Invariance for Rough Differential Equations

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Invariance for Rough Differential Equations

Laure Coutin, Nicolas Marie

To cite this version:

Laure Coutin, Nicolas Marie. Invariance for Rough Differential Equations. Stochastic Processes and

their Applications, Elsevier, 2016, 127 (7), pp.2373-2395. �10.1016/j.spa.2016.11.002�. �hal-01519399�

INVARIANCE FOR ROUGH DIFFERENTIAL EQUATIONS

LAURE COUTIN* AND NICOLAS MARIE**

Abstract. In 1990, in Itô’s stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset C ofR^d(d∈N^∗) for stochastic differential equa- tions (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato’s results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed.

Contents

1. Introduction 1

2. Preliminaries 4

3. An invariance theorem for rough differential equations 6

3.1. Sufficient condition of invariance 8

3.2. Necessary condition of invariance : compact case 10 3.3. Necessary condition of invariance : convex case 13 4. A comparison theorem for rough differential equations 17 5. Invariance for differential equations driven by a fractional Brownian

motion 18

5.1. Fractional Brownian motion 18

5.2. A logistic equation driven by a fractional Brownian motion 20

Appendix A. Tangent and normal cones 20

References 21

MSC2010 : 60H10

1. Introduction

The invariance of a nonempty closed convex subset of R^d (d∈ N^∗) for a (ordi- nary) differential equation was solved by Nagumo in [26], see also [2] for a simple proof. It was obtained by Aubin and Da Prato in [5] for stochastic differential equations. More explicit results in the special case of polyhedrons have been estab- lished in Milian [25]. In [10], Cresson, Puig and Sonner have introduced a stochastic generalization of the well-known Hodgkin-Huxley neuron model satisfying the as- sumptions of the stochastic viability theorem of Milian [25]. On the viability and the invariance of sets for stochastic differential equations, see also Milian [24], Gau- tier and Thibault [15], and Michta [23].

In [6], the results of [5] were extended by Aubin and Da Prato to the stochas- tic differential inclusions. The case of stochastic controlled differential equations was studied by Da Prato and Frankowska in [12] or more recently by Buckdahn,

Key words and phrases. Viability theorem ; Comparison theorem ; Rough differential equations

; Fractional Brownian motion ; Logistic equation.

1

Quicampoix, Rainer and Teichmann in [8]. An unified approach which provides a viability theorem for stochastic differential equations, backward stochastic differ- ential equations and partial differential equations is developed in Buckdahn et al.

[7].

The invariance of a subset of R^d for a stochastic differential equation driven by a α-Hölder continuous process withα∈(1/2,1)has been already studied by several authors in the fractional calculus framework developed by Nualart and Rascanu in [29]. In Ciotir and Rascanu [9] and Nie and Rascanu [27], the authors have proved a sufficient and necessary condition for the invariance of a closed subset ofR^d for a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H ∈(1/2,1). In [22], Melnikov, Mishura and Shevchenko have proved a sufficient condition for the invariance of a smooth and nonempty subset of R^d for a stochastic differential equation driven by a mixed process containing both a Brownian motion and aα-Hölder continuous process withα∈(1/2,1).

The rough paths theory introduced by T. Lyons in 1998 in the seminal paper [20]

provides a natural and powerful framework to study differential equations driven by α-Hölder signals with α ∈ (0,1]. The theory and its applications are widely studied by many authors. For instance, see the book of Friz and Victoir [14], the nice introduction of Friz and Hairer [13], or the approach of Gubinelli [16].

The main purpose of this article is to extend the viability theorem of Aubin and Da Prato [5] and to provide a comparison theorem for the rough differential equations.

The paper deals also with an application of the viability theorem to stochastic differential equations driven by a fractional Brownian motion of Hurst parameter greater than1/4.

LetT >0be arbitrarily chosen, and consider the differential equation

(1) yt=y0+

Z t 0

b(ys)ds+ Z t

0

σ(ys)dws;t∈[0, T]

where,y0∈R^d,b(resp. σ) is a continuous map fromR^dinto itself (resp. Md,e(R)), and w: [0, T]→R^e is aα-Hölder continuous signal withe∈N^∗and α∈(0,1].

At Section 2, some definitions and results on rough differential equations are stated in order to take Equation (1) in that sense. Section 3 deals with a viability theorem for Equation (1) taken in the sense of rough paths (see Friz and Victoir [14]) and a convex or compact set. At Section 4, a comparison theorem for the rough dif- ferential equations is proved by using the viability results of Section 3. At Section 5, the viability theorem is applied to stochastic differential equations driven by a fractional Brownian motion of Hurst parameter greater than 1/4. Finally, Appen- dix A is a brief survey on convex analysis.

For the sake of readability, all results are proved on [0, T], but they can be ex- tended onR+ via some usual localization arguments.

The results established in this paper could be applied in stochastic analysis it- self, and in other sciences as neurology. On the one hand, in stochastic analysis, one could study the viability of rough differential inclusions as in Aubin and Da Prato [6] in Itô’s calculus framework, or could also compare the viability condition for rough differential equations to the reflecting boundary conditions for Ito’s sto- chastic differential equations (see Lions and Sznitman [19]). On the other hand,

together with J.M. Guglielmi who is neurologist at the American Hospital of Paris, we are studying a fractional Hodgkin-Huxley neuron model, that extends the model of Cresson et al. [10], in order to model injured nerves membrane potential in some neuropathies.

The following notations are used throughout the paper.

Notations (general) :

• The Euclidean scalar product onR^d is denoted byh., .i, and the Euclidean norm on R^d is denoted by k.k. The canonical basis of R^d is denoted by (ek)_k∈

J1,dK. For everyx∈R^d, itsj-th coordinate with respect to(ek)_k∈

J1,dK

is denoted by x^(j) for everyj∈J1, dK.

• For every x0∈R^d andr∈R+,Bd(x0, r) :={x∈R^d :kx−x0k6r}.

• The interior, the closure and the frontier of a set S ⊂R^d are respectively denoted by int(S),S and∂S.

• For every k∈J1, dK,Dk:={x∈R^d:x^(k)>0}.

• For a nonempty closed set S ⊂R^d, and everyx∈R^d, ΠK(x)denotes the set of best approximations ofxby the elements ofK :

(2) Π_S(x) :=

x^∗∈S:kx−x^∗k= inf

y∈Skx−zk

.

• The distance betweenx∈R^mand a nonempty closed setS⊂R^d is d_S(x) := inf

y∈Skx−yk.

• The space of the matrices of size d×e is denoted byMd,e(R). The (eu- clidean) matrix norm on Md,e(R) is denoted by k.k_Md,e(R). If d = e, then Md(R) := Md,e(R). The canonical basis of Md,e(R) is denoted by (ek,l)_(k,l)∈

J1,dK×J1,eK.

• Let E and F be two vector spaces. The space of the linear maps from E into F is denoted by L(E, F). IfE =F, thenL(E) :=L(E, F).

• The space of the continuous functions from [0, T] into R^d is denoted by C⁰([0, T],R^d)and equipped with the uniform normk.k∞,T.

• The space of the continuous functionslfrom(0, t₀)into]0,∞[witht₀>0, and such that

lim

t→0⁺

t^β

l(t) = 0 ;∀β >0, is denoted by St₀.

Notations (rough paths). See Friz and Victoir [14], Chapters 5, 7, 8 and 9 :

• ∆T :={(s, t)∈R²+: 06s < t6T}.

• The space of the α-Hölder continuous functions from [0, T] into R^d is de- noted by C^α-Höl([0, T],R^d) and equipped with the α-Hölder semi-norm k.kα-Höl,T :

kxkα-Höl,T := sup

(s,t)∈∆T

kxt−xsk

|t−s|^α ;∀x∈C^α-Höl([0, T],R^d).

• The step-N signature of x∈C^1-Höl([0, T],R^d)withN ∈N^∗ is denoted by SN(x):

SN(x)t:=

1,

Z

0<u<t

dxu, . . . , Z

0<u1<···<uN<t

dxu₁⊗ · · · ⊗dxu_N

;∀t∈[0, T].

• The step-N free nilpotent group overR^d is denoted byG^N(R^d): G^N(R^d) :={SN(γ)1 ;γ∈C^1-Höl([0,1],R^d)}.

• The space of the geometric α-rough paths from [0, T] into G^[1/α](R^d) is denoted by GΩα,T(R^d):

GΩα,T(R^d) :={S_[1/α](x) ;x∈C^1-Höl([0, T],R^d)}^dα-Höl,T

where, dα-Höl,T is theα-Hölder distance for the Carnot-Carathéodory met- ric.

2. Preliminaries

The purpose of this section is to provide the appropriate formulation of Equa- tion (1) in the rough paths framework. At the end of the section, a convergence result for the Euler scheme associated to Equation (1) is stated, and a definition of invariant sets for rough differential equations is provided.

The definitions and propositions stated in the major part of this section come from Lyons and Qian [21], Friz and Victoir [14], or Friz and Hairer [13].

First, the signal w is α-Hölder continuous with α ∈ (0,1]. In addition, w has to satisfy the following assumption.

Assumption 2.1. There existsw∈GΩα,T(R^e)such that w⁽¹⁾ =w.

LetW : [0, T]→R^e+1 be the signal defined by : Wt:=te1+

e+1

X

k=2

w_t^(k−1)ek ;∀t∈[0, T].

By Friz and Victoir [14], Theorem 9.26, there exists at least oneW∈GΩα,T(R^e+1) such that W⁽¹⁾ =W.

Let us state the conditions the collection of vector fields of a rough differential equation has to satisfy in order to get at least the existence of solutions.

Notation. For everyγ >0,bγcis the largest integer strictly smaller thanγ.

Definition 2.2. Consider γ >0,l, m∈N^∗ and a nonempty closed setV ⊂R^l. A map f : R^l → M_l,m(R) is γ-Lipschitz continuous (in the sense of Stein) from V intoM_l,m(R)if and only if :

(1) f|_V ∈C^bγc(V,M_l,m(R)).

(2) f, Df, . . . , D^bγcf are bounded on V.

(3) D^bγcf is(γ−bγc)-Hölder continuous fromR^lintoL(V^⊗bγc,Ml,m(R))(i.e.

there exists C >0 such that for everyx, y∈V,

kD^bγcf(y)−D^bγcf(x)k_L(V⊗bγc,Ml,m(R))6Cky−xk^γ−bγc).

The set of all such maps is denoted by Lip^γ(V,Ml,m(R)).

The map f is locally γ-Lipschitz continuous from R^l into M_l,m(R), if for every nonempty compact setK⊂R^l,f isγ-Lipschitz continuous fromK intoMl,m(R).

The set of all such maps is denoted by Lip^γ_loc(R^l,Ml,m(R)).

In the sequel, bandσsatisfy the following assumption.

Assumption 2.3. There existsγ∈(1/α,[1/α] + 1)such that : (1) b∈Lip^γ−1_loc (R^d) andσ∈Lip^γ−1_loc (R^d,Md,e(R)).

(2) b (resp. σ) is Lipschitz continuous fromR^d into itself (resp. Md,e(R)).

(3) D^[1/α]b (resp. D^[1/α]σ) is (γ −[1/α])-Hölder continuous from R^d into L((R^d)^⊗[1/α],R^d)(resp. L((R^d)^⊗[1/α],Md,e(R))).

Letfb,σ:R^d→ Md,e+1(R)be the map defined by : f_b,σ(x) :=

d

X

k=1

b^(k)(x)e_k,1+

e+1

X

l=2 d

X

k=1

σ_k,l(x)e_k,l ;∀x∈R^d.

In the rough paths framework, dy=fb,σ(y)dWwithy0∈R^das initial condition is the appropriate formulation of Equation (1).

Under the Assumptions 2.1 and 2.3, by Friz and Victoir Theorem 10.26, Exer- cice 10.55 and Exercice 10.56, the rough differential equation dy=f_b,σ(y)dWwith y0 ∈ R^d as initial condition has at least one solution y on [0, T]. Precisely, there exists a sequence(Wⁿ)n∈N of elements ofC^1-Höl([0, T],R^e+1)such that

(3) lim

n→∞dα-Höl,T(S_[1/α](Wⁿ),W) = 0 and

(4) lim

n→∞ky−yⁿk_∞,T = 0

where, for every n ∈ N, yⁿ is the solution on [0, T] of the ordinary differential equationdyⁿ=fb,σ(yⁿ)dWⁿ withy0 as initial condition.

Moreover, if b and σ satisfy the following assumption, the solution of Equation (1) is unique and denoted byπ_fb,σ(0, y₀,W).

Assumption 2.4. b∈Lip^γ_loc(R^d)andσ∈Lip^γ_loc(R^d,Md,e(R)).

Let us now define the Euler scheme for Equation (1) and state a convergence result.

Let D := (t0, . . . , tn) be a dissection of [0, T] with n ∈ N^∗. The Euler scheme ybⁿ := (ybⁿ_t0, . . . ,yb_tⁿ_n)for Equation (1) along the dissectionD is defined by

ybⁿ_t

k :=E^Wtk−1,tk ◦ · · · ◦E^Wt0,t1y0 ;∀k∈J1, nK with

E^gx:=x+Ef_b,σ(x, g) and

Ef_b,σ(x, g) :=

[1/α]

X

k=1 e+1

X

i1,...,ik=1

fb,σ,i₁. . . fb,σ,i_kI(x)g^{(k),i1,...,ik}

for everyg∈G^[1/α](R^e+1)andx∈R^d, and whereIdenotes the identity map from R^d into itself.

Theorem 2.5. LetD:= (t₀, . . . , t_n)be a dissection of[0, t]witht∈[0, T]andn∈ N^∗. Under the Assumptions 2.3 and 2.4, there exists a constant C >0 depending only on α,γ,f_b,σ andkWkα-Höl,T such that

kπf_b,σ(0, y0;W)t−yb_tⁿk6Ct|D|^θ−1 where, θ:= (bγc+ 1)α >1 and|D| is the mesh ofD.

See Friz and Victoir [14], Theorem 10.30.

Finally, let us state a definition of invariant sets for Equation (1).

LetS be a subset ofR^d.

Definition 2.6. A functionϕ: [0, T]→R^d is viable inS if and only if, ϕ(t)∈S ; ∀t∈[0, T].

The following definition provides a natural extension of the notion of invariant set in the rough paths theory setting.

Definition 2.7. .

(1) The subset S is invariant for(σ,w)if and only if, for any initial condition y₀ ∈ S, every solution on [0, T] of the rough differential equation dy = σ(y)dw is viable inS.

(2) The subsetSis invariant for(b, σ,W)if and only if, for any initial condition y0 ∈ S, every solution on [0, T] of the rough differential equation dy = fb,σ(y)dWis viable in S.

3. An invariance theorem for rough differential equations Consider a nonempty closed set K ⊂ R^d. For every map ϕ: R^d → M_d,m(R) withm∈N^∗, consider

Kϕ:=

m

\

k=1

{x∈R^d:∀x^∗∈ΠK(x), ϕ.,k(x)∈TK(x^∗)^◦◦}, (5)

and then

Kb,±σ :=Kb∩Kσ∩K−σ. (6)

The invariance of K for(b, σ,W)is studied in this section under the two following assumptions on the mapsbandσ, and the signal w.

Assumption 3.1. K⊂K_b,±σ.

Assumption 3.2. There exists λ, µ ∈]0,∞[, β ∈ (0,2α∧1), t0 ∈ (0, T], l ∈ St₀

and a countable setBe⊂∂B_e(0,1)such that{±ek;k∈J1, eK} ⊂ Be,Be=∂B_e(0,1) and

−µ= inf

δ∈Be

lim inf

t→0⁺

hδ, wti t^βl(t) 6 sup

δ∈Be

lim inf

t→0⁺

hδ, wti t^βl(t) =−λ.

Consider also the following stronger assumption on the mapsb andσ: Assumption 3.3. K_b,±σ=R^d.

Now, let us state the main result of the paper ; the invariance theorem.

Theorem 3.4. Under the Assumptions 2.1 and 2.3 onb andσ: (1) Under Assumption 3.3, K is invariant for (b, σ,W).

(2) When K is convex :

(a) Under Assumption 3.1, K is invariant for(b, σ,W).

(b) Under the Assumptions 2.4 and 3.2, if K is invariant for (b, σ,W), then Assumption 3.1 is fulfilled.

(3) When K is compact and b ≡0, under Assumption 3.2, if K is invariant for(σ,w), then Assumption 3.1 is fulfilled.

Remark 3.5. .

(1) By Remark A.3, for any map ϕ:R^d → Md,m(R)with m∈N^∗, int(K)⊂ Kϕ. So, in particular, int(K) ⊂ K_b,±σ. Therefore, Assumption 3.1 is satisfied if and only if ∂K⊂Kb,±σ.

(2) If K is convex, then K_ϕ=

m

\

k=1

{x∈R^d:ϕ_.,k(x)∈T_K(p_K(x))}

for every ϕ:R^d → Md,m(R) withm∈N^∗, and wherepK(x) is the unique projection ofx∈R^d on K.

(3) As in Aubin and Da Prato[5], whenKis not convex, the sufficient condition involves allx∈R^d, and not only allx∈K (see the statement of Theorem 1.5 and its remark page 601).

(4) Assumption 3.1 for some usual convex subsets of R^d :

• WhenK is a vector subspace of R^d, Assumption 3.1 means that b(K)⊂K

and

σ_.,k(K)⊂K ; ∀k∈J1, eK.

• When K is the unit ball ofR^d, Assumption 3.1 means that for every x∈R^d such thatkxk= 1,

hx, b(x)i60 and

hσ.,k(x), xi= 0 ; ∀k∈J1, eK.

• Consider the polyhedron K=\

i∈I

{x∈R^d:hsi, x−aii60}

where, I⊂Nis a nonempty finite set, and (ai)_i∈I and(si)_i∈I are two families of elements of R^d such that si 6= 0 for every i ∈ I. Here, Assumption 3.1 means that for every x ∈ K and i ∈ I such that hsi, x−aii= 0,

hsi, b(x)i60 and

hsi, σ.,k(x)i= 0 ;∀k∈J1, eK.

These conditions on b and σ are quite natural, and the same as in Milian[25]or Cresson et al. [10], where the driving signal of the main equation is a Brownian motion.

(5) Assumption 3.2 is close to the notion of "signal rough at time 0" stated at [13], Chapter 6.

(6) Almost all the paths of thee-dimensional fractional Brownian motion satisfy Assumption 3.2 (see Proposition 5.2).

At Subsection 3.1, the invariance of K for (b, σ,W) is proved under Assumption 3.3, and under Assumption 3.1 when K is convex. At Subsection 3.2, whenK is compact and b≡0, under Assumption 3.2, the necessity of Assumption 3.1 to get the invariance of K for (σ,w) is proved. At Subsection 3.3, when K is convex, under the Assumptions 2.4 and 3.2, the necessity of Assumption 3.1 to get the invariance ofK for(b, σ,W)is proved.

3.1. Sufficient condition of invariance. The main purpose of this subsection is to prove the invariance of K for (b, σ,W) under Assumption 3.3, and under Assumption 3.1 whenKis convex (Theorem 3.4.(1,2.a)). As in Aubin and Da Prato [5], the proof deeply relies on the fact that t ∈[0, T]7→d²_K(yt) has a nonpositive epiderivative (see J.P. Aubin et al. [4], Section 18.6.2), where y is the solution of dy = fb,σ(y)dW with α = 1 and y0 ∈ K as initial condition (see Lemma 3.6).

Finally, whenKis convex and compact, Corollary 3.7 allows to relax the regularity assumptions onb andσ.

Lemma 3.6. Let K be nonempty closed subset ofR^d. Under the Assumptions 2.3 and 3.3 with α = 1, the solution y on [0, T] of the ordinary differential equation dy=fb,σ(y)dW withy0∈K as initial condition is viable in K.

Proof. In order to show thaty is viable inK in a second step, as in Aubin and Da Prato [5], the following inequality is proved in a first step :

(7) lim inf

h→0⁺

d²_K(yt+h)−d²_K(yt)

h 60.

Step 1. Fort∈[0, T] andh >0,

(8) y_t+h−y_t=b(y_t)h+σ(y_t)(w_t+h−w_t) +R_t,h with

Rt,h:=

Z t+h t

[b(ys)−b(yt)]ds+ Z t+h

t

[σ(ys)−σ(yt)]dws.

Sincey (resp. b) is Lipschitz continuous from[0, T](resp. R^d) intoR^d, there exists C1>0such that

Z t+h t

[b(ys)−b(yt)]ds

6C1h².

The function w is Lipschitz continuous from [0, T] into R^e, so there exists w˙ ∈ L^∞([0, T],R^e)such that :

ws=w0+ Z s

0

˙

wudu ;∀s∈[0, T].

Then,

Z t+h t

[σ(y_s)−σ(y_t)]dw_s

=

Z t+h t

[σ(y_s)−σ(y_t)] ˙w_sds 6 kwk˙ _∞,T

Z t+h t

kσ(ys)−σ(yt)k_Md,e(R)ds.

Since y (resp. σ) is Lipschitz continuous from [0, T] (resp. R^d) into R^d (resp.

Md,e(R)), there existsC2>0such that : Z t+h

t

kσ(ys)−σ(yt)k_Md,e(R)ds6C2h². Therefore,

(9) kRt,hk6C3h²

withC₃:=C₁+C₂kwk˙ _∞,T.

Fory^∗_t ∈ΠK(yt)andy^∗_t+h∈ΠK(yt+h)arbitrarily chosen : d²_K(y_t+h) = ky_t+h−y^∗_t+hk²

6 kyt+h−y^∗_tk²

= kyt+h−ytk²+ 2hyt+h−yt, yt−y_t^∗i+kyt−y_t^∗k².

So,

(10) d²_K(y_t+h)−d²_K(y_t)6kyt+h−y_tk²+ 2hyt+h−y_t, y_t−y_t^∗i.

The function y is Lipschitz continuous from[0, T] into R^d, so there existsC4>0 such that :

kyt+h−ytk²6C4h². By Equality (8) and Inequality (9) :

hyt+h−y_t, y_t−y_t^∗i6C₃h² + hb(yt), yt−y^∗_tih+

e

X

k=1

hσ.,k(yt), yt−y_t^∗i(w_t+h^(k) −w^(k)_t ).

Moreover,

hb(yt), yt−y_t^∗ih+

e

X

k=1

hσ.,k(yt), yt−y^∗_ti(w_t+h^(k) −w^(k)_t )60

because y_t−y_t^∗∈T_K(y^∗_t)^◦ by Proposition A.4, andb(y_t),±σ.,k(y_t)∈T_K(y_t^∗)^◦◦ for every k∈J1, eKby Assumption 3.3. So,

hyt+h−yt, yt−y^∗_ti6C3h². Therefore, by Inequality (10) :

d²_K(yt+h)−d²_K(yt)6C5h² withC5:= 2C3+C4. This achieves the first step.

Step 2. Consider the functionϕ: [0, T]→R+ defined by : ϕ(t) :=d²_K(y_t) ;∀t∈[0, T].

Assume that there exists τ ∈(0, T] such thatϕ(τ)>0. Sinceϕ is continuous on [0, T], the set

{t∈[0, τ) :∀s∈(t, τ], ϕ(s)>0}

is not empty, and its infimum is denoted by t∗. Moreover, if ϕ(t∗) > 0, then there exists t∗∗ ∈[0, t∗) such thatϕ(t)>0 for every t∈(t∗∗, t∗]. So, necessarily, ϕ(t_∗) = 0.

By Inequality (7), for everyt∈[t_∗, τ], D_↑ϕ(t)(1) := lim inf

h→0⁺,u→1

ϕ(t+hu)−ϕ(t)

h 60.

So, by Aubin [2] :

0< ϕ(τ) =ϕ(τ)−ϕ(t_∗)60.

There is a contradiction, then ϕ is nonpositive on [0, T]. Since ϕ([0, T]) ⊂ R+, necessarily :

ϕ(t) =d²_K(y_t) = 0 ; ∀t∈[0, T].

In other words,y is viable inK.

Via Lemma 3.6, let us prove Theorem 3.4.(1,2).

Proof. Theorem 3.4.(1,2a). Theorem 3.4.(1) is proved at the first step, and Theo- rem 3.4.(2a) is proved at the second step.

Step 1. Assume that α ∈ (0,1] and K_b,±σ = R^d. Since K is a closed subset of R^d, every solution on [0, T] of the rough differential equation dy = fb,σ(y)dW withy0∈Kas initial condition is viable inKby Lemma 3.6 together with Equality (4).

Step 2. Assume thatα= 1,Kis convex andK⊂K_b,±σ(see (6) for a definition).

Letybe the solution on[0, T]of the ordinary differential equationdy=fb,σ(y)dW with y0∈K as initial condition. Consider the mapsB:=b◦pK,S :=σ◦pK and F :R^d → Md,e+1(R)such that :

F(x) :=

d

X

k=1

B^(k)(x)e_k,1+

e+1

X

l=2 d

X

k=1

S_k,l(x)e_k,l; ∀x∈R^d.

Since fb,σ (resp. pK) is Lipschitz continuous from R^d into Md,e+1(R)(resp. K), F is Lipschitz continuous from R^d into Md,e+1(R). So, the ordinary differential equation dY =F(Y)dW with y₀∈K as initial condition has a unique solutionY on[0, T].

SinceK_B,±S=R^d,Y is viable inK by Lemma 3.6.

Therefore, the solution y of the ordinary differential equation dy = fb,σ(y)dW with y0 as initial condition coincides withY on[0, T] because fb,σ coincides with F onK. In particular,y is viable inK.

Assume now that α ∈ (0,1]. Since K is a closed subset of R^d, every solution on [0, T] of the rough differential equationdy =fb,σ(y)dWwith y0 ∈K as initial

condition is viable inK by Equality (4).

Corollary 3.7. Under the Assumptions 2.1 and 3.1, ifKis a nonempty convex and compact subset ofR^d,b∈Lip^γ−1_loc (R^d)andσ∈Lip^γ−1_loc (R^d,M_d,e(R))withγ >1/α, then all the solutions of the rough differential equationdy=f_b,σ(y)dWwithy₀∈K as initial condition are defined on [0, T]and viable in K.

Proof. Since f_b,σ is locally(γ−1)-Lipschitz continuous from R^d into M_d,e+1(R), there exists τ ∈ (0, T] such that the rough differential equation dy = fb,σ(y)dW withy0∈K as initial condition has at least one solutiony on[0, τ).

Since b and σ satisfy Assumption 3.1, by Theorem 3.4.(2) applied to(b, σ,W) on [0, τ[:

yt∈K;∀t∈[0, τ[.

So, y is bounded on [0, τ[ by at least one continuous function from [0, T] into R^d because Kis a bounded subset of R^d.

Therefore, by Friz and Victoir [14], Theorem 10.21, y is defined on [0, T], and

by Theorem 3.4.(2), it is viable in K.

3.2. Necessary condition of invariance : compact case. WhenKis compact andb≡0, the purpose of this subsection is to prove that under Assumption 3.2, if K is invariant for(σ,w), then Assumption 3.1 is fulfilled (Theorem 3.4.(3)).

Lemma 3.8. Under Assumption 3.2, forT₀:=t₀∧T,

M := max

k∈J1,eK

sup

t∈[0,T0]

w_t^(k) t^βl(t)

<∞.

Proof. By Assumption 3.2 : min

k∈J1,eK

lim inf

t→0⁺

w_t^(k)

t^βl(t) = min

k∈J1,eK

lim inf

t→0⁺

hek, wti t^βl(t)

> inf

δ∈Be

lim inf

t→0⁺

hδ, wti t^βl(t) =−µ.

Moreover, the function

t∈(0, T0]7−→ w^(k)_t t^βl(t) is continuous. So, there existsr >0such that :

min

k∈J1,eK

inf

t∈[0,T0]

w^(k)_t

t^βl(t)>−rµ.

Similarly, there existsR >0such that : max

k∈J1,eK

sup

t∈[0,T0]

w_t^(k)

t^βl(t) 6Rµ.

This achieves the proof.

Proposition 3.9. Under Assumption 3.2 :

−µ= inf

δ∈∂Be(0,1)lim inf

t→0⁺

hδ, wti

t^βl(t) 6 sup

δ∈∂B_e(0,1)

lim inf

t→0⁺

hδ, wti t^βl(t) =−λ.

Proof. Letδ∈∂Be(0,1)be arbitrarily chosen. SinceBe=∂Be(0,1), there exists a sequence(δn)_n∈N of elements ofBe such that :

n→∞lim kδ−δnk= 0.

By Lemma 3.8 :

M := max

k∈J1,eK

sup

t∈[0,T0]

w_t^(k) t^βl(t)

<∞.

For every n∈Nandt∈(0, T₀], hδ, wti

t^βl(t) = hδ−δn, wti

t^βl(t) +hδn, wti t^βl(t) 6 kδ−δnk · kwtk

t^βl(t)+hδn, wti t^βl(t) 6 kδ−δ_nkM+hδ_n, w_ti

t^βl(t) . So, for every n∈N, by Assumption 3.2 :

lim inf

t→0⁺

hδ, wti

t^βl(t) 6kδ−δ_nkM + lim inf

t→0⁺

hδn, wti t^βl(t) 6kδ−δ_nkM −λ

−−−−→

n→∞ −λ.

Since the right hand side of the previous inequality is not depending onδ : sup

δ∈∂Be(0,1)

lim inf

t→0⁺

hδ, w_ti t^βl(t) 6−λ.

Moreover, by Assumption 3.2 and sinceBe⊂∂B(0,1) :

−λ= sup

δ∈Be

lim inf

t→0⁺

hδ, w_ti

t^βl(t) 6 sup

δ∈∂Be(0,1)

lim inf

t→0⁺

hδ, w_ti t^βl(t).

Therefore,

sup

δ∈∂Be(0,1)

lim inf

t→0⁺

hδ, wti t^βl(t) =−λ.

Similarly,

inf

δ∈∂Be(0,1)

lim inf

t→0⁺

hδ, w_ti t^βl(t) =−µ.

Proof. Theorem 3.4.(3). Let y be a solution of the rough differential equation dy=σ(y)dw withy₀∈K as initial condition, and assume thaty is viable inK.

Consider s ∈ T_K(y₀)^◦ and ε > 0. Since K is a nonempty compact subset of R^d, by Proposition A.5, there existsδε>0such that for every x∈K∩Bd(y0, δε),

hx−y₀, si6εkx−y₀k.

Since y is continuous, there exists t_ε∈[0, T] such thaty([0, t_ε])⊂B_d(y₀, δ_ε). So, for every t∈[0, tε],

hyt−y0, si6εkyt−y0k.

Then,

lim sup

t→0⁺

hy_t−y₀, si

t^βl(t) 6εlim sup

t→0⁺

ky_t−y₀k t^βl(t) .

For everyt∈[0, T], by Theorem 2.5 applied with the dissection(0, t)of[0, t], there exists a constantC >0, depending onT but not ont, such that :

ky_t−y₀−σ(y₀)w_tk 6Ct|(0, t)|^{(bγc+1)α−1} 6CT^(bγc−1)αt^2α. Sinceβ <2α,

lim sup

t→0⁺

hσ(y0)w_t, si

t^βl(t) 6εlim sup

t→0⁺

kσ(y0)w_tk t^βl(t) . Therefore, by duality inR^d :

(11) lim inf

t→0⁺

hu(s), w_ti

t^βl(t) >0 ;∀s∈T_K(y₀)^◦ where, u:T_K(y₀)^◦→R^eis the map defined by

u(s) :=−σ(y₀)^τs;∀s∈T_K(y₀)^◦, and σ(y₀)^τ is the transpose of the matrixσ(y₀).

Assume that there exists s∈TK(y0)^◦ such thatu(s)6= 0, and put v(s) := u(s)

ku(s)k ∈∂Be(0,1).

By Inequality (11) :

lim inf

t→0⁺

hv(s), wti t^βl(t) >0.

There is a contradiction with Assumption 3.2 by Proposition 3.9. So, necessarily : u(s) = 0 ; ∀s∈TK(y0)^◦.

Therefore, since (ek)_k∈

J1,eKis a basis of R^e :

hσ.,k(y0), si= 0 ;∀k∈J1, eK.

In particular, ±σ.,k(y0) ∈ TK(y0)^◦◦ for every k ∈ J1, eK. This achieves the proof

because y0∈K has been arbitrarily chosen.

3.3. Necessary condition of invariance : convex case. WhenKis convex, the purpose of this subsection is to prove that under the Assumptions 2.4 and 3.2, ifK is invariant for (b, σ,W), then bandσ satisfy Assumption 3.1 (Theorem 3.4.(2b)).

First, that result is proved for the half-hyperplanes.

Lemma 3.10. Under the Assumptions 2.1, 2.3, 2.4 and 3.2, if there exists ν ∈ J1, dK such thatD_ν={x∈R^d:x^(ν)>0} is invariant for (b, σ,W), then

b^(ν)(x−x^(ν)eν)>0 andσν,.(x−x^(ν)eν) = 0 for every x∈R^d.

Proof. Letyb: [0, T]→R^d be the map defined by : yb_t:=E^W0,ty₀;∀t∈[0, T].

For everyt∈[0, T],(y₀,by_t)coincides with the Euler scheme for the rough differen- tial equation dy=f_b,σ(y)dWwithy₀∈R^d as initial condition along the dissection Dt:= (0, t)of[0, t].

In a first step, it is proved that if there existsy0∈Dν such that

(12) lim inf

t→0⁺

yb^(ν)_t t^βl(t) <0,

then Dν is not invariant for (b, σ,W). In a second step, it is proved that if there exists y₀ ∈∂D_ν such thatσ_ν,.(y₀)6= 0 or b^(ν)(y₀)<0, then by satisfies Inequality (12).

Step 1. Assume that there existsy₀∈D_ν such that : lim inf

t→0⁺

yb^(ν)_t t^βl(t) <0.

For everyt∈[0, T], by Theorem 2.5 applied with the dissection(0, t)of[0, t], there exists a constantC₁>0, depending onT but not ont, such that

kπ_fb,σ(0, y₀;W)_t−by_tk 6C₁t|D_t|^θ−1

=C1t^θ withθ:= (bγc+ 1)α >1. So,

π_f^(ν)

b,σ(0, y0;W)t

t^βl(t) 6C₁t^θ−β

l(t) + by^(ν)_t

t^βl(t) ;∀t∈(0, T₀].

Moreover, since θ >1> β andl∈ S_t0 : lim inf

t→0⁺

"

C1

t^θ−β

l(t) + by_t^(ν) t^βl(t)

#

= lim inf

t→0⁺

yb_t^(ν) t^βl(t).

Therefore, by Inequality (12) : lim inf

t→0⁺

π^(ν)_f

b,σ(0, y0;W)t

t^βl(t) <0.

In conclusion, there existst₁∈[0, T₀]such that : π^(ν)_f

b,σ(0, y0;W)t₁<0.

The pathπ^(ν)_f

b,σ(0, y0;W)is not viable inDν.

Step 2. Let us show that if there exists y0 ∈ ∂Dν such that σν,.(y0) 6= 0 or

b^(ν)(y₀)<0, thenby satisfies Inequality (12).

Case 1. Assume that there existsy0∈∂Dν such thatσν,.(y0)6= 0. By Lemma 3.8 : M := max

k∈J1,eK

sup

t∈[0,T0]

w_t^(k) t^βl(t)

<∞.

On the one hand, since kσ_ν,.(y₀)k⁻¹σ_ν,.(y₀)∈∂B_e(0,1) andB_e=∂B_e(0,1), there exists a sequence(u_n)_n∈Nof elements ofB_esuch that :

n→∞lim

un− σν,.(y0) kσν,.(y0)k

= 0.

So, there exists n₀∈Nsuch that for everyn∈N∩[n₀,∞[, kσν,.(y0)−vnk6 λ

4eMkσν,.(y0)k

and vn := kσν,.(y0)kun where, λ is defined in Assumption 3.2. Then, for every n∈N∩[n0,∞[,

sup

t∈[0,T0]

1

t^βl(t)|hσν,.(y₀)−v_n, w_ti| 6 eMkσν,.(y₀)−v_nk 6 λ

4kσ_ν,.(y₀)k.

(13)

On the other hand, by the definition of the Euler scheme for Equation (1), there exists C2>0such that :

|E_f^(ν)

b,σ(y₀,W0,t)− hσν,.(y₀), w_ti|6C₂(t^2α∨t) ;∀t∈[0, T].

Sincel∈ St₀ andβ∈(0,2α∧1): lim

t→0⁺

1 t^βl(t)|E_f^(ν)

b,σ(y0,W0,t)− hσν,.(y0), wti| ≤ C2 lim

t→0⁺

t^2α−β l(t)

= 0.

(14)

For every n ∈N∩[n₀,∞[, by Assumption 3.2, Inequality (13) and Equality (14) together :

lim inf

t→0⁺

yb_t^(ν)

t^βl(t) = lim inf

t→0⁺

hσν,.(y0), wti t^βl(t) 6 sup

t∈[0,T0]

1

t^βl(t)|hσν,.(y0)−vn, wti|+ lim inf

t→0⁺

hvn, wti t^βl(t) 6 λ

4kσν,.(y0)k −λkσν,.(y0)k=−3λ

4 kσν,.(y0)k.

So,

lim inf

t→0⁺

yb^(ν)_t t^βl(t) <0.

By the first step of the proof, it means that Dν is not invariant for (b, σ,W). In other words, ifK is invariant for(b, σ,W), then

(15) σν,.(x) = 0 ; ∀x∈∂Dν.

Case 2. Assume that D_ν is invariant for (b, σ,W) and there existsy₀∈∂D_ν such that b^(ν)(y0)< 0. By the first case, since Dν is invariant for (b, σ,W), Equation (15) is true.

Lett∈[0, T]be arbitrarily chosen.

• Ifα∈(1/2,1), then

yb_t^(ν) =b^(ν)(y₀)t+hσν,.(y₀), w_ti

=b^(ν)(y0)t.

• Ifα∈(0,1/2], then

yb^(ν)_t =b^(ν)(y0)t+hσν,.(y0), wti+

[1/α]

X

k=2 e

X

i1,...,ik=1

[σi₁. . . σi_kI(y0)]^(ν)w^{(k),i1,...,ik}. Consider k ∈ J2,[1/α]K and i₁, . . . , i_k ∈J1, eK. There exists a real family (ρ_{i2,...,ik−1,l1,...,lk−1}(y₀))_{(l1,...,lk−1)∈}

J1,dK

k−1 such that : [σ_i1. . . σ_ikI(y₀)]^(ν)=

d

X

l1,...,lk−1=1

σ_l1,i1(y₀)∂_l^k−1

1,...,lk−1σ_ν,ik(y₀)ρ_{i2,...,ik−1,l1,...,lk−1}(y₀).

Consider the set

I_k−1:={(l₁, . . . , l_k−1)∈J1, dK

k−1:∃κ∈J1, k−1K, l_κ6=ν}.

By (15) together with Schwarz’s lemma :

∂^k−1_l

1,...,lk−1σν,.(x) = 0 ;∀x∈∂Dν,∀(l1, . . . , l_k−1)∈I_k−1. Then,

[σi1. . . σi_kI(y0)]^(ν) = σν,i1(y0)∂ν,...,ν^k−1 σν,i_k(y0)ρi2,...,ik−1,ν,...,ν(y0) + X

l₁,...,l_k−1∈I_k−1

σl₁,i₁(y0)∂_l^k−1

1,...,lk−1σν,i_k(y0)ρi₂,...,i_k−1,l₁,...,l_k−1(y0)

= 0.

So,

yb_t^(ν)=b^(ν)(y₀)t.

Sincel∈ St0 and1> β, there existst₂∈(0, T₀]such that : 0< t^1−β

l(t) <1 ;∀t∈(0, t2].

Therefore,

lim inf

t→0⁺

yb_t^(ν)

t^βl(t) 6lim inf

t→0⁺

yb_t^(ν) t 6b^(ν)(y₀)<0.

This achieves the proof because there is a contradiction by the first step of the

proof.

Via Lemma 3.10, let us prove that Assumption 3.1 is necessary to get the invariance ofK for(b, σ,W).

Proof. Theorem 3.4.(2b). In a first step, it is proved that if the half-space Ha,s:={x∈R^d:hs, x−ai60}

witha∈R^d ands∈R^d\{0}is invariant for(b, σ,W), then hb(x), si60

and

hσ.,k(x), si= 0 ;∀k∈J1, eK

for every x ∈ ∂Ha,s. In a second step, this result is used to show that if K is invariant for(b, σ,W), thenK⊂Kb,±σ.

Step 1. Let y0 ∈ Ha,s be arbitrarily chosen. Since s ∈ R^d\{0}, there exists ν ∈J1, dKsuch thats^(ν)6= 0. Consider the mapU :R^d →R^d defined by :

U(x) :=−(x−a) + (x^(ν)−a^(ν)− hx−a, si)eν ;∀x∈R^d. The mapU is one to one from R^d into itself, and

U⁻¹(x) =−x+a− 1

s^(ν)(x^(ν)− hx, si)eν ;∀x∈R^d.

Moreover, U|_Ha,s (resp. U|_∂Ha,s) is one to one from H_a,s (resp. ∂H_a,s) into D_ν (resp. ∂Dν) where,Dν is defined in Lemma 3.10. For everyx, h∈R^d,

DU(x).h= −h+ (h^(ν)− hh, si)e_ν

= MUh with

MU :=−I+eν,ν−

d

X

k=1

s^(k)eν,k.

Consider the maps B:R^d→R^d andS:R^d→ Md,e(R)defined by B(x) := MUb(U⁻¹(x))

= (eν,ν−I)b(U⁻¹(x))− hb(U⁻¹(x)), sieν

and

S(x) :=M_Uσ(U⁻¹(x))

= (eν,ν−I)σ(U⁻¹(x))−

e

X

k=1

hσ.,k(U⁻¹(x)), sieν,k

for every x∈R^d. LetF :R^d→ Md,e+1(R)be the map defined by : F(x) :=

d

X

k=1

B^(k)(x)e_k,1+

e+1

X

l=2 d

X

k=1

S_k,l(x)e_k,l; ∀x∈R^d.

Since U⁻¹ ∈ L(R^d) and b and σ satisfy assumptions 2.3 and 2.4, B and S also.

So, by Friz and Victoir [14], Theorem 10.26, Exercice 10.55 and Exercice 10.56, the rough differential equation dz=F(z)dWwithU(y0)as initial condition has a unique solution z on[0, T]. By the (rough) change of variable formula, for every t∈[0, T],

U⁻¹(zt) = y0+ Z t

0

M_U⁻¹dzs

= y₀+ Z t

0

f_b,σ(U⁻¹(z_s))dW_s. Therefore, since dy=fb,σ(y)dW has a unique solution, (16) πF(0, .;W) =U(πf_b,σ(0, U⁻¹(.);W)).

Assume that H_a,s is invariant for (b, σ,W). Since U|Ha,s is one to one from H_a,s into D_ν, by Equality (16),D_ν is invariant for(B, S,W). So, by Lemma 3.10 :

B^(ν)(x−x^(ν)eν)>0andSν,.(x−x^(ν)eν) = 0

for everyx∈R^d. Letk∈J1, eKandx∈∂Ha,sbe arbitrarily chosen. SinceU|∂H_a,s

is one to one from ∂Ha,s into∂Dν, U(x)∈∂Dν. Therefore, by construction ofB and S :

hb(x), si=−B^(ν)(U(x))60

and

hσ.,k(x), si= −

d

X

j=1

σj,k(x)s^(j)

= −S_ν,k(U(x)) = 0.

Step 2. Assume that there exists y0 ∈ ∂K such that y0 6∈ K_b,±σ. Then, there exists s∈NK(y0)such that :

(17) hb(y0), si>0or (∃k∈J1, eK:hσ.,k(y0), si 6= 0).

Consider the half-space

H_y0,s:={x∈R^d:hs, x−y₀i60}.

By the first step of the proof, (17) implies that there exists t ∈ [0, T] such that πf_b,σ(0, y0;W)t 6∈ Hy₀,s. Moreover, since y0 ∈ ∂K and s ∈ NK(y0), K ⊂ Hy₀,s. Therefore,πf_b,σ(0, y0;W)t6∈K. This achieves the proof by contraposition.

4. A comparison theorem for rough differential equations In this section, a comparison theorem for rough differential equations is proved by using the viability theorem of Section 3.

Consider a nonempty setI⊂J1, dK, and

K:={(x₁, x₂)∈(R^d)²:∀i∈I, x⁽ⁱ⁾₁ 6x⁽ⁱ⁾₂ }.

The following comparison theorem is a consequence of Theorem 3.4.

Proposition 4.1. Forj ∈ {1,2}, consider b_j :R^d →R^d and σ^j :R^d → M_d,e(R) satisfying assumptions 2.3 and 2.4, and the mapf^j :R^d→ Md,e+1(R)defined by :

f^j(x) :=

d

X

k=1

b^(k)_j (x)ek,1+

e+1

X

l=2 d

X

k=1

σ_k,l^j (x)ek,l ; ∀x∈R^d.

Under the Assumptions 2.1 and 3.2, the two following conditions are equivalent : (1) For every (y₀¹, y²₀)∈ K,i ∈ I and t ∈ [0, T], (y¹_t)⁽ⁱ⁾ 6(y_t²)⁽ⁱ⁾ wherey^j is

the solution of the rough differential equation dy^j = f^j(y^j)dW with y^j₀ as initial condition forj ∈ {1,2}.

(2) For every (x₁, x₂)∈K andi∈I, if x⁽ⁱ⁾₁ =x⁽ⁱ⁾₂ , then b⁽ⁱ⁾₁ (x₁)6b⁽ⁱ⁾₂ (x₂)

and

σ_i,k¹ (x₁) =σ_i,k² (x₂) ;∀k∈J1, eK.

Proof. The set K is (isomorphe to) a nonempty closed convex polyhedron of R^2d. Indeed,

K∼=\

i∈I

{x∈R^2d :hsi, xi60}

withsi:=ei−ed+i for everyi∈I. Let F:R^2d → M2d,e+1(R)be the map defined by :

F(x1, x2) :=

e+1

X

l=1 d

X

k=1

[f_k,l¹ (x1)ek,l+f_k,l² (x2)ed+k,l] ; ∀(x1, x2)∈(R^d)². Since bj and σ^j satisfy Assumption 2.3 for j ∈ {1,2}, B := F.,1 and S :=

(F.,k)_k∈

J2,e+1K also.