Sensitivities via Rough Paths

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Sensitivities via Rough Paths

Nicolas Marie

To cite this version:

Nicolas Marie. Sensitivities via Rough Paths. ESAIM: Probability and Statistics, EDP Sciences, 2015,

19, pp.515-543. �10.1051/ps/2015001�. �hal-01519403�

URL: http://www.emath.fr/ps/

SENSITIVITIES VIA ROUGH PATHS

Nicolas MARIE ¹

Abstract . Motivated by a problematic coming from mathematical finance, the paper deals with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations. These regularity results together with the Malliavin calculus are applied to the sensitivities analysis of stochastic differential equations driven by multidimensional Gaussian processes with continuous paths as the fractional Brownian motion. The well known results on greeks in the Itô stochastic calculus framework are extended to stochastic differential equations driven by a Gaussian process which is not a semi-martingale.

1991 Mathematics Subject Classification. 60H10.

.

Contents

1. Introduction 2

2. Regularity of the Itô map : existing and additional results 6

2.1. Differentiability of the Itô map with respect to x 0 and V 8

2.2. Differentiability of the Itô map with respect to the driving signal 15

2.3. Application to the Gaussian stochastic analysis 18

3. Sensitivity analysis of Gaussian rough differential equations 19

4. Application to mathematical finance and simulations 24

4.1. Calculation of sensitivities in a fractional stochastic volatility model 24

4.2. Simulations 25

Appendix A. Fractional Brownian motion 28

References 30

Corresponding author : Nicolas MARIE.

Address : Laboratoire ISTI. ESME Sudria.

51 Boulevard de Brandebourg, 94200, Ivry-sur-Seine.

E-mail addresses: marie@esme.fr, nicolas.marie.math@gmail.com (N. MARIE).

Tel.: (+33)7 70 01 83 51.

Keywords and phrases: Rough paths, Rough differential equations, Malliavin calculus, Sensitivities, Mathematical finance, Gaussian processes

1

Laboratoire ISTI - ESME Sudria - 51 Boulevard de Brandebourg, 94200, Ivry-sur-Seine

c EDP Sciences, SMAI 1999

Acknowledgements. Many thanks to Laure Coutin and Laurent Decreusefond for their advices. This work was supported by the ANR Masterie.

1. Introduction

Motivated by a problematic coming from mathematical finance, the paper deals with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations (RDEs). These regularity results together with the Malliavin calculus are applied to the sensitivities analysis of stochastic differential equations (SDEs) driven by multidimensional Gaussian processes with continuous paths as the fractional Brownian motion.

First of all, some notions of mathematical finance are reminded.

Consider a probability space (Ω, A, P ), a d-dimensional Brownian motion B and F := (A t ; t ∈ [0, T ]) the filtration generated by B (d ∈ N

and T > 0).

Consider the financial market consisting of d + 1 assets (one risk-free asset and d risky assets) over the filtered probability space (Ω, A, F , P ). At the time t ∈ [0, T ], the deterministic price of the risk-free asset is denoted by S _t ⁰ , and the prices of the d risky assets are given by the random vector S _t := (S ¹ _t , . . . , S ^d _t ).

In a first place, assume that the process S is the solution of a stochastic differential equation, taken in the sense of Itô :

S _t = x + Z t

0

µ (S _u ) du + Z t

0

σ (S _u ) dB _u ; x ∈ R ^d

where, µ : R ^d → R ^d and σ : R ^d → M d ( R ) are some (globally) Lipschitz continuous functions.

Let P

∼ P be the risk-neutral probability measure of the market (i.e. such that S

:= S/S ⁰ is a ( F , P

)- martingale).

Theorem 1.1. Consider an option of payoff h ∈ L ² (Ω, A T , P

). Then, there exists an admissible strategy ϕ such that :

∀t ∈ [0, T ], V t (ϕ) = E

S _t ⁰

S ₀ ⁰ h

A t

P

-a.s.

where V (ϕ) is the associated wealth process.

Theorem 1.1 is a consequence of the stochastic integral representation of the discounted claim (see N.H. Bing- ham and R. Kiesel [1], Lemma 6.1.2 and Theorem 6.1.5).

With the notations of Theorem 1.1, V _T (ϕ) = E

(S ⁰ _T /S ₀ ⁰ h). It is the price of the option, and when h := F (S _T ) with some function F : R ^d → R + , it is possible to get the existence and an expression of the sensitivities of V T (ϕ) to perturbations of the initial condition and of the volatility function σ for instance :

∆ := ∂ _x E

[F(S _T ^x )] and V := ∂ _σ E

[F (S _T ^σ )] .

In finance, these sensitivities are called the greeks. For instance, ∆ involves in the ∆-hedging which provides the admissible strategy of Theorem 1.1 (see [13], Subsection 4.3.3). However, these quantities don’t involve in finance only. They could also be used in pharmacokinetics as mentioned at [19], Section 5.

The greeks have been deeply studied by several authors. In [8], E. Fournié et al. have established the existence

of the greeks and have provided expressions of them via the Malliavin calculus by assuming that σ satisfied

a uniform elliptic condition (see Theorem 1.2). In [11], E. Gobet and R. Münos have extended these results

by assuming that σ only satisfied a hypoelliptic condition. On the computation of greeks in the Black-Scholes

model, see P. Malliavin and A. Thalmaier [17], Chapter 2. On the sensitivities in models with jumps, see

N. Privault et al. [23] and [7]. Finally, via the cubature formula for the Brownian motion, J. Teichmann has

provided some estimators of the Malliavin weights for the computation of greeks (see J. Teichmann [26]). On

the regularity of the solution map of SDEs taken in the sense of Itô, see H. Kunita [12].

At the following theorem, δ is the divergence operator associated to the Brownian motion B (see D. Nualart [22], Section 1.3).

Theorem 1.2. Assume that b and σ are differentiable, of bounded and Lipschitz continuous derivatives, and F ∈ L ² ( R ^d ; R ⁺ ).

(1) If σ satisfies the uniform elliptic condition (i.e. there exists ε > 0 such that for every a, b ∈ R ^d , b ^T σ ^T (a)σ(a)b > εkbk ² ), then ∆ exists and

∆ = E

F (S _T )δ(h ^∆ ) where, h ^∆ is an adapted d-dimensional stochastic process.

(2) Let ˜ σ : R ^d → M d ( R ) be a function such that for every ε belonging to a closed neighborhood of 0, σ + ε˜ σ satisfies the uniform elliptic condition. Then V exists and

V = E

F (S _T )δ(h

) where, h

is an (anticipative) d-dimensional stochastic process.

See [8], propositions 3.2 and 3.3 for a proof.

Under some technical assumptions stated at Subsection 2.3, the main purpose of this paper is to extend Theorem 1.2 to the following SDE, taken in the sense of rough paths introduced by T. Lyons in [15] :

X t = x + Z t

0

µ (X s ) ds + Z t

0

σ (X s ) dW s ; x ∈ R ^d

where, W is a centered d-dimensional Gaussian process with continuous paths of finite p-variation (p > 1), and the functions µ and σ satisfy the following assumption.

Assumption 1.3. µ and σ are [p] + 1 times differentiable, bounded and of bounded derivatives.

Subsections 2.1 and 2.2 deal with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations. In particular, the continuous differentiability of the Itô map with respect to the collection of vector fields is proved, and completes the existing results of regularity with respect to the initial condition and to the driving signal (see P. Friz and N. Victoir [10], chapters 4 and 11). In order to apply the (probabilistic) integrability results coming from T. Cass, C. Litterer and T. Lyons [3], some tailor-made upper-bounds are provided for each derivative. Subsection 2.3 reminds some definitions and results related to the good geometric rough path over a Gaussian process having a covariance function satisfy- ing the technical Assumption 2.9, called enhanced Gaussian process by P. Friz and N. Victoir. The results of subsections 2.1 and 2.2 are applied together with the results coming from [3] in order to show the (probabilistic) integrability of the solution of a Gaussian RDE and their derivatives. The main problem is solved at Section 3 by using the results of Section 2 together with the Malliavin calculus. Some simulations of ∆ and V are provided at Subsection 4.2.

The fractional Brownian motion (fBm) introduced in [18] by B.B. Mandelbrot and J.W. Van Ness has been studied by several authors in order to generalize the Brownian motion classically used to model the prices process of the risky assets. For instance, the regularity of the paths of the process and its memory are both controlled by the Hurst parameter H of the fBm. However, the fBm is not a semi-martingale if H 6= 1/2 (see. [22], Proposition 5.1.1). In [24], L.C.G. Rogers has shown the existence of arbitrages if the prices process of the assets is modeled by a fBm. In order to bypass that difficulty, several approaches have been studied.

For instance, in [4], P. Cheridito assumed that the prices process was modeled by a mixed fractional Brownian

motion which is a semi-martingale depending on a fBm. At Subsection 4.1, the prices of the risky assets are

modeled by a fractional SDE, in which the volatility is modeled by another one. The results of Section 3 are applied in order to show the existence and provide an expression of the sensitivity of the price of the option with respect to the collection of vector fields of the equation of the volatility.

The paper uses many results on rough paths and rough differential equations coming from [10] and, T. Lyons and Z. Qian [16]. The paper also uses results of Malliavin calculus coming from [22].

The notations, short definitions and results used throughout the paper are stated below. However, the original results of the literature are cited throughout the paper.

Notations (general) :

• R ^e and R ^d (e, d ∈ N

) are equipped with their Euclidean norms, both denoted by k.k.

• The canonical basis of R ^d is denoted by (e 1 , . . . , e d ). With respect to that basis, for k = 1, . . . , d, the k-th component of any vector u ∈ R ^d is denoted by u ^k .

• The closed ball of R ^d with respect to k.k, of center a ∈ R ^d and of radius r > 0, is denoted by B(a, r).

• The usual matrix (resp. operator) norm on M e,d ( R ) (resp. L( R ^e ; R ^d )) is denoted by k.k

(resp. k.k

).

• Consider 0 6 s < t 6 T . The set of all the dissections of [s, t] is denoted by D s,t . In particular, D _T := D _0,T .

• ∆ _T := {(u, v) ∈ R ² : 0 6 u < v 6 T }.

• The space of continuous (resp. continuously differentiable) functions from [s, t] into R ^d is denoted by C ⁰ ([s, t]; R ^d ) (resp. C ¹ ([s, t]; R ^d )) and equipped with the uniform norm k.k

.

• Differentiability means differentiability in the sense of Fréchet (see H. Cartan [2], Chapter I.2).

• Consider two Banach spaces E and F . Let ϕ : E → F be a map derivable at point x ∈ E, in the direction h ∈ E. The derivative of ϕ at point x, in the direction h, is denoted by :

D _h ϕ(x) := lim

ε→0

ϕ(x + εh) − ϕ(x)

ε in F.

• Consider three Banach spaces E, F and G, and a differentiable map ϕ : E × F → G. At point (x, y) ∈ E × F, the Fréchet derivative of ϕ(x, .) (resp. ϕ(., y)) is denoted by ∂ y f (x, y) (resp. ∂ x ϕ(x, y)).

Notations (rough paths) :

• Consider p > 1 and α ∈]0, 1]. The space of continuous functions of finite p-variation (resp. α-Hölder continuous functions) from [s, t] into R ^d is denoted by

C ^p-var [s, t]; R ^d :=







y ∈ C ⁰ [s, t]; R ^d

: sup

D={r

|D|−1

X

k=1

ky r

− y _r

k ^p < ∞







(resp. C ^α-höl ([s, t]; R ^d ), which is a subset of C ^1/α-var ([s, t]; R ^d )) and is equipped with the p-variation distance d p-var;s,t (resp. the α-Hölder distance d α-höl;s,t ). See [10], chapters 5 and 8 about these spaces.

• Consider N ∈ N

and y : [0, T ] → R ^d a continuous function of finite 1-variation. The step-N tensor algebra over R ^d is denoted by

T ^N ( R ^d ) := R ⊕ R ^d ⊕ · · · ⊕ ( R ^d )

,

the step-N signature of y is denoted by

S N (y) :=

1,

Z . 0

dy r , . . . , Z

0<r

<···<r

<.

dy r

⊗ · · · ⊗ dy r

,

and the step-N free nilpotent group over R ^d is denoted by G ^N ( R ^d ) :=

S _N (y) ₁ ; y ∈ C ^1-var ([0, 1]; R ^d ) . See [10], Chapter 7.

• For k = 0, . . . , N, the (k + 1)-th component of any X ∈ T ^N ( R ^d ) is denoted by X ^k .

• The space of geometric p-rough paths is denoted by GΩ p,T ( R ^d ) :=

S _[p] (y); y ∈ C ^1-var ([0, T ]; R ^d ) ^d

,

and is equipped with the p-variation distance d p-var;T , or with the uniform distance d

, associated to the Carnot-Carathéodory distance. See [10], Chapter 9.

• The closed ball of GΩ p,T ( R ^d ) with respect to d p-var;T , of center Y ∈ GΩ p,T ( R ^d ) and of radius r > 0, is denoted by B _p,T (Y, r).

• For every Y ∈ GΩ p,T ( R ^d ), ω Y,p : (s, t) ∈ ∆ ¯ T 7→ kY k ^p _p-var;s,t is a control. See [10], Chapter 1 about some properties of the controls.

• Consider q > 1 such that 1/p + 1/q > 1, Y ∈ GΩ p,T ( R ^d ) and h ∈ GΩ q,T ( R ^e ). The geometric p-rough path over (Y ¹ , h ¹ ) provided at [10], Theorem 9.26 is denoted by S _[p] (Y ⊕ h). The translation of Y by h provided at [10], Theorem 9.34 is denoted by T h Y .

• Consider γ > 0. The space of collections of γ-Lipschitz (resp. locally γ-Lipschitz) vector fields on R ^e is denoted by Lip ^γ ( R ^e ; R ^d ) (resp. Lip ^γ _loc ( R ^e ; R ^d )) (see [10], Definition 10.2). Lip ^γ ( R ^e ; R ^d ) is equipped with the γ-Lipschitz norm k.k _lip

such that, for every V ∈ Lip ^γ ( R ^e ; R ^d ),

kV k _lip

:= max n

kV k

, kDV k

, . . . , kD

V k

, kD

V k

o .

• The closed ball of Lip ^γ ( R ^e ; R ^d ) with respect to k.k lip

, of center V ∈ Lip ^γ ( R ^e ; R ^d ) and of radius r > 0, is denoted by B Lip

(V, r).

• Consider ε > 0, a compact interval I included in [0, T ], a control ω : ¯ ∆ _T → R + and Y ∈ GΩ _p,T ( R ^d ).

Put

M ε,I,ω := sup

D={r_k} ∈ D_I ω r_k, r_k+1

6 ε

|D|−1

X

k=1

ω (r k , r k+1 ) ,

M ε,I,p (Y ) := M ε,I,ω

and

N ε,I,p (Y ) := sup {n ∈ N : τ n 6 sup(I)}

where, τ 0 := inf(I) and for every n ∈ N , τ n+1 := inf

t ∈ I : kY k ^p _p-var;τ

_,t > ε and t > τ n ∧ sup(I).

In the sequel, I := [0, T ].

• Consider γ > p and V ∈ Lip ^γ _loc ( R ^e ; R ^d ) satisfying the p-non explosion condition (i.e. V and D ^[p] V are respectively globally Lipschitz continuous and (γ − [p])-Hölder continuous on R ^e ). The unique solution of dX = V (X )d W with the initial condition X 0 ∈ G ^[p] ( R ^e ) or X 0 ∈ R ^e , is denoted by π V (0, X 0 ; W ).

• By [10], Exercice 10.55, if V is a collection of affine vector fields and ω : ¯ ∆ T → R ⁺ is a control satisfying k W k p-var;s,t 6 ω ^1/p (s, t) for every (s, t) ∈ ∆ ¯ T , there exists a constant C 1 > 0, not depending on X 0 ∈ R ^e and W , such that :

kπ V (0, x 0 ; W )k

6 C 1 (1 + kx 0 k)e ^C

^M

.

By [10], Theorem 10.36, if V ∈ Lip ^γ ( R ^e ; R ^d ), there exists a constant C ₂ > 0, not depending on X ₀ ∈ G ^[p] ( R ^e ), V and W , such that for every (s, t) ∈ ∆ ¯ T ,

kπ V (0, X 0 ; W )k p-var;s,t 6 C 2

kV k _lip

k W k p-var;s,t ∨ kV k ^p _lip

k W k ^p _p-var;s,t .

By [10], Theorem 10.47, if V ∈ Lip ^γ ( R ^e ; R ^d ), there exists a constant C ₃ > 0, not depending on V and W , such that for every (s, t) ∈ ∆ ¯ _T ,

Z

V ( W )d W

_p-var;s,t 6 C 3 kV k _lip

k W k p-var;s,t ∨ k W k ^p _p-var;s,t .

Notations (Gaussian stochastic analysis) :

• For every t ∈ [0, T ], [0, t] is equipped with the Borel σ-algebra B t generated by the usual topology on [0, t].

• R ^d is equipped with the Borel σ-algebra generated by the usual Euclidean topology on R ^d , and G ^[p] ( R ^d ) is equipped with the Borel σ-algebra generated by the Carnot-Carathéodory topology on G ^[p] ( R ^d ).

These σ-algebras are both denoted by B.

• Let W be a d-dimensional centered Gaussian process with continuous paths. Its Cameron-Martin space is denoted by

H ¹ :=

h ∈ C ⁰ ([0, T ]; R ) : ∃Z ∈ W s.t. ∀t ∈ [0, T ], h t = E (W t Z) with

W := span {W t , t ∈ [0, T ]} ^L

2

(see [10], Subsection 15.2.2 and Section 15.3), its reproducing kernel Hilbert space is denoted by H, and the Wiener integral with respect to W defined on H is denoted by W (see [22], Section 1.1).

• The Malliavin derivative associated to W is denoted by D for the R ^d -valued (resp. H-valued) random variables, and its domain in L ² (Ω) (resp. L ² (Ω; H )) is denoted by D ^1,2 (resp. D ^1,2 (H)) (see [22], Section 1.2).

• For the R ^d -valued random variables, the divergence operator associated to D is denoted by δ, and its domain in L ² (Ω; H) is denoted by dom(δ) (see [22], Section 1.3).

• The vector space of the R ^d -valued (resp. H -valued) random variables locally derivable in the sense of Malliavin is denoted by D ^1,2 loc (resp. D ^1,2 loc (H )) (see [22], Subsection 1.3.5).

2. Regularity of the Itô map : existing and additional results

This section deals with the regularity of the Itô map associated to RDEs. On one hand, the results on the continuity and the differentiability of the Itô map with respect to the initial condition and to the driving signal coming from [10], Chapter 11 are stated. In addition, the continuous differentiability of the Itô map with respect to the collection of vector fields is proved. On the other hand, in order to apply the integrability results coming from [3], some tailor-made upper-bounds are provided for each derivative.

First, the existing continuity results of the Itô map and of the rough integral are synthesized.

Theorem 2.1. Consider R > 0 :

(1) The Itô map (X 0 , W , V ) 7→ π V (0, X 0 ; W ) is uniformly continuous from

G ^[p] ( R ^e ) × B p,T (1, R) × Lip ^γ ( R ^e ; R ^d ) into GΩ p,T ( R ^d ).

(2) The map

J : ( W , V ) 7−→

Z

V ( W )d W is uniformly continuous from

B p,T (1, R) × Lip ^γ−1 ( R ^d ; R ^d ) into GΩ p,T ( R ^d ).

In each case, the uniform continuity holds true if B p,T (1, R) and GΩ p,T ( R ^d ) are equipped with the uniform distance d

.

See [10], corollaries 10.39,40,48 for a proof.

Remark. Consider x 0 ∈ R ^e , W ∈ GΩ p,T ( R ^d ) and V := (V 1 , . . . , V d ) a collection of affine vector fields on R ^e . By [10], Theorem 10.53, π V (0, x 0 ; W ) t belongs to the ball B(0; R(x 0 , W )) of R ^e for every t ∈ [0, T ], where

R(x ₀ , W ) := C(1 + kx ₀ k)e ^CkWk

and C > 0 is a constant not depending on x ₀ and W . Moreover, for every x ˜ ₀ ∈ R ^e and every f W ∈ GΩ _p,T ( R ^d ), k x ˜ 0 k 6 kx 0 k and kf W k p-var;T 6 k W k p-var;T = ⇒ R(˜ x 0 , f W ) 6 R(x 0 , W ).

So, if V ˆ ∈ Lip ^γ ( R ^e ; R ^d ) is the collection of vector fields satisfying V ˆ ≡ V on B(0; R(x 0 , W )), then π V (0, .) ≡ π V ˆ (0, .) on the set B(0, kx 0 k) × B p,T (1, k W k p-var;T ).

Therefore, by Theorem 2.1, the map π V (0, .) is uniformly continuous from

B(0, kx 0 k) × B p,T (1, k W k p-var;T ) into C ^p-var ([0, T ]; R ^e ).

The uniform continuity holds true if B _p,T (1, k W k _p-var;T ) and C ^p-var ([0, T ]; R ^e ) are equipped with the uniform distance d

.

The following technical corollary of [10], Theorem 9.26 allows to apply the integrability results of [3] to differ- ential equations having a drift term.

Corollary 2.2. Consider p > q > 1 such that 1/p + 1/q > 1, Y ∈ GΩ _p,T ( R ^d ), h ∈ GΩ _q,T ( R ^e ) and ε > 0. There exists a constant C > 0, depending only on p and q, such that :

M ε,I,p [S [p] (Y ⊕ h)] 6 C[khk ^p _q-var;T + M ε,I,p (Y )].

Proof. On one hand, for every (s, t) ∈ ∆ ¯ _T ,

ω Y,p (s, t) = kY k p-var;s,t 6 kS _[p] (Y ⊕ h)k p-var;s,t .

On the other hand, since p/q > 1 and, ω _Y,p and ω _h,q are two controls :

ω = kY k ^p _p-var + khk ^p _q-var = ω _Y,p + ω ^p/q _h,q

is also a control.

Then, by [10], Proposition 7.52, there exists a constant C > 1, depending only on p and q, such that for every (s, t) ∈ ∆ ¯ _T ,

S _[p] (Y ⊕ h)

p

p-var;s,t 6 Cω(s, t).

In conclusion,

M _ε,I,p

S _[p] (Y ⊕ h)

6 C sup

D={r_k} ∈ D_I ω r_k, r_k+1

6 ε

|D|−1

X

k=1

ω(r _k , r _k+1 )

6 C h

khk ^p _q-var;T + M ε,I,p (Y ) i

by the super-additivity of the control ω ^p/q _h,q .

2.1. Differentiability of the Itô map with respect to x 0 and V

In order to prove the continuous differentiability of the Itô map of RDEs with respect to the collection of vector fields, it has to be shown for ODEs first.

Proposition 2.3. Consider γ > 1, x ₀ ∈ R ^e and a continuous function w : [0, T ] → R ^d of finite 1-variation.

The map V 7→ π _V (0, x ₀ ; w) is continuously differentiable from

Lip ^γ ( R ^e ; R ^d ) into C ^1-var ([0, T ]; R ^e ).

Proof. In a first step, the derivability of the Itô map with respect to the collection of vector fields is established at every points and in every directions of Lip ^γ ( R ^d ; R ^e ). In a second step, via [10], Proposition B.5, the contin- uous differentiability of the partial Itô map is proved.

Step 1. Consider V, V ˜ ∈ Lip ^γ ( R ^e ; R ^d ), ε ∈]0, 1], x ^V := π _V (0, x ₀ ; w) and y ^{V, V ˜} the solution of the following ODE :

y ^V, _t ^{V ˜} = Z t

0

hDV (x ^V _s ), y _s ^{V, V ˜} idw s + Z t

0

V ˜ (x ^V _s )dw s . (1)

For every t ∈ [0, T ],

x ^V _t ^{+ε V ˜} − x ^V _t

ε − y _t ^{V, V ˜} = Z t

0

"

V (x ^V _s ^{+ε V ˜} ) − V (x ^V _s )

ε − hDV (x ^V _s ), y _s ^{V, V ˜} i

# dw s +

Z t 0

h V ˜ (x ^V _s ^{+ε V ˜} ) − V ˜ (x ^V _s ) i dw _s

= P t (ε) + Q t (ε) + R t (ε) where,

P _t (ε) := ε

Z t

0

h

V (x ^V _s ^{+ε V ˜} ) − V (x ^V _s ) − hDV (x ^V _s ), x ^V _s ^{+ε V ˜} − x ^V _s i i dw _s ,

Q t (ε) :=

Z t 0

h V ˜ (x ^V _s ^{+ε V ˜} ) − V ˜ (x ^V _s ) i

dw s and R t (ε) :=

Z t 0

hDV (x ^V _s ), ε

(x ^V _s ^{+ε V ˜} − x ^V _s ) − y ^V, _s ^{V ˜} idw s .

Firstly, since V is continuously differentiable on R ^e , by [10], Lemma 4.2 : kP t (ε)k 6 ε

kwk 1-var;T sup

t∈[0,T]

kV (x ^V _t ^{+ε V ˜} ) − V (x ^V _t ) − hDV (x ^V _t ), x ^V _t ^{+ε V ˜} − x ^V _t ik

6 η(ε)ε

kwk 1-var;T kx ^{V +ε V ˜} − x ^V k

where, η(ε) → 0 when ε → 0.

By [10], Theorem 3.18 :

kP (ε)k

6 M ₃ (ε) := 2η(ε)e ^3M

^M

M ₂ k V ˜ k

kwk 1-var;T (2) with

M ₁ := kV k _lip

+ k V ˜ k _lip

> kV + ε V ˜ k _lip

∨ kV k _lip

and M ₂ := kwk 1-var;T .

Secondly, since V ˜ is continuously differentiable and of bounded derivative on R ^e , it is a collection of Lips- chitz continuous vector fields. Then, by [10], Theorem 3.18 :

kQ(ε)k

6 M 4 (ε) := 2εe ^3M

^M

M 2 k V ˜ k ² _lip

kwk 1-var;T . (3) Thirdly,

kR t (ε)k 6 kV k lip

Z t 0

x ^V _s ^{+ε V ˜} − x ^V _s ε − y ^V, _s ^{V ˜}

kdw s k. (4)

Therefore, by inequalities (2), (3) and (4) :

x ^V _t ^{+ε V ˜} − x ^V _t ε − y _t ^{V, V ˜}

6 M ₃ (ε) + M ₄ (ε) + kV k lip

Z t 0

x ^V _s ^{+ε V ˜} − x ^V _s

ε − y _s ^{V, V ˜}

kdw s k.

In conclusion, by the Gronwall lemma :

x ^{V +ε V ˜} − x ^V

ε − y ^{V, V ˜}

6 [M ₃ (ε) + M ₄ (ε)] e

−−−→ ε→0 0.

Step 2. The solution of Equation (1) satisfies :

D V ˜ x ^V = π _A (0, 0; .) ◦ J [F _V, V ˜ (.), .] ◦ (π _V (0, x ₀ ; .), .)(w)

where, A : R ^e → L(L( R ^e ) × R ^e ; R ^e ) and F _V, V ˜ : R ^e × R ^d → L( R ^e × R ^d ; L( R ^e ) × R ^e ) are two collections of vector fields, respectively defined by :

A(a)(L, b) := L.a + b and

F _V, V ˜ (a, a

)(b, b

) := (hDV (a), .ib

; ˜ V (a)b

)

for every a, b ∈ R ^e , a

, b

∈ R ^d and L ∈ L( R ^e ).

Firstly, by the second point of Theorem 2.1, the map J is uniformly continuous on every bounded sets of C ^1-var [0, T ]; R ^e × R ^d

× C ^1-var [0, T ]; R ^e × R ^d .

Secondly, the map (V, V , a) ˜ 7→ F _V, V ˜ (a) is uniformly continuous on every bounded sets of Lip ^γ R ^e ; R ^d

× Lip ^γ R ^e ; R ^d

× R ^e × R ^d

by construction.

Thirdly, the maps π A (0, 0; .) and V 7→ π V (0, x 0 ; w) are respectively uniformly continuous on every bounded sets of

C ^1-var ([0, T ]; L( R ^e ) × R ^e ) and Lip ^γ R ^e ; R ^d by Theorem 2.1 and its remark.

Therefore, by composition, the map (V, V ˜ ) 7→ D V ˜ x ^V is uniformly continuous on every bounded sets of Lip ^γ R ^e ; R ^d

× Lip ^γ R ^e ; R ^d .

In conclusion, by [10], Proposition B.5, the map V 7→ π V (0, x 0 ; w) is continuously differentiable from Lip ^γ ( R ^e ; R ^d ) into C ^1-var ([0, T ]; R ^e ).

Theorem 2.4. Consider W ∈ GΩ p,T ( R ^d ) :

(1) Let V := (V 1 , . . . , V d ) be a collection of γ-Lipschtiz vector fields on R ^e . The map x 0 7→ π V (0, x 0 ; W ) is continuously differentiable from

R ^e into C ^p-var ([0, T ]; R ^e ) .

For every t ∈ [0, T ], the Jacobian matrix of π V (0, .; W ) t at point x 0 ∈ R ^e is denoted by J _t←0 ^x

^,

.

Moreover, for every ε > 0, there exists a constant C 1 > 0 only depending on p, γ, ε and kV k

, such that for every x 0 ∈ R ^e ,

kJ _.←0 ^x

^,

k

6 C 1 e ^C

^M

⁽

⁾ .

(2) For every t ∈ [0, T ], J _t←0 ^x

^,

is an invertible matrix. Moreover, for every ε > 0, there exists a constant C 2 > 0 only depending on p, γ, ε and kV k

, such that for every x 0 ∈ R ^e ,

k(J _.←0 ^x

^,

)

k

6 C 2 e ^C

^M

⁽

⁾ .

(3) Consider x ₀ ∈ R ^e . The map V 7→ π _V (0, x ₀ ; W ) is continuously differentiable from Lip ^γ ( R ^e ; R ^d ) into C ^p-var ([0, T ]; R ^e ).

Moreover, for every R > 0 and V, V ˜ ∈ B

(0, R), there exists two constants η > 0 and C ₃ > 0, depending (continuously) on R but not on W , such that :

k∂ V π V (0, x 0 ; W ). V ˜ k

6 C 3 e ^C

^M

⁽

⁾ .

Proof. See [10], theorems 11.3-6 for a proof of the continuous differentiability of the Itô map with respect to the initial condition, and see [3], Corollary 3.4 about the upper-bound provided at the first point for kJ _.←0 ^x

^,

k

; x 0 ∈ R ^e .

Let I be the identity matrix of M e ( R ). The proofs of the points 1 and 2 are similar because if w : [0, T ] → R ^d is a continuous function of finite 1-variation, then

J _t←0 ^x

^,w = I + Z t

0

hDV [π V (0, x 0 ; W ) s ], J _s←0 ^x

^,w idw s and (J _t←0 ^x

^,w )

= I −

Z t 0

hDV [π _V (0, x ₀ ; W ) _s ], (J _s←0 ^x

^,w )

idw s

as mentioned at the proof of [10], Proposition 4.11.

The proof of the third point is detailed. In a first step, the continuous differentiability of the Itô map with respect to the collection of vector fields is proved. In a second step, in order to apply the integrability results coming from [3], a tailor-made upper-bound for the derivative of the Itô map with respect to V is provided.

Step 1. Since W ∈ GΩ p,T ( R ^d ), there exists a sequence (w ⁿ , n ∈ N ) of functions belonging to C ^1-var ([0, T ]; R ^d ) and satisfying :

n→∞ lim d _p-var;T

S _[p] (w ⁿ ) _0,. , W

= 0. (5)

Consider n ∈ N , W ⁿ := S _[p] (w ⁿ ) _0,. , x ₀ ∈ R ^e , a := (x ₀ , 0),

X 0 :=

1, a, . . . , a

[p]!

∈ T ^[p] R ^e+1

and V, V ˜ ∈ Lip ^γ ( R ^e ; R ^d ).

By Proposition 2.3, the map π . (0, x 0 ; w ⁿ ) is continuously differentiable from Lip ^γ ( R ^e ; R ^d ) into C ^1-var ([0, T ]; R ^e ).

In particular, ∂ V π V (0, x 0 ; w ⁿ ). V ˜ = ϕ( W ⁿ , V, V ˜ ) with

ϕ(., V, V ˜ ) := π _A (0, 0; .) ◦ J (.; F _V, V ˜ ) ◦ π _F

(0, X ₀ ; .)

where,

A : R ^e −→ L(L( R ^e ) × R ^e ; R ^e ),

F _V, V ˜ : R ^e × R ^d −→ L( R ^e × R ^d ; L( R ^e ) × R ^e ) and F V : R ^e −→ L( R ^d ; R ^e × R ^d )

are three collections of vector fields, respectively defined by : A(a)(L, b) := L.a + b,

F _V, V ˜ (a, a

)(b, b

) := (hDV (a), .ib

; ˜ V (a)b

) and

F _V (a)b

:= (V (a)b

, b

)

for every a, b ∈ R ^e , a

, b

∈ R ^d and L ∈ L( R ^e ).

Consider ε ∈]0, 1]. By the Taylor formula applied to π _. (0, x ₀ ; W ⁿ ) between V and V + ε V ˜ , and [10], Defi- nition 10.17 :

π _{V +ε} V ˜ (0, x 0 ; W ) − π V (0, x 0 ; W ) = lim

n→∞

Z ε 0

ϕ( W ⁿ , V + θ V , ˜ V ˜ )dθ (6) uniformly.

Via the Lebesgue theorem and [10], Proposition B.1, let show that the derivative of π . (0, x 0 ; W ) at point V , in the direction V ˜ , exists in C ^p-var ([0, T ]; R ^e ) equipped with the norm k.k p-var;T and coincides with ϕ( W , V, V ˜ ).

On one hand, by the continuity results of Theorem 2.1 :

∀θ ∈]0, 1], ϕ( W ⁿ , V + θ V , ˜ V ˜ ) −−−−→

n→∞ ϕ( W , V + θ V , ˜ V ˜ ) in C ^p-var ([0, T ]; R ^e ) equipped with k.k

.

On the other hand, by applying successively [10], theorems 10.47 and 10.36, for every θ ∈]0, 1] and every (s, t) ∈ ∆ ¯ T ,

ω ₁ ^1/p (s, t; n; θ) :=

Z

F _{V +θ} V , ˜ V ˜

h

π F

(0, X 0 ; W ⁿ ) i

dπ F

(0, X 0 ; W ⁿ ) _p-var;s,t 6 ω ^1/p ₂ (s, t; n)

with

ω ₂ ^1/p (s, t; n) := ω ^1/p ₃ (s, t; n) ∨ ω 3 (s, t; n) ∨ ω ₃ ^p (s, t; n) and

ω 3 (s, t; n) := η 1 k W ⁿ k ^p _p-var;s,t where η 1 > 0 is depending on V and V ˜ , but not on W ⁿ and θ.

By [10], Exercice 10.55, there exists a constant C 4 > 0, not depending on W ⁿ and θ, such that :

ϕ( W ⁿ , V + θ V , ˜ V ˜ )

6 C 4 exp





 C 4 sup

D={r_k} ∈ D_I ω₂ r_k, r_k+1;n

6 1

|D|−1

X

k=1

ω 2 (r k , r k+1 ; n)







= C ₄ exp





 C ₄ sup

D={r_k} ∈ D_I ω₃ r_k, r_k+1;n

6 1

|D|−1

X

k=1

ω ₃ (r _k , r _k+1 ; n)





 ,

because

ω 2 (.; n) ≡ ω 3 (.; n) when ω 2 (.; n) 6 1.

By the super-additivity of the control ω 3 (.; n) :

ϕ( W ⁿ , V + θ V , ˜ V ˜ )

6 C 4 e ^η

^C

.

In the right-hand side of that inequality, since η ₁ and C ₄ are not depending on W ⁿ and θ, and since sup

n∈

k W ⁿ k ^p _p-var;T < ∞

by (5) :

sup

θ∈[0,1]

sup

n∈N

ϕ( W ⁿ , V + θ V , ˜ V ˜ )

< ∞ in C ^p-var ([0, T ]; R ^e ) equipped with k.k

.

Therefore, by the Lebesgue theorem and Inequality (6) : π _{V +ε} V ˜ (0, x 0 ; W ) − π V (0, x 0 ; W ) =

Z ε 0

ϕ( W , V + θ V , ˜ V ˜ )dθ.

Since θ 7→ ϕ( W , V + θ V , ˜ V ˜ ) is continuous from

[0, 1] into C ^p-var ([0, T ]; R ^e ) (equipped with k.k p-var;T )

by Theorem 2.1 ; by [10], Proposition B.1, the derivative of π . (0, x 0 ; W ) at point V , in the direction V ˜ , exists in C ^p-var ([0, T ]; R ^e ) equipped with k.k p-var;T and coincides with ϕ( W , V, V ˜ ).

Finally, as at the second step of the proof of Proposition 2.3, via [10], Proposition B.5 and Lemma 4.2, the map π . (0, x 0 ; W ) is continuously differentiable from

Lip ^γ ( R ^e ; R ^d ) into C ^p-var ([0, T ]; R ^e ).

Step 2. Consider R > 0 and V, V ˜ ∈ B Lip

(0, R).

By applying successively [10], theorems 10.47 and 10.36, for every (s, t) ∈ ∆ ¯ _T , ω ₄ ^1/p (s, t) :=

Z

F _V, V ˜ [π F

(0, X 0 ; W )] dπ F

(0, X 0 ; W ) _p-var;s,t 6 ω ^1/p ₅ (s, t)

with

ω ^1/p ₅ (s, t) := ω ₆ ^1/p (s, t) ∨ ω 6 (s, t) ∨ ω ^p ₆ (s, t) and

ω 6 (s, t) := η 2 k W k ^p _p-var;s,t where η 2 > 0 is depending on R (continuously), but not on W .

By [10], Exercice 10.55, there exists a constant C 5 > 0, not depending on R and W , such that :

∂ V π V (0, x 0 ; W ). V ˜

6 C 5 exp





 C 5 sup

D={r_k} ∈ D_I ω₅ r_k, r_k+1

6 1

|D|−1

X

k=1

ω 5 (r k , r k+1 )







= C 5 exp





 C 5 sup

D={r_k} ∈ D_I ω6 r_k, r_k+1

6 1

|D|−1

X

k=1

ω 6 (r k , r k+1 )





 ,

because

ω 5 ≡ ω 6 when ω 5 6 1.

However,

sup

D={r_k} ∈ D_I ω₆ r_k, r_k+1

6 1

|D|−1

X

k=1

ω ₆ (r _k , r _k+1 ) = η ₂ M _η

2

,I,p ( W ).

Therefore,

∂ V π V (0, x 0 ; W ). V ˜

6 C 3 e ^C

^M

⁽

⁾

with C 3 := C 5 (1 ∨ η 2 ) and η := η ₂

.

Notations. In the sequel, the matrices J _t←0 ^x

^,

and (J _t←0 ^x

^,

)

will be respectively denoted by J _0←t

and J _t←0

for the sake of simplicity. Moreover, for every (s, t) ∈ ∆ ¯ T , put

J _s←t

:= J _s←0

J _0←t

and J _t←s

:= J _t←0

J _0←s

. Then,

J _s←t

J _t←s

= J _t←s

J _s←t

= I.

At the following corollary, the upper-bounds provided at the previous theorem are extended to RDEs having a drift term.

Corollary 2.5. Consider m ∈ N

, p > q > 1 such that 1/p + 1/q > 1, h : [0, T ] → R ^m a continuous function of finite q-variation, W ∈ GΩ p,T ( R ^d ) and W ^h := S _[p] ( W ⊕ h) :

(1) Let V := (V 1 , . . . , V d+m ) be a collection of γ-Lipschitz vector fields on R ^e . For every ε > 0, there exists a constant C 1 > 0 depending only on p, q, γ, ε and kV k

, such that for every x 0 ∈ R ^e ,

kJ _.←0

k

6 C ₁ exp h C ₁ h

khk ^p _q-var;T + M _ε,I,p ( W ) ii .

(2) Consider x ₀ ∈ R ^e . For every R > 0 and V, V ˜ ∈ B

(0, R), there exists two constants ε > 0 and C ₂ > 0, depending on R but not on h and W , such that :

k∂ _V π _V (0, x ₀ ; W ^h ). V ˜ k

6 C ₂ exp h C ₂ h

khk ^p _q-var;T + M _ε,I,p ( W ) ii .

Proof. By Corollary 2.2, there exists a constant C 3 > 0, depending only on p and q, such that for every ε > 0, M ε,I,p ( W ^h ) 6 C 3

h khk ^p _q-var;T + M ε,I,p ( W ) i .

Therefore, by Theorem 2.4 :

(1) Let V ∈ Lip ^γ ( R ^e ; R ^d+m ) be arbitrarily chosen. For every ε > 0, there exists a constant C ₄ > 0 depending only on p, γ, ε and kV k _lip

, such that for every x ₀ ∈ R ^e ,

kJ _.←0

k

6 C ₄ e ^C

^M

⁽

⁾ 6 C 1 exp h

C 1

h khk ^p _q-var;T + M ε,I,p ( W ) ii

with C ₁ := C ₄ (1 ∨ C ₃ ).

(2) Let x ₀ ∈ R ^e be arbitrarily chosen. For every R > 0 and V, V ˜ ∈ B _Lip

(0, R), there exists two constants ε > 0 and C 5 > 0, depending on R but not on W ^h , such that :

k∂ V π V (0, x 0 ; W ^h ). V ˜ k

6 C 5 e ^C

^M

⁽

⁾ 6 C ₂ exp h

C ₂ h

khk ^p _q-var;T + M _ε,I,p ( W ) ii

with C ₂ := C ₅ (1 ∨ C ₃ ).

2.2. Differentiability of the Itô map with respect to the driving signal

First of all, the notion of differentiability introduced by P. Friz and N. Victoir on GΩ p,T ( R ^d ) is reminded.

Definition 2.6. Consider a Banach space F, p > q > 1 such that 1/p + 1/q > 1, and an open set U of GΩ p,T ( R ^d ). The map ϕ : GΩ p,T ( R ^d ) → F is continuously differentiable in the sense of Friz-Victoir on U if and only if, for every Y ∈ U , the map

h ∈ C ^q-var ([0, T ]; R ^d ) 7−→ ϕ(T _h Y ) ∈ F is continuously differentiable.

With the notations of Definition 2.6, if ϕ is continuously differentiable from U into F in the sense of Friz-Victoir, then

∀Y ∈ U , ψ ^Y : h ∈ C ^q-var ([0, T ]; R ^d ) 7−→ ψ ^Y (h) = ϕ(T _h Y ) is derivable at every points and in every directions of C ^q-var ([0, T ]; R ^d ).

Notation. For every continuous function h : [0, T ] → R ^d of finite q-variation, D _h ^FV ϕ(Y ) := D _h ψ ^Y (0)

= lim

ε→0

ϕ(T _εh Y ) − ϕ(T ₀ Y )

ε .

In the sequel, D ^FV is called the Friz-Victoir (directional) derivative operator.

Theorem 2.7. Consider a collection V := (V 1 , . . . , V d ) of γ-Lipschitz vector fields on R ^e and x 0 ∈ R ^e . The map W 7→ π V (0, x 0 ; W ) is continuously differentiable from

GΩ _p,T ( R ^d ) into C ^p-var ([0, T ]; R ^e ) in the sense of Friz-Victoir.

Moreover, for every W ∈ GΩ p,T ( R ^d ) and every continous function h : [0, T ] → R ^d of finite q-variation, D _h ^{F V} π V (0, x 0 ; W ) =

Z . 0

J _.←s

V [π V (0, x 0 ; W ) s ] dh s .

(Duhamel principle).

Consider W ∈ GΩ p,T ( R ^d ) and a control ω : ¯ ∆ T → R + satisfying :

∀(s, t) ∈ ∆ ¯ T , k W k p-var;s,t 6 ω ^1/p (s, t).

(1) There exists a constant C ₁ > 0, not depending on W and ω, such that for every continous function h : [0, T ] → R ^d of finite q-variation,

kD

_h π V (0, x 0 ; W )k

6 C 1 exp h

C 1 (khk ^p _q-var;T + M 1,I,ω ) i .

(2) There exists a constant C 2 > 0, not depending on W and ω, such that for every continous function h : [0, T ] → R ^d of finite q-variation,

kD _h

π V (0, x 0 ; W )k p-var;T 6 C 2 exp h C 2

h khk ^p _q-var;T + ω(0, T ) ii .

Proof. See [10], theorems 11.3-6 and Exercice 11.9 for a proof of the first part.

Consider a continuous function h : [0, T ] → R ^d of finite q-variation, W ^h := S _[p] ( W ⊕ h), a := (x ₀ , 0, 0) and

X 0 :=

1, a, . . . , a

[p]!

∈ T ^[p] R ^e+2 .

By [10], Theorem 11.3 :

D ^FV _h π V (0, x 0 ; W ) = π A (0, 0; .) ◦ J (., F ) ◦ π G (0, X 0 ; .)( W ^h ) where,

A : R ^e −→ L(L( R ^e ) × R ^e ; R ^e ),

F : R ^e × R ^d × R ^d −→ L( R ^e × R ^d × R ^d ; L( R ^e ) × R ^e ) and G : R ^e −→ L( R ^d × R ^d ; R ^e × R ^d × R ^d ) are three collections of vector fields, respectively defined by :

A(a)(L, b) := L.a + b,

F(a, a

, a

)(b, b

, b

) := (hDV (a), .ib

; V (a)b

) and G(a)(b

, b

) := (V (a)b

, b

, b

)

for every a, b ∈ R ^e , a

, b

, a

, b

∈ R ^d and L ∈ L( R ^e ).

By applying successively [10], theorems 10.47 and 10.36, for every (s, t) ∈ ∆ ¯ T , ω ^1/p ₁ (s, t) :=

Z

F

π G (0, X 0 ; W ^h )

dπ G (0, X 0 ; W ^h ) _p-var;s,t 6 ω ^1/p ₂ (s, t)

with

ω ^1/p ₂ (s, t) := ω ₃ ^1/p (s, t) ∨ ω 3 (s, t) ∨ ω ^p ₃ (s, t) and, by [10], Proposition 7.52 :

ω 3 (s, t) := ε 1

khk ^p _q-var;s,t + ω(s, t)

> ε 2 k W ^h k ^p _p-var;s,t

where, ε ₁ , ε ₂ > 1 are two constants not depending on W , ω and h.

On one hand, by [10], Exercice 10.55, there exists a constant C ₃ > 0, not depending on W , ω and h, such that :

D ^FV _h π V (0, x 0 ; W )

6 C 3 exp





 C 3 sup

D={r_k} ∈ D_I ω2 r_k, r_k+1

6 1

|D|−1

X

k=1

ω 2 (r k , r k+1 )







= C ₃ exp





 C ₃ sup

D={r_k} ∈ D_I ω₃ r_k, r_k+1

6 1

|D|−1

X

k=1

ω ₃ (r _k , r _k+1 )







6 C 1 exp h C 1

khk ^p _q-var;T + M 1,I,ω

i

with C 1 := C 3 ε 1 , because

ω ₂ ≡ ω ₃ when ω ₂ 6 1 (7)

and

∀(s, t) ∈ ∆ ¯ T , ω(s, t) 6 ω 3 (s, t).

On the other hand, by [10], Theorem 10.53, there exists a constant C 4 > 0, not depending on W , ω and h, such that for every (s, t) ∈ ∆ ¯ T satisfying ω 2 (s, t) 6 1,

kD _h ^FV π V (0, x 0 ; W ) s,t k 6 C 4

1 + kD _h ^FV π V (0, x 0 ; W ) s k

ω ₂ ^1/p (s, t)e ^C

^ω

^(s,t) 6 C 4

1 + kD _h ^FV π V (0, x 0 ; W )k

ω ^1/p ₃ (s, t)e ^C

^ω

^(0,T)

by (7).

Therefore, by the super-additivity of the control ω 3 , there exists a constant C 2 > 0, not depending on W , ω and h, such that :

kD _h ^FV π _V (0, x ₀ ; W )k p-var;T 6 C ₂ exp h C ₂ h

khk ^p _q-var;T + ω(0, T ) ii .

At the following corollary, the upper-bounds provided at the previous theorem are extended to RDEs having a drift term.

Corollary 2.8. Consider m ∈ N

, p > q > 1 such that 1/p + 1/q > 1, r ∈ [1, p[ such that 1/p + 1/r > 1, g : [0, T ] → R ^m a continuous function of finite r-variation, W ∈ GΩ p,T ( R ^d ), W ^g := S _[p] ( W ⊕ g), V :=

(V ₁ , . . . , V _d+m ) a collection of γ-Lipschitz vector fields on R ^e and x ₀ ∈ R ^e . There exists a constant C > 0, not depending on g and W , such that for every continuous function h : [0, T ] → R ^d+m of finite q-variation,

kD

_h π V (0, x 0 ; W ^g )k

6 C exp h C h

khk ^p _q-var;T + kgk ^p _r-var;T + M 1,I,p ( W ) ii .

Proof. Let h : [0, T ] → R ^d+m be a continuous function of finite q-variation. By Corollary 2.2, there exists a constant C 1 > 0, depending only on p and r, such that :

M 1,I,p ( W ^g ) 6 C 1

h kgk ^p _r-var;T + M 1,I,p ( W ) i

.

Then, by Theorem 2.7, there exists a constant C ₂ > 0, not depending on W ^g and h, such that : kD ^FV _h π _V (0, x ₀ ; W ^g )k

6 C ₂ exp h

C ₂ h

khk ^p _q-var;T + M _1,I,p ( W ^g ) ii 6 C exp h

C h

khk ^p _q-var;T + kgk ^p _r-var;T + M _1,I,p ( W ) ii

with C := C ₂ (1 ∨ C ₁ ).

2.3. Application to the Gaussian stochastic analysis

Consider a d-dimensional stochastic process W and the probability space (Ω, A, P ), where Ω is the vector space of continuous functions from [0, T ] into R ^d , A is the σ-algebra generated by cylinder sets of Ω, and P is the probability measure induced by the process W on (Ω, A).

In order to prove Corollary 2.15 which is crucial at Section 3, the existing results on Gaussian rough paths proved by P. Friz and N. Victoir in [9], and by T. Cass, C. Litterer and T. Lyons in [3] have to be stated first.

Consider the two following technical assumptions on the stochastic process W .

Assumption 2.9. W is a d-dimensionnel centered Gaussian process with continuous paths. Moreover, its covariance function c _W is of finite 2D ρ-variation with ρ ∈ [1, 2[ (see [10], Definition 5.50).

Assumption 2.10. There exists p > q > 1 such that : 1

p + 1

q > 1 and H ¹ , → C ^q-var ([0, T ]; R ^d ).

Example. By [10], Proposition 15.5, Proposition 15.7 and Exercice 20.2, the fractional Brownian motion of Hurst parameter H ∈]1/4, 1/2] satisfies assumptions 2.9 and 2.10.

Theorem 2.11. Consider a stochastic process W satisfying Assumption 2.9, and p > 2ρ. For almost every ω ∈ Ω, there exists a geometric p-rough path W (ω) over W (ω) satisfying :

(1) There exists a deterministic constant C > 0, only depending on ρ, p and kc W k ρ-var;[0,T ]

, such that : E

e ^CkWk

< ∞.

(generalized Fernique theorem).

(2) Let (W ⁿ , n ∈ N ) be a sequence of linear approximations, or of mollifier approximations, of the process W . W is the limit in p-variation, in L ^r (Ω) for every r > 1, of the sequence (S 3 (W ⁿ ), n ∈ N ) (universality).

W is the enhanced Gaussian process over W . See [10], Theorem 15.33 for a proof.

Proposition 2.12. Consider a stochastic process W satisfying assumptions 2.9 and 2.10, W the enhanced Gaussian process over W , and the Cameron-Martin’s space H ¹ ⊂ Ω of the process W . Then,

∀ω ∈ Ω, ∀h ∈ H ¹ , W (ω + h) = T h W (ω).

See [10], Lemma 15.58 for a proof.

Proposition 2.13. For every geometric p-rough path Y and every ε > 0,

M ε,I,p (Y ) 6 ε [2N ε,I,p (Y ) + 1] .

See [3], Proposition 4.6 for a proof.

Theorem 2.14. Consider a stochastic process W satisfying assumptions 2.9 and 2.10, and W the enhanced Gaussian process over W . Then,

∀C, ε, r > 0, Ce ^CN

⁽

⁾ ∈ L ^r (Ω).

See [3], Theorem 6.4 and Remark 6.5 for a proof.

Corollary 2.15. Consider x ₀ ∈ R ^e , V := (V ₁ , . . . , V _d+1 ) and V ˜ := ( ˜ V ₁ , . . . , V ˜ _d+1 ) two collections of γ-Lipschitz vector fields on R ^e , a stochastic process W satisfying assumptions 2.9 and 2.10, W the enhanced Gaussian process over W , W ^g := S _[p] ( W ⊕ g) with g := Id _[0,T] , and a continuous function h : [0, T ] → R ^d+1 of finite q-variation.

kJ _.←0

k

, k∂ V π V (0, x 0 ; W ^g ). V ˜ k

and kD _h

π V (0, x 0 ; W ^g )k

belong to L ^r (Ω) for every r > 0.

Proof. It is a straightforward consequence of corollaries 2.5 and 2.8, of Proposition 2.13 (deterministic results),

and of Theorem 2.14 (probabilistic result).

3. Sensitivity analysis of Gaussian rough differential equations

This section solves the problem stated in the introduction of the paper by using the deterministic results on RDEs of subsections 2.1 and 2.2, the probabilistic results on Gaussian RDEs of Subsection 2.3 and the Malliavin calculus.

Assume that W , µ and σ defined in the introduction satisfy the following assumption.

Assumption 3.1. The process W satisfies assumptions 2.9 and 2.10, and C ₀ ¹ [0, T ]; R ^d

⊂ H ¹ .

Moreover, there exists a constant C > 0 such that :

∀h ∈ C ₀ ¹ [0, T ]; R ^d

, khk _H

6 Ck hk ˙

.

The functions µ and σ satisfy Assumption 1.3 and, for every a ∈ R ^d , σ(a) is an invertible matrix. Moreover, the function σ

: R ^d → M d ( R ) is bounded.

Example. The fractional Brownian motion B ^H of Hurst parameter H ∈]1/4, 1[ satisfies Assumption 3.1.

Indeed, it has been stated at Subsection 2.3 that B ^H satisfies assumptions 2.9 and 2.10. Moreover, by the first point of L. Decreusefond and S. Ustunel [5], Theorem 3.3 :

C ₀ ¹ [0, T ]; R ^d

⊂ H ¹ .

Consider h ∈ C ₀ ¹ ([0, T ]; R ^d ). By the second point of [5], Theorem 3.3 : khk H

= kJ H ( ˙ h)k H

= k hk ˙ _L

([0,T])

6 T ^1/2 k hk ˙

.

Assume also that the function F : R ^d → R satisfies the following assumption.

Assumption 3.2. The function F is continuously differentiable from R ^d into R . Moreover, there exists two constants C > 0 and N ∈ N

such that, for every a ∈ R ^d ,

|F(a)| 6 C(1 + kak) ^N and kDF (a)k

6 C(1 + kak) ^N .

The following results are solving, at least partially, the problem stated in the introduction of the paper.

Notations :

• Under Assumption 3.1, the enhanced Gaussian process over W is denoted by W , W ^g := S _[p] ( W ⊕ g) with g := Id [0,T] , and V := (V ₁ , . . . , V _d+1 ) is the collection of vector fields defined by :

V (a)(b, c) := µ(a)c + σ(a)b for every a, b ∈ R ^d and c ∈ R .

• Let S p ⊂ Lip ^γ ( R ^d ; R ^d ) be the space of functions from R ^d into M d ( R ), [p] + 1 times differentiable, bounded and of bounded derivatives.

• For every x ∈ R ^d , E [F (X _T )] is denoted by f _T (x, σ).

Lemma 3.3. Let I = (I ¹ , . . . , I ^d ) be the map from H into H ¹ such that : I ⁱ (h) := E

W ⁱ (h ⁱ )W ⁱ

∈ H ¹

for every h ∈ H := H 1 ⊕ · · · ⊕ H d and i = 1, . . . , d. I is an isometry from H into H ¹ .

Proof. On one hand, the linearity of I : H → H ¹ is a straightforward consequence of the linearity of W : H → L ² (Ω).

On the other hand, by construction of W and of the scalar products on H and H ¹ :

hI(h), I(g)i H

=

d

X

i=1

h E

W ⁱ (h ⁱ )W ⁱ

; E

W ⁱ (g ⁱ )W ⁱ i _H

i

=

d

X

i=1

E

W ⁱ (h ⁱ )W ⁱ (g ⁱ )

= hh, gi H

for every functions h, g ∈ H .

The following lemma extends [10], Proposition 20.5 to Gaussian RDEs having a drift term.

Lemma 3.4. For every x ₀ ∈ R ^d and almost every ω ∈ Ω, the map h 7→ π _V [0, x ₀ ; W ^g (ω + h)] is continuously differentiable from

H ¹ into C ^p-var [0, T ]; R ^d . For every t ∈ [0, T ], π V (0, x 0 ; W ^g ) t ∈ D ^1,2

and for every h ∈ H ¹ ,

hDπ V (0, x 0 ; W ^g ) t , I

(h)i H = D

_(h,0) π V (0, x 0 ; W ^g ) t

= Z t

0

J _t←s

σ [π _V (0, x ₀ ; W ^g ) _s ] dh _s .

Proof. By Proposition 2.12, for almost every ω ∈ Ω and every h ∈ H ¹ , W ^g (ω + h) = S _[p] [ W (ω + h) ⊕ g]

= S _[p] [T _h W (ω) ⊕ g]

= T _(h,0) S _[p] [ W (ω) ⊕ g]

= T _(h,0) W ^g (ω).