
t
12
13
Xrn , dKrn , a.s. for all n ∈ N∗ ,
18
19
Kt − Ks ≤ Vt − Vs , a.s.
20
20
21
and
22
t
23
25
22
t
s
23
Xr , dKr , a.s.
ϕ (Xr ) dr ≤
24
24
25
s
26
27
28
29
30
31
32
33
34
35
36
37
38
26
A.2. Complements on tightness
Theorem 45. The family {X n : n ∈ N∗ } is tight in C(R+ ; Rd ) if and only if, for every T ≥ 0,
%
$
(j ) lim sup P(n) X n ≥ N = 0,
N∞
$
40
42
43
44
45
46
47
27
If Xtn : t ≥ 0 , n ∈ N∗ , is a family of continuous stochastic processes then the following result
is a consequence of the Arzelà–Ascoli theorem (see, e.g., Theorem 7.3 in [6]).
We recall the notations:
X n T := sup Xtn : t ∈ [0, T ] ,
mXn (ε; [0, T ]) := sup Xtn − Xsn : t, s ∈ [0, T ] , |t − s| ≤ ε .
39
41
15
17
then
19
21
14
16
s
17
18
3
law
n ∈ N∗ ,
10
12
2
4
5
8
n ∈ N, be C
2
[0, T ] ; Rd
1
(jj)
0
n≥1
ε0
n≥1
N∞
30
31
32
33
34
35
36
37
38
40
∀a > 0.
41
42
Moreover, tightness yields that for all T > 0
$
lim
29
39
%
lim sup P(n) (mXn (ε; [0, T ]) ≥ a) = 0,
28
43
%
sup P(n) X n T ≥ N = 0.
n≥1
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Without using the above theorem, it can be proved the following criterion for tightness which
is well adapted to our needs. The proof can be found in E. Pardoux and A. Răşcanu [31] (Proposition 1.53) and we will give the sketch of the proof.
1
Proposition 46. Let Xtn : t ≥ 0 , n ∈ N∗ , be a family of Rd -valued continuous stochastic processes defined on probability space (, F, P). Suppose that for every T ≥ 0, there exist α > 0
and b ∈ C (R+ ) with b(0) = 0, such that
4
8
(j )
11
12
13
(jj)
lim sup P({X0n ≥ N }) = 0,
N→∞ n∈N∗
$
%
n
α
E 1 ∧ sup Xt+s − Xtn ≤ ε · b(ε), ∀ε > 0, n ≥ 1, t ∈ [0, T ] .
0≤s≤ε
14
15
16
17
18
Then the family
: n ∈ N∗ }
is tight in C(R+
25
26
27
1
36
sup
39
40
(i − 1)T
.
and ti =
Nk
45
46
47
22
23
24
26
27
sup |z (ti + s) − z (ti )| ≤ γk , ∀k ∈ N∗
28
29
30
31
is compact in C([0, T ]; Rd ).
From Markov’s inequality and (jj)
32
33
34
P Xn ∈
/ Kε
Nk
≤ P({X0n > M}) +
P({ sup Xtni +s − Xtni > γk })
k∈N∗ i=1
0≤s≤εk
Nk
Nk × εk × b(εk )
ε
εk × b(εk ) ε
< +
=
≤ ε.
+
2
γkα
2
γkα
∗
∗
k∈N i=1
k∈N
35
36
37
38
39
40
41
The proof is complete now.
2
43
44
13
25
41
42
12
21
2
1≤i≤Nk 0<s≤εk
37
38
11
18
34
35
10
17
Kε = z ∈ C([0, T ]; Rd ) : |z (0)| ≤ M,
30
33
9
20
T
Let γk = (k−1)/α and εk 0 be such that b(εk ) ≤ k . Let Nk =
4 T
εk
2
Applying Theorem Arzelá–Ascoli we see that the set
29
32
7
19
1
ε
28
31
6
16
ε
sup P({X0n ≥ M}) < .
2
n∈N∗
21
24
5
15
; Rd ).
Proof. We fix ε, T > 0. From (j ), there exists M = Mε ≥ 1 such that
20
23
3
14
{X n
19
22
2
8
2
1
9
10
39
42
43
Proposition
47. Let g : R+ → R+ be a continuous function satisfying g(0) = 0
and G :
C R+ ; Rd → R+ be a mapping which is bounded on compact subsets of C R+ ; Rd . Let X n ,
Y n , n ∈ N∗ , be random variables with values in C R+ ; Rd . If {Y n : n ∈ N∗ } is tight and for all
n ∈ N∗
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45
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(i) X0n ≤ G (Y n ) , a.s.
(ii) mXn (ε; [0, T ]) ≤ G (Y n ) g (mY n (ε; [0, T ])) , a.s., ∀ ε, T > 0,
3
4
5
6
7
4
Proof. Let δ > 0 be arbitrary. Then there exists a compact set Kδ ⊂ C [0, ∞[ ; Rd such that for
all n ∈ N∗
5
P Yn ∈
/ Kδ < δ .
8
9
Define Nδ = sup G (x). Then
12
x∈Kδ
11
12
P X n > Nδ < δ.
13
14
13
14
0
15
15
Let a > 0 be arbitrary. There exists ε0 > 0 such that
17
19
23
28
29
30
31
32
33
34
35
36
19
21
P (mXn (ε; [0, T ]) ≥ a) ≤ P g (mY n (ε; [0, T ])) ≥
a
Nδ ,
Y n ∈ Kδ
/ Kδ ) ≤ δ
+ P (Y n ∈
and the result follows.
41
42
43
44
47
24
25
28
A.3. Itô’s stochastic integral
29
In this subsection we consider {Bt : t ≥ 0} to be a k-dimensional Brownian
motion on
a
stochastic basis (which is supposed to be complete and right-continuous) , F, P, {Ft }t≥0 .
Let Sd [0, T ] be the space of p.m.c.s.p. X : × [0, T ] → Rd and d (0, T ) the space of
p.m.c.s.p. X : × [0, T ] → Rd such that
30
31
32
33
34
35
T
|Xt |2 dt < ∞, P-a.s.
0
36
37
38
39
Write Sd (and d ) for space of p.m.c.s.p. X : × [0, T ] → Rd such that the restriction of X to
[0, T ] belongs to Sd (respectively to d ).
If X ∈ Sd×k and B is an Rk -Brownian motion, then the stochastic process {(Xt , Bt ) : t ≥ 0}
can be seen as a random variable with values in the space C(R+ , Rd×k ) × C(R+ , Rk ). The law
of this random variable will be denoted L (X, B).
45
46
23
27
39
40
22
26
2
37
38
18
20
25
27
17
Consequently for all n ∈ N∗ ,
24
26
16
a
sup g (mx (ε; [0, T ])) <
, ∀ 0 < ε < ε0 .
N
δ
x∈Kδ
18
22
7
10
11
21
6
9
10
20
2
3
then {X n : n ∈ N∗ } is tight.
8
16
1
40
41
42
43
44
45
Proposition 48. (See Corollary 2.13 in [31].) Let X, X̂ ∈ Sd [0, T ], B, B̂ be two Rk -Brownian
motions and g : R+ × Rd → Rd×k be a function such that
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g (·, y) is measurable ∀ y ∈ Rd ,
y → g (t, y) is continuous dt-a.e.
1
2
1
2
3
4
3
If
4
5
5
L (X, B) = L(X̂, B̂) on C(R+ , Rd+k ),
6
6
7
8
7
then
8
9
L X, B,
10
11
12
·
g (s, Xs ) dBs = L X̂, B̂,
0
·
9
g(s, X̂s )d B̂s on C(R+ , Rd+k+d ).
10
11
12
0
13
14
13
We present now a continuity property of the mapping
14
15
15
T
16
(X, B) −→
17
18
16
Xs dBs .
17
18
0
19
20
21
22
19
−→ Rk
FtB,X
−→ Rd×k
Given B : × R+
and X : × R+
be two stochastic processes, let
be
the natural filtration generated jointly by B and X.
For the proof of the next proposition see Proposition 2.4 in [9] or Proposition 2.14 in [31].
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24
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28
29
30
33
Proposition 49. Let
: × R+
stochastic processes such that
→ Rk
and
X, X n , X̃ n
: × R+
→ Rd×k
be continuous
29
30
31
n
n Then B n , {FtB ,X } , n ≥ 1, and B, {FtB,X } are Brownian motions and as n → ∞
t
t
sup Xsn dBsn −→ Xs dBs −→ 0
t∈[0,T ] 37
0
32
33
34
35
in probability.
(50)
36
37
0
38
38
A.4. A forward stochastic inequality
39
40
40
Let X, X̂ ∈ Sd be two semimartingales defined by
41
42
42
t
43
Xt = X0 + Kt +
44
45
t
Gs dBs , t ≥ 0,
0
46
47
25
28
t∈[0,T ]
36
24
27
-Brownian motion ∀ n ≥ 1;
(i) B̃ n is Ft
(ii) L(B̃ n , X̃ n ) = L (B n , X n ) on C(R+ , Rk × Rd×k ), for all n ≥ 1;
2
T (iii) 0 Xsn − Xs ds + sup Btn − Bt → 0, in probability, as n → ∞, for all T > 0.
35
41
22
26
B̃ n ,X̃ n
34
39
21
23
B, B n , B̃ n
31
32
20
X̂t = X̂0 + K̂t +
43
Ĝs dBs , t ≥ 0,
0
(51)
44
45
46
where
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♦ K, K̂ ∈ Sd ;
♦ K· (ω), K̂· (ω) ∈ BV loc R+ ; Rd , K0 (ω) = K̂0 (ω) = 0, P-a.s.;
♦ G, Ĝ ∈ d×k .
1
2
3
4
5
6
7
4
Assume that there exist p ≥ 1 and λ ≥ 0 and V a bounded variation p.m.c.s.p., with V0 = 0,
such that, as measures on R+ ,
8
Xt − X̂t , dKt − d K̂t +
9
1
2
mp + 9pλ |Gt − Ĝt |2 dt ≤ |Xt − X̂t |2 dVt .
12
15
16
e−pVt |X
|p
t − X̂t
EFt p/2 ≤ Cp,λ p/2 ,
1 + δe−2Vt |Xt − X̂t |2
1 + δ||e−V (X − X̂)||2[t,s]
In particular for δ = 0
25
E ||e
−V
p
(X − X̂)||[t,s]
≤ Cp,λ e
−pVt
|Xt − X̂t | ,
p
P-a.s.,
24
25
26
∀r ≥ 0,
29
32
33
p
EFt 1 ∧ ||e−V (X − X̂)||[t,s] ≤ Cp,λ 1 ∧ |e−Vt (Xt − X̂t )|p ,
36
39
42
43
44
45
46
47
31
32
33
34
35
36
for all 0 ≤ t ≤ s.
37
References
39
40
41
28
30
Corollary 51. If assumption (52) is satisfied with λ > 1 and p ≥ 1, then there exists a positive
constant Cp,λ depending only on (p, λ) such that P-a.s.
35
38
27
29
we obtain:
34
37
21
23
1
r
(1 ∧ r) ≤ 1/2 ≤ 1 ∧ r,
2
1 + r2
28
20
22
As a consequence of the above theorem, since
27
31
16
18
for all 0 ≤ t ≤ s.
26
30
15
19
Ft
21
24
12
17
20
23
11
14
P-a.s.
19
22
9
13
p
||e−V (X − X̂)||[t,s]
17
18
7
10
Theorem 50. (See Corollary 6.74 in [31].) Let p ≥ 1. If the assumption (52) is satisfied with
λ > 1, then there exists a positive constant Cp,λ such that for all δ ≥ 0, 0 ≤ t ≤ s:
13
14
6
8
(52)
10
11
5
38
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