A mixed-step algorithm for the approximation of the stationary regime of a diffusion

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A mixed-step algorithm for the approximation of the stationary regime of a diffusion

Gilles Pagès, Fabien Panloup

To cite this version:

Gilles Pagès, Fabien Panloup. A mixed-step algorithm for the approximation of the stationary regime of a diffusion. Stochastic Processes and their Applications, Elsevier, 2014, vol. 124 (1), p. 522-565.

�10.1016/j.spa.2013.07.011�. �hal-00756056v2�

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

Gilles Pag`es ^∗ , Fabien Panloup ^† March 8, 2013

Abstract

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called mixed-step scheme where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson-Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we end by giving some sufficient

“asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.

Keywords: stochastic differential equation; stationary process; steady regime; ergodic diffusion; Cen- tral Limit Theorem; Euler scheme; Poisson equation ; Richardson-Romberg extrapolation.

AMS classification (2000): 60G10, 60J60, 65C05, 65D15, 60F05.

1 Introduction

In a series of papers ([11, 13, 19, 20, 17, 18]) going back to [10], have been investigated the prop- erties of an Euler scheme with decreasing step as a tool for the numerical approximation of the [steady/stationary] regime of a diffusion, possibly with jumps, satisfying some [stability/mean revert- ing] conditions. The purpose of the present paper is to propose and investigate a variant of the original procedure sharing the similar properties in terms of convergence and rate but with a lower complexity, especially in its functional form, i.e. when trying to compute the expectation of a functional of the process (over a finite time interval [0, T ]) with respect to the stationary distribution of the process.

∗

Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires, UMR 7599, UPMC, Case 188, 4 pl. Jussieu, F-75252 Paris Cedex 5, France, E-mail: [email protected]

†

Institut de Math´ ematiques de Toulouse, Universit´ e Paul Sabatier & INSA Toulouse, 135, av. de Rangueil, F-31077

Toulouse Cedex 4, France, E-mail: [email protected]

In this paper we will focus on the case of Brownian diffusions. We consider an R ^d -valued diffusion process (X _t ) solution to

(SDE) ≡ dX _t = b(X _t )dt + σ(X _t )dW _t , (1.1) where (W _t ) _t≥0 is a q-dimensional standard Brownian motion (SBM ) and the coefficients b and σ are Lipschitz continuous functions from R ^d to R ^d and M _d,q respectively (where M _d,q denotes the set of d × q-matrices). Under these assumptions, strong existence and uniqueness hold for the SDE starting from any R ^d -valued r.v. independent of W and (X _t ) _t≥0 is a homogeneous Markov process with semi- group (P _t ) _t≥0 . We will denote by P _µ , the distribution of the whole process (X _t ) _t≥0 (supported by the set C ( R ₊ , R ^d ) of continuous functions from R ₊ to R ^d ) when starting from X ₀ with distribution µ. We also assume throughout the paper that (X _t ) _t≥0 has a unique invariant distribution ν . Except in the one-dimensional case, ν is usually not explicit and the numerical computation of ν or P _ν (which, in particular, is fundamental, to estimate the asymptotic behavior of ergodic processes) then requires some specific numerical methods.

Let us briefly describe the discrete and continuous time Euler schemes with decreasing step result- ing from the time discretization of the diffusion (X _t ) _{t ≥ 0} . First we introduce a non-decreasing sequence (Γ _n ) _n≥1 sequence of discretization times starting from Γ ₀ = 0 and we assume that the step sequence defined as its increments by γ _n := Γ _n − Γ _n−1 , n ≥ 1, is nonincreasing and satisfies

n → lim + ∞ γ _n = 0 and Γ _n = X n

k=1

γ _k −−−−−→ ^n→+∞ + ∞ . (1.2) The discrete time Euler scheme ( ¯ X _Γ

) _n≥0 (with Brownian increments) is recursively defined at dis- cretization times Γ _n by ¯ X ₀ = x ₀ and

X ¯ _Γ

= ¯ X _Γ

+ γ _n+1 b( ¯ X _Γ

) + σ( ¯ X _Γ

)(W _Γ

− W _Γ

). (1.3) If we introduce the notation

t = Γ _N(t) with N (t) = min { n ≥ 0, Γ n+1 > t } (1.4) so that t = Γ _k if and only if t ∈ [Γ _k , Γ _k+1 ), the stepwise constant Euler scheme also reads

∀ t ∈ R ₊ , X ¯ _t = ¯ X _t .

The idea at the origin of [10] was to make the guess, mimicking the pointwise ergodic theorem, that the weighted empirical measure

ν _n (ω, dx) = 1 Γ _n

X n

k=1

γ _k δ X ¯

(ω) (dx) (1.5)

or more generally

ν _n ^η (ω, dx) = 1 H n

X n

k=1

η _k δ X ¯

(ω) (dx)

where (η n ) n ≥ 1 is a sequence of positive weights such that H n = η 1 + · · · + η n → ∞ would weakly

converge a.s. to the invariant distribution ν under appropriate assumptions on both step and weight

sequences (γ _n ) _{n ≥ 1} and (η _n ) _{n ≥ 1} and on the drift b and σ. Typically, some mean-reverting assumptions

stated through the existence of an ”essentially quadratic” twice differentiable coercive function V : R ^d → (0; + ∞ ) for which there exists a ∈ (0, 1) such that

A V ≤ β − αV ^a

where A stands for the infinitesimal generator of the diffusion. In fact, as far as the weighted empirical measures of the Euler scheme are concerned, slightly more stringent conditions are required.

The a.s weak convergence of ν n ^η toward ν as well as its rate of convergence (CLT or a.s depending on the step rate of decay) has been extensively investigated in the above cited references, including much more general setting than Brownian diffusions. Basically when the step goes to 0 fast enough the empirical measure behaves like the empirical measure of the diffusion itself and satisfies the standard CLT (see e.g. [2] and [18] for results on the diffusion itself). When the convergence of (γ _n ) to 0 is too slow, the whole procedure is slowed down and satisfies an a.s. (or at least in probability) rate of convergence property. The critical rate is obtained with γ _n = n ⁻

with a biased CLT at rate n

.

Then, motivated by problems arising in Finance, pricing of exotic derivatives (Asian and barrier options) in stationary (stochastic) volatility models, this approach has been extended to functionals and led to investigate (when η _n = γ _n ) the behaviour of the empirical measures (ν ^(n),T ( ¯ X(ω), dα)) or (ν ^(n),T (ξ(ω), dα)) where ¯ X and ξ denote respectively the stepwise constant and continuous time Euler schemes and, for a c` adl` ag function Y : R ₊ → R ^d , ν ^(n),T (Y, dα) is defined on the set of probability measures on the Skorokhod set D ([0, T ], R ^d ), T > 0, denoting a fixed horizon,

ν ^(n),T (Y, dα) = 1 Γ _n

X n

k=1

γ _k δ _Y

(dα), n ≥ 1 where, by definition, for every c` adl` ag function α ∈ D ( R ₊ , R ^d )

α ^(t),T (s) = α(t + s), s ∈ [0, T ].

What we call continuous time Euler scheme here (also sometimes known in the literature as the genuine Euler scheme) is the process (ξ _t ) _t≥0 defined as interpolation of the stepwise constant Euler scheme, each driving term (time and Brownian motion) being interpolated in its own scale:

∀ n ∈ N , ∀ t ∈ [Γ _n , Γ _n+1 ), ξ _t = ¯ X _Γ

+ (t − Γ _n )b( ¯ X _Γ

) + σ( ¯ X _Γ

)(W _t − W _Γ

). (1.6) A convenient and synthetic form for the genuine Euler scheme is to write it as an Itˆo process satisfying the following pseudo-diffusion equation

ξ _t = X ₀ + Z t

0

b(ξ _s )ds + Z t

0

σ(ξ _s )dW _s . (1.7)

Such an approximation looks more accurate than the former one, especially when dealing with func-

tionals of the process, as it has been emphasized – in the constant step framework – in the literature

on several problems related to the Monte Carlo estimation of (a.s. continuous) functionals of a dif-

fusion (with a finite horizon) (see e.g. [4], Chapter 5). This follows from the classical fact that the

L ^p -convergence rate of this scheme for the sup norm is √ γ instead of √ γ log(1/γ) for its stepwise

constant counterpart (where γ stands for the step). On the other hand, the simulation of a functional

of (ξ _t ) _{t ∈ [τ,τ+T ]} is deeply connected with the simulation of functionals of the Brownian bridge so that

it is only possible for few specific such functionals (like running maxima, running means,. . . ).

These empirical measures on path spaces have in turn been extensively investigated in [17, 18].

However, their practical implementation suffers in practice from a significant drawback: one cannot prevent the complexity of the computation of F ( ¯ X ^(Γ

^),T ) to go to infinity as n grows since more and more iterations of the Euler schemes are needed to “cover” a laps of time equal to T . In practice, within the usual range of accuracy requested, no storing difficulty has been encountered. Furthermore in many standard situations where the functional F only involves running maxima or time integrals some recursive procedures make possible not to store the whole path of “length” T . It remains that the computational cost remains high.

As concerns the computation of the (marginal) invariant distribution, this led us naturally to consider a mixed step procedure where the decreasing step Euler scheme would be treated as if it were a constant step Euler scheme with step T > 0. This means, to implement and investigate the convergence of the sequence of empirical measures

¯

µ _n (ω, dx) = 1 n

n − 1

X

k=0

δ _ξ

_(ω) (dx) , n ≥ 1,

having in mind that the genuine Euler scheme can always be simulated at a locally finite number of additional times. Our aim is to prove that its empirical measure a.s. weakly converges toward ν under the same assumptions as ν _n with the same rate structure (however an additional uniqueness assumption of ν with respect to P _T is requested as one could expect). Of course the asymptotic variance/bias will differ: as for ν _n it was related to the continuous Poisson equation f − ν(f ) = −A ϕ while in this new framework, it will be related to the Poisson equation involving the pseudo-infinitesimal generator f − ν(f) = ^{Id −} _T ^P

(ϕ). Furthermore, the fact that we only “sample” the scheme at times nT will imply a control of the discretization error on intervals of length T . More precisely, we will rely here on an extension of the classical weak error results “` a la Talay-Tubaro” (see [24]) in our decreasing step framework, also known as the P DE method. This part is self-contained since we will directly establish by probabilistic methods the spatial regularity of the objects to be plugged in the P DE.

We will also investigate the functional counterpart of ¯ µ _n (ω, dx), denoted ¯ µ ⁽ⁿ⁾ (ω, dα), and defined on (the Borel σ-field of) C ( R ₊ , R ^d ) by

∀ n ∈ N , µ ¯ ⁽ⁿ⁾ (ω, dα) = 1 n

n−1 X

k=0

δ _ξ

(ω) (dα) where for any t > 0, (ξ ^(t) _s ) _{s ≥ 0} is defined by ξ _s ^(t) = ξ _t+s .

The second aim of the paper is to illustrate how to implement a specific Romberg extrapolation method in this framework, in order to “erase” partially the slowing effect due to the time discretization of the underlying diffusion. We will show that, provided the decreasing step sequence and the constant pseudo-step T are consistent, this allows for a significant speeding-up of the procedure (as well as an extension of the range of the CLT as a rule for the convergence rate). Furthermore in order to control the asymptotic variance it seems natural to consider consistent Brownian increments for the two schemes, as emphasized in [16] in a constant step framework. However, we will see that the situation is more involved when dealing with long run behaviour. In fact, the best strategy depends on the long run behaviour of the duplicated diffusion system

( dX _t = b(X _t )dt + σ(X _t )dW _t

dX _t ^(θ) = b(X _t ^(θ) )dt + σ(X _t ^(θ) )(θdW _t + √

1 − θ ² df W _t )

where W f and W are independent and θ ∈ [ − 1, 1]. We will see that the strategy is actually optimal with θ = 1 (i.e. with consistent Brownian increments) but under an additional assumption: we will need to assume that the invariant measure of the duplicated diffusion (with ρ = 1) is unique (see Sections 3.4 and 3.5 for more details).

The paper is organized as follows. Section 2 is devoted to some additional notations and to some background on some existing results which are required for our study. In Section 3 are stated the main results of the paper including the functional case, the Romberg extrapolation and a brief review of some conditions of uniqueness of the invariant distribution of the above duplicated system. Section 4, 5 and 6 are devoted to the proof of the main theorems, including new results on the weak error.

Finally, in section 7, we give some asymptotic confluence conditions which ensure the existence of regular solutions to Poisson-type equations (see condition ( P k,T )).

2 Background and Notations

2.1 Notations h x, y i = P

i x _i y _i will denote the canonical inner product and | x | = p

h x, x i will denote Euclidean norm of a vector x ∈ R ^d . For every k ∈ N , we denote by C ^k ( R ^d ) the set of functions f : R ^d → R whose derivatives up to order k are continuous and by C ^k,α ( R ^d ) (α ∈ (0, 1]) the subset of C ^k ( R ^d ) such that for every k ₁ , . . . , k _d ∈ N such that k ₁ + . . . + k _d = k, ∂ ^k

x

··· x

is a (locally) α-H¨older continuous function.

Let A = [a ij ] ∈ M _d,q be an R -valued matrix with d rows and q columns. A ^∗ will denote the transpose of A, Tr(A) = P

i a _ii its trace and k A k := p

Tr(AA ^∗ ) = ( P

ij a ² _ij )

. If d = q, one writes A x ^⊗2 for x ^∗ Ax.

We denote by A the infinitesimal generator of (X _t ) _t≥0 defined for every f ∈ C ² ( R ^d ) by

∀ x ∈ R ^d , A f (x) = h∇ f (x), b(x) i + 1

2 Tr(σ ^∗ (x)D ² f (x)σ(x))

where for every x ∈ R ^d , D ² f (x) denotes the Hessian matrix of f defined by (D ² f) _ij (x) = ∂ _x ²

_x

f(x), i, j ∈ { 1, . . . , d } . We also denote by ( A ^(k) ) _{k ≥ 1} the sequence of operators recursively defined by A ⁽¹⁾ = A and

∀ f ∈ C ^2k ( R ^d ), A ^(k+1) f = A ( A ^(k) f )

as long as b and σ are smooth enough. As concerns the discretized process (ξ _t ) _t≥0 , we also introduce the following associated operators ¯ A ^(k) recursively defined for f ∈ C ^2k ( R ^d ), for every x ∈ R ^d and k ∈ N by,

A ¯ f (., x) = h∇ f (.), b(x) i + 1

2 Tr(σ ^∗ (x)D ² f (.)σ(x)) and, A ¯ ^(k+1) f (., x) = ¯ A ( ¯ A ^(k) f (., x)).

We denote by ( F t ) _t≥0 the usual augmentation of (σ(W _s , 0 ≤ s ≤ t)) _t≥0 by P -negligible sets. For t ≥ 0, we also set E _t [ . ] = E [ . |F t ].

When necessary, we will adopt the more precise notations ¯ X ^x,(h

⁾ or ξ ^x,(h

⁾ for the stepwise constant

or genuine continuous-time Euler schemes respectively, in order to specify that these processes start

from x ∈ R ^d at time 0 and that their discretization step sequence is (h _n ) _{n ≥ 1} .

2.2 Convergence results

In this part, we recall some a.s.-convergence results for (¯ µ _n (ω, dx)) _{n ≥ 1} to the invariant distribution.

To this end, we denote by EQ ( R ^d ) the set of positive C ² -functions V : R ^d → R such that there exists ρ > 0 such that

lim inf

| x |→ + ∞ | x | ^{− ρ} V (x) > 0, |∇ V | ² ≤ CV and D ² V is bounded.

Note that lim _|x|→+∞ | x | ^{− ρ} V (x) > 0 implies that lim _|x|→+∞ V (x) = + ∞ which is a sufficient assump- tion for a part of the results. The interest of this slightly more restrictive assumption is to say that a function g has polynomial growth if and only if there exists r > 0 such that | g | ≤ CV ^r .

Then, for any symmetric d × d matrix S, set λ ⁺ _S := max(0, λ ₁ , . . . , λ _d ) where λ ₁ , . . . , λ _d denote the eigenvalues of S. Let a ∈ (0, 1] and p ∈ [1, + ∞ ). We introduce the following mean-reverting assumption with intensity a:

(S a,p ) : There exists a function V ∈ EQ ( R ^d ) such that:

(i) ∃ C _a > 0 such that | b | ² + Tr(σσ ^∗ ) ≤ C _a V ^a .

(ii) There exist β ∈ R and ρ > 0 such that h∇ V, b i + λ _p Tr(σσ ^∗ ) ≤ β − ρV ^a , where λ p := 1

2 sup

x ∈ R

λ ⁺

D

V (x)+(p − 1)

. The function V is then called a Lyapunov function for the diffusion (X _t ) _{t ≥ 0} . If (S _a,p ) holds for every p ∈ [1, ∞ ), we denote it by (S _{a, ∞} ). Note that (S _{a, ∞} ) holds if | b | ² ≤ C a V ^a , h∇ V, b i ≤ β − ρV ^a and Tr(σσ ^∗ ) = o(V ^a ).

We finally introduce a uniqueness assumption for the invariant distribution.

(S ^ν _T ): ν is an invariant distribution for (P _t ) _{t ≥ 0} and the unique one for P

. Then, the following result follows from Lemma 3.3 from [18]:

P ^ROPOSITION 2.1. Assume (S a,p ) holds with p > 2 and ain(0, 1]. Then, (i) The sequence of empirical measures (¯ µ _n (ω, dx)) _{n ≥ 1} satisfies

sup

n≥1 µ ¯ _n (ω, V

^{+a − 1} ) < + ∞ a.s. (2.8) In particular, (¯ µ _n (ω, dx)) _{n ≥ 1} is a.s. tight.

(ii) Assume (S ^ν _T ). Then, a.s., for every continuous function f such that f (x) = o(V

^+a−1 (x)) as

| x | → + ∞ . Then, µ ¯ _n (ω, f ) −−−−−→ ^n→+∞ ν(f ) a.s.

2.3 Smoothness and growth of solutions to parabolic PDE’s

Let T ∈ (0, ∞ ) and let g ∈ C ² ( R ^d , R ). Set for every (t, x) ∈ [0, T ] × R ^d , u _g (t, x) := E [g(X _T ^x _{− t} )] where X ^x denotes the solution to (1.1) starting from x ∈ R ^d . Owing to Itˆo’s formula and to the commutation property between A and P _{T − t} , u _g is a solution to the parabolic P DE:

( ∂ _t u _g (t, x) + A u _g (t, x) = 0 (t, x) ∈ [0, T ] × R ^d ,

u _g (T, x) = g(x), x ∈ R ^d . (2.9)

In fact, the main point is to obtain smoothness (and growth control) properties for u _g . This is the

purpose of the following proposition.

P ROPOSITION 2.2. Let m be a positive integer. Assume that b and σ are C ^m,α -functions (α ∈ (0, 1]) on R ^d with bounded derivatives. Assume that g is a C ^m -function on R ^d such that g and its derivatives of order l ≤ m have polynomial growth. Then, x 7→ u _g (t, x) is a C ^m -function on R ^d and t 7→ u _g (t, x) is a C ^⌊m/2⌋ -function on R ₊ . Furthermore, there exists r > 0 such that ∀ t ∈ [0, T ],

| u g (t, x) | ≤ C T (1 + | x | ^r ) and | ∂ ^ℓ

x

...x

u g (t, x) | ≤ C T (1 + | x | ^r ), (2.10) for every ℓ ∈ { 1, . . . , m } and β ₁ , . . . , β _d ∈ N ^d such that β ₁ + . . . + β _d = ℓ. Finally, in the particular case where g and its derivatives are bounded, u and its derivatives are also bounded.

We provide references and a direct self-contained probabilistic proof in the appendix.

3 Main results

3.1 Marginal case

Let f : R ^d → R be a continuous function. Before stating the first result of this paper about the rate of convergence of (¯ µ _n (ω, f )) _n≥1 , we need to introduce the following “co-boundary” assumption on f (where m is an integer):

( P m,T ): There exists a C ^m -function g

: R ^d → R such that f (x) − ν (f ) = g

(x) − P

g

(x)

T , x ∈ R ^d . (3.11)

Furthermore, if g

and its derivatives are bounded functions, we will say that f satisfies ( P _m,T ^b ).

Likewise, if g

and its derivatives have polynomial growth, we will denote the assumption by ( P _m,T ^pol ).

R EMARK 3.1. Assumption ( P m,T ) can be viewed as the existence of a regular solution to a discrete version of the Poisson equation f − ν(f ) = −A g. In Section 7, we give some criteria which ensure such condition when the diffusion is asymptotically confluent (see Proposition 7.8). We refer to [21, 22, 23]

for results on the Poisson equation itself in an elliptic setting. Since T is a fixed positive real number throughout the paper, we will usually write g instead of g

in order to alleviate the notations.

T HEOREM 3.1. Assume (S _a,p ) holds with a ∈ (0, 1] and p ∈ (2, + ∞ ]. Assume (S ^ν _T ). Let m ∈ N and let α ∈ (0, 1] such that b and σ are C ^m,α -functions on R ^d with bounded derivatives. Let f : R ^d → R be a Borel function satisfying ( P m,T ^b ) if p < + ∞ or ( P _m,T ^pol ) if p = + ∞ . Then,

(i) If m = 4 and √ ¹ n

P n

k=1 γ _{N (kT )} −−−−−→ ^{n → + ∞} 0,

√ nT 1 n

X n

k=1

f (ξ _(k−1)T ) − ν (f )

!

n → = ⇒ N + ∞ (0; ˆ σ _T ² )

with σ ˆ ² _T = 1 T

Z

g

² (x) − (P

g

(x)) ² ν (dx).

(ii) If m = 5 and √ ¹ n

P n

k=1 γ _{N(kT )} −−−−−→ ^{n → + ∞} + ∞ and γ _{N (nT )} − γ _{N((n+1)T )} = o(γ _{N(nT )} ) as n → + ∞ , P n n

k=1 γ _{N(kT )} 1 n

X n

k=1

f (ξ _(k−1)T ) − ν (f )

!

−→ P m

= 1 2T

Z

R

Z T

0

E [Φ _g (s, X _s ^x )]dsν(dx)

as n → + ∞ , where “ −→ ^P ” denotes the convergence in probability and

Φ _g (t, x) = hA (∂ _x u _g )(t, x) − ∂ _x ( A u _g )(t, x), b(x) i + A (∂ _x ²

u _g )(t, x) − ∂ _x ²

( A u _g )(t, x)

σ(x) ^{⊗ 2} (3.12) where u(t, x) = E [g(X _T ^x _{− t} )] and ∂ _x u _g and ∂ _x ²

u _g denote the gradient and the Hessian matrix of x 7→

u _g (t, x) respectively.

(iii) If m = 5 and q

T n

P n

k=1 γ _{N(kT )} −−−−−→ ^{n → + ∞} β ₀ ∈ (0, + ∞ ) and γ _{N(nT )} − γ _{N ((n+1)T )} = o(γ _{N(nT )} ) as n → + ∞ ,

√ nT 1 n

X n

k=1

f(ξ _{(k − 1)T} ) − ν(f )

!

n→+∞ = ⇒ N (β ₀ m

; ˆ σ ² _T ).

R EMARK 3.2. Owing to the stationarity property, one can check that ˆ σ ² _T can also be written ˆ

σ _T ² = 1 T

Z E

h

(g

(X _T ^x ) − E [g

(X _T ^x )]) ² i

ν(dx). (3.13)

3.2 Comparison with the original procedure

Throughout this section, we compare (¯ µ _n (ω, dx)) _{n ≥ 1} with the original procedure (ν _n (ω, dx)) _{n ≥ 1} defined by (1.5).

Assume that γ _k = γ 1 k ^{− ρ} with ρ ∈ (0, 1). Then, Γ n = γ 1 n ^{1 − ρ} + O(1) and N (t) = γ ⁻

1 1−ρ

1 t

+ O(1) . It follows that

γ _{N(nT )} = γ

1 1−ρ

1 T ⁻

n ⁻

+ O(n ⁻

) and thus, that

√ 1 n

X n

k=1

γ _{N(kT )} −−−−−→ ^n→+∞ 0 ⇐⇒ ρ > 1 3 .

Likewise, since γ _{N (nT )} − γ _{N((n+1)T )} = O(n ⁻

), γ _{N (nT )} − γ _{N((n+1)T )} = o(γ _{N(nT )} ) for every ρ ∈ (0, 1).

At time t _n = nT , we deduce that the error is of order (t _n ) ⁻

if ρ > 1/3 and of order (t _n ) ⁻

is ρ ≤ 1/3. Now, from a numerical point of view, we must compute the error in terms of the number N of discretization times. The error is then of order

 

 Γ ⁻

1 2

N ∝ N ⁻

if ρ ∈ (1/3, 1) Γ ⁻

ρ 1−ρ

N ∝ N ^{− ρ} if ρ ∈ (0, 1/3].

(3.14) This means that we exactly retrieve the error orders obtained for the original procedure (ν _N (ω, dx)) _{N ≥1} (see [10]). Thus, as announced in the abstract, the mixed-step algorithm has the same order of rate of convergence as the original procedure. In particular, the error order is minimized for ρ = 1/3 and is proportional to N ⁻

. In order to go further in the comparison with the previous algorithm, it is now natural to compare the variance ˆ σ ² _T with

ˆ σ ₀ ² =

Z

| σ ^∗ ∇ g

(x) | ² ν(dx) = − 2 Z

g

(x) A g

(x)ν (dx).

which denotes that of the original procedure where g

is a solution to the Poisson equation: f − ν(f ) =

−A g

. As a first example, we focus on a very particular case: the one-dimensional Ornstein-Uhlenbeck

0 1 2 3 4 5 0

2 4 6 8 10 12

Figure 1: T 7→ σ ˆ ² _T for the O.U. process with f (x) = x ²

process dX _t = − ¹ ₂ X _t + dW _t with the function f (x) = x ² . In this case, closed forms are available, namely

ˆ

σ ² ₀ = 4 and ∀ T > 0, σ ˆ _T ² = 2T (1 + e ^{− T} ) 1 − e ^{− T} ≥ 4.

T → σ ˆ ² _T is a continuous increasing function on R ₊ with linear growth (see Figure 1).

In a general setting, some of the above properties can be preserved. For instance, under the (nice) assumptions of Section 7, it can be shown that for every C ² -function f such that f , ∇ f and D ² f are bounded, the following properties hold:

T lim → 0 σ ˆ ² _T = ˆ σ ² ₀ , lim

T → 0 ∂ _T σ ˆ _T ² = 0 and lim

T → + ∞ ∂ _T ˆ σ ² _T = Z

(f − ν(f )) ² ν(dx). (3.15) These properties are proved in Appendix B. Roughly speaking, (3.15) says that the variance is close to that of the original procedure near 0 and that, asymptotically, T 7→ σ ˆ ² _T increases linearly with T with a gradient being the variance of f under the invariant distribution.

3.3 Functional case

We now consider the problem of the computation of E _ν (F ) = R

E [F(X ^x )]ν(dx) where F : C ([0, T ], R ^d ) → R is a Lipschitz continuous functional (with a slight abuse of notation, for α ∈ C ( R ₊ , R ^d ), we will write F (α) instead of F (α _t , 0 ≤ t ≤ T )). We denote by f

the function defined for every x ∈ R ^d by

f _F (x) = E [F(X ^x )].

Like in the marginal case, the rate of convergence of the procedure is strongly relateded to the weak error over a finite horizon, i.e. to the error between E [F (X ^x )] and E [F (ξ ^x,h )]. However, by contrast with the marginal case, there is no general expansion of such an error in the functional setting. That is why we introduce the following assumption:

(C _F (α)) (α ∈ ] ¹ ₂ , 1]): For any sequence of positive numbers h := (h _k ) ^k _k=1

such that P k

k=1 h _k = T ,

E [F(X ^x )] − E [F (ξ ^x,h )] ≤ C(1 + | x | ^r ) k h k ^α _∞

where k h k ∞ = max ^k _k=1

h _k and r is a positive number.

In fact, when F is a Lipschitz continuous functional, the above assumption is true with α = 1/2 and r = 1 (since E [ sup

0 ≤ t ≤ T | X _t ^x − ξ _t ^x,h | ] = O( √

h)). Case α ∈ (1/2, 1] can be of interest for some particular functionals. For instance, if F(α) = f ( R T

0 α _s ds) where f is Lispchitz continuous, it can be shown that (C _F (1)) holds (see [12]). The below theorem is then divided into two parts, respectively without or with this additional assumption:

T HEOREM 3.2. Assume (S _{a, ∞} ) holds with an a ∈ (0, 1]. Assume (S ^ν _T ). Let α ∈ (0, 1] such that b and σ are C ^4,α -functions on R ^d with bounded derivatives. Let F : C ([0, T ], R ^d ) → R be a Lipschitz continuous function such that f

satisfies ( P _4,T ^pol ) with a function denoted by g

. Then,

(i) If 1

√ n X n

k=1

p γ _{N (kT )} −−−−−→ ^{n → + ∞} 0,

√ n

¯

µ ⁽ⁿ⁾ (F ) − E _ν (F )

= √ n 1

n X n

k=1

F (ξ ^{((k − 1)T )} ) − E _ν (F )

!

n → = ⇒ N + ∞ 0; e σ _F ²

with σ e ² _F = Z

F (X ^x ) − E [F (X ^x )] + 1

T (g

(X _T ^x ) − E [g

(X _T ^x )]) 2

ν (dx).

(ii) If (C _F (α)) holds for an α ∈ ] ¹ ₂ , 1] and √ ¹ n

P n

k=1 γ _{N(kT )} α n → + ∞

−−−−−→ 0, the above weak rate of convergence of µ ¯ ⁽ⁿ⁾ remains true.

R EMARK 3.3. Let α ∈ ₁

2 , 1

and set γ _n = γ ₁ n ^{− ρ} . Owing to the computations of Section 3.2, we have

√ 1 n

X n

k=1

(γ _{N (kT )} ) ^{α n→} −−−−−→ ^{+ ∞} 0 ⇐⇒ ρ > 1 2α + 1

and the (weak) rate of convergence, written this time in terms of the number N = N (nT ) of dis- cretization times of the Euler scheme, is of order

Γ ⁻ _N

∼ CN ⁻

if ρ ∈ 1

2α + 1 , 1

.

Note in particular, when α = 1/2, this leads to an “optimal” error which is O(N

^+ε ) for every ε > 0.

Again, the rate order Γ ⁻ _N

is the same as that obtained for the weighted empirical measure ν ^{(N )} (ξ(ω), dα) originally introduced in [17]. However, for a given functional F : C ([0, T ], R ^d ) → R , the computation of ¯ µ ⁽ⁿ⁾ (ω, F ) is (a priori much) less demanding in terms of complexity than that of ν ^(N) (ξ(ω), dα).

More precisely, the complexity of (¯ µ ⁽ⁿ⁾ (ω, F )) _{n ≥ 1} is linear in terms of the number N of discretization times, i.e. the number of operations for the computation of ¯ µ ⁽ⁿ⁾ (ω, F ) is proportional to

κ _E × N (nT ) + X n

k=1

κ

× (N (kT ) − N ((k − 1)T ) = (κ _E + κ _F ) × N (nT )

where κ _E denotes the computational complexity of one iteration of the Euler scheme and κ

× m the complexity of the computation of F (ξ) where ξ ∈ C ([0, T ], R ^d ) is a typical path of an Euler scheme with m time steps. For the original procedure a similar computation leads to

κ _E × N (nT ) +

N(nT X ) − 1

k=0

κ

(N (Γ _k + T) − N (Γ _k )) ≈ κ _E × N (nT ) + κ _F X n

ℓ=1

(N(ℓT ) − N ((ℓ − 1)T )) ²

which grows to infinity infinitely faster. However in some particular cases (fortunately often those of interest) the functional F can be computed recursively (see [18] for details) so that our estimate κ _F × m should be replaced by κ

× c ₀ which makes the resulting global complexity comparable to our new approach. As a conclusion, our new procedure yields a better control of the complexity for all (computable) functionals F .

In practice, F (ξ ^{(kT )} ) is not always computable since the genuine Euler scheme is non-constant on any time interval. If we replace the genuine Euler scheme by its stepwise constant c` adl` ag counterpart, it can be shown that the CLT established in (i) still holds for the same functional F under the slightly more stringent assumption that for every ε > 0, √ ¹

n

P n

k=1 (γ _{N (kT )} )

^{− ε n→+∞} −−−−−→ 0.

3.4 The Richardson-Romberg extrapolation

In this section, we want to perform and analyze a Richardson-Romberg (RR) extrapolation to cancel the first order term in the expansion of the time discretization. Note that we will only consider this problem in the marginal case since we do not have any explicit expansion of the weak error in the functional case. As concerns discretization schemes with decreasing step this procedure has been in- troduced in [14] (see Chapter V). However, our proposal is slightly different since we will design our two schemes in a more consistent way.

Let θ ∈ [0, 1] and let (W, W f ) denote a 2q-dimensional SBM. Denote again by ( F t ) _t≥0 the usual aug- mentation of (σ((W, W f ) _s , 0 ≤ s ≤ t)) _{t ≥ 0} . Then, we define W ^θ by

W ^θ = θW + p

1 − θ ² W f so that h W, W ^θ i t = θt.

We denote by (ξ _t ^(θ) ) _t≥0 the Euler scheme built with the increments of W ^θ and a step sequence ( e γ _n ) _n≥1 defined for every n ≥ 1 by

e

γ _{2n − 1} = e γ _2n = γ _n 2 . In the sequel, we set Γ e _n = P n

k=1 γ e _k (note that e Γ _2n = Γ _n ) and for every t ≥ 0, N e (t) = inf { k ≥ 0, Γ e _k+1 >

t } .

We also denote by (¯ µ ^(θ) n (ω, dy)) _n≥1 , the sequence of empirical measures related to (ξ ^(θ) _t ) _t defined for every n ≥ 1 by

¯

µ ^(θ) _n (ω, f ) = 1 n

X n

k=1

f (ξ _(k ^(θ) _{− 1)T} (ω)).

Finally, we consider the sequence of (weighted) random measures ( µ e ^(θ) n (ω, dy)) n ≥ 1 defined by the extrapolation of ¯ µ _n and ¯ µ ^(θ) _n in proportion with scaling factor of their respective step sequences:

e

µ ^(θ) _n (ω, f) = 2¯ µ ^(θ) _n (ω, f ) − µ ¯ _n (ω, f).

In the sequel of this section, we are then interested in the rate of convergence of ( µ e ^(θ) _n (ω, f)) _{n ≥ 1} toward ν(f ). To this aim , we introduce X ^(θ) := (X, X ^(θ) ) solution to the system of duplicated SDE’s

( dX _t = b(X _t )dt + σ(X _t )dW _t

dX _t ^(θ) = b(X _t ^(θ) )dt + σ(X _t ^(θ) )dW _t ^θ (3.16)

starting from (X 0 , X ₀ ^(θ) ) R ^2d -valued random vector supposed to be independent of σ(W, W f )). Since b and σ are Lipschitz continuous functions, the R ^2d -valued process X ^(θ) is a Feller Markov diffusion process whose semi-group is denoted by (Q ^(θ) _t ) _t≥0 . In the theorem below, we need that Q ^(θ) _T has a unique invariant distribution ν ^(θ) . It is clear that, if P _T has a unique invariant distribution ν, then ν ^(θ) belongs to C ν,ν := { µ ∈ P ( R ^2d ), µ ₁ = µ ₂ = ν } where P ( R ^2d ) denotes the set of probabilities on R ^2d and µ ₁ = µ( . × R ^d ) and µ ₂ = µ( R ^d × . ) (first and second marginals of µ). Since C ν,ν is a weakly compact and convex subset of the Banach space of signed measures on ( R ^2d , B ( R ^2d )), the existence of an invariant distribution ν ^(θ) for Q ^(θ)

follows from the Kakutani fixed-point Theorem applied to µ 7→ µQ ^(θ) _T (in fact dealing with the temporal mean of (Q ^(θ) _t ) _t≥0 would yield an invariant distribution for the whole semi-group if needed). Moreover,we know that uniqueness of ν ^(θ) implies that of ν still owing to the Kakutani Theorem since, if ν and ˜ ν were two invariant distributions of P _T then µ 7→ µQ ^(θ) _T would have another fixed distribution on C ν,˜ ν . The reverse is not true but uniqueness of ν ^(θ) being important to this problem, it will be discussed in Section 3.5 below.

Let us now state the main result of this section:

T HEOREM 3.3. Assume (S _{a, ∞} ) holds with an a ∈ (0, 1] and assume that Tr(σσ ^∗ (x)) = o(V ^a (x)) as

| x | → + ∞ . Let θ ∈ [0, 1] and assume that Q ^(θ) _T admits a unique invariant distribution ν ^(θ) . Assume that b and σ are C ^9,α -functions on R ^d with bounded existing derivatives, α ∈ (0, 1]. Let f : R ^d → R be a Borel function satisfying ( P _9,T ^pol ) and denote by g

the solution to (3.11). Then, assume that ϕ ⁽¹⁾ defined by ϕ ⁽¹⁾ (x) = ¹ ₂ R T

0 E [Φ _g

T

(X _s ^x )]ds satisfies ( P _5,T ^pol ) and denote by g _ϕ

the associated solution to (3.11). Assume that (γ n ) n ≥ 1 satisfies P + ∞

k=1 γ _N(kT ² ₎ = + ∞ and γ _{N(nT )} − γ _{N((n+1)T )} = o(γ _N ² _{(nT )} ).

Then, (i) If √ ¹

nT

P _n

k=1 γ _N(kT ² ₎ −−−−−→ ^n→+∞ 0, then

√ nT e

µ ^(θ) _n (ω, f ) − ν(f ) _n→+∞

= ⇒ N

0, (ˆ σ _T ^(θ) ) ² where, (ˆ σ _T ^(θ) ) ² = 5ˆ σ ² _T − 4

T Z

g

(x)g

(y) − P

g

(x)P

g

(y)

ν ^(θ) (dx, dy). (3.17) (ii) If

q T n

P n

k=1 γ _N ² _{(kT )} −−−−−→ ^n→+∞ β 0 ∈ (0, + ∞ ], then ^P

ⁿ

γ

e

µ ^(θ) n (ω, f ) − ν(f )

is tight with bounded (resp. subgaussian) weak limits if β ₀ = + ∞ (resp. β ₀ < + ∞ ).

Furthermore, if (γ _n ) _{n ≥ 1} is such that N (nT ) = nT for every n ≥ 1, then we have the following more precise result:

– If β ₀ = + ∞ , P n n

k=1 γ ² _{N(kT )}

µ e ^(θ) _n (ω, f ) − ν (f ) P

−→ m e

= − 1 2T

Z

R

Z _T

0

E 1

T Φ _g

ϕ(1)

(X _s ^x ) + χ _g (s, X _s ^x )

dsν(dx)

as n → + ∞ , where χ g is given by (4.30).

– If β ₀ ∈ (0, + ∞ ),

√ nT e

µ ^(θ) _n (ω, f ) − ν(f) _n→+∞

= ⇒ N

β 0 m e

, (ˆ σ ^(θ) _T ) ²

.

R ^EMARK 3.4. • Using the stationarity property, (ˆ σ _T ^(θ) ) ² can be written as follows:

(ˆ σ ^(θ) _T ) ² = 1 T

Z

R

× R

Var x,y

2g

(X _T ^(θ) ) − g

(X T )

ν ^(θ) (dx, dy).

• Owing to the definition of N (t), we have for every k ∈ N , C _T ¹ γ _N(k+1)T ² ≤

γ

X

l=N(kT )+1

γ _l ³ ≤ C _T ² γ _N ² _{(kT )} , (3.18) so that

X n

k=1

γ _N(kT ² ₎ −−−−−→ ^n→+∞ + ∞ ⇐⇒ X

k ≥ 1

γ _k ³ = + ∞ .

If γ _n = γ ₁ n ^−ρ with ρ ∈ (0, 1), this condition is satisfied if and only if ρ < 1/3. Using the computations of Subsection 3.2, one checks that in this case, γ _{N(nT )} − γ _{N ((n+1)T )} = o(γ _N(nT ² ₎ ), and that,

√ 1 n

X n

k=1

γ _N(kT ² ₎ −−−−−→ ^{n → + ∞} 0 ⇐⇒ ρ > 1 4 .

In fact, the Romberg extrapolation implies that the discretization error is lower than in Theorem 3.1 and thus, that the rate of convergence in (i) can be preserved for smaller values of ρ. More precisely, after N discretization times, the error is of order

( N ⁻

if ρ ∈ (1/5, 1/3) N ^−ρ if ρ ∈ (0, 1/5].

The order of the error attains its minimum at ρ = 1/5 and is then proportional to N ⁻

.

• If the additional assumption on the uniqueness of the invariant distribution for Q ^(θ) _T fails, tightness results are preserved. In particular, in (i), we would have only obtained that √

n e

µ ^(θ) _n (ω, f ) − ν (f ) is a tight sequence with Gaussian limits.

• The last part of the above theorem requires an additional assumption on the steps: N (nT ) = nT . In fact, if this assumption fails, some edge terms must be managed, especially in the expansion of the weak error relative to this problem, which become non-negligible at the second order. However, it seems that this assumption could be avoided owing to a sharper and tedious study of these edge terms. This refinement will not be tackled in the paper.

An important numerical question relative to Theorem 3.3 is the variance (ˆ σ g ^(θ) ) ² . When one performs a Romberg extrapolation, it is important to compare the variance of the modified scheme with that of the original one. In [16], this problem is tackled for the standard Euler scheme on a finite interval and it is shown that the variance of the original scheme can be preserved by taking θ = 1, i.e. by considering the same underlying Brownian motion for both schemes. Here, we obtain a similar result for the invariant distribution up to a uniqueness assumption for the invariant distribution when θ = 1. In that case, the distribution ν ∆ defined by ν ∆ (h) = R

h(x, x)ν (dx) is clearly an invariant

distribution for Q ⁽⁰⁾

. If this is the only one, we can minimize the variance. This is the purpose of the

next proposition.

P ^ROPOSITION 3.3. Assume (S ^ν _T ). Let f ∈ L ² (ν).

(i) For every θ ∈ [0, 1], σ ˆ _T ² ≤ (ˆ σ ^(θ) _T ) ² .

(ii) If ν _∆ is the unique invariant distribution of Q ⁽¹⁾

, then (ˆ σ _T ⁽¹⁾ ) ² = ˆ σ ² _T . Proof. (i) Owing to the stationarity property, we have

Z

g(x)g(y) − P

g(x)P

g(y)ν ^(θ) (dx, dy)

= Z

E _x,y h

g(X _T ) − E [g(X _T )]

g(X _T ^(θ) ) − E [g(X _T ^(θ) )] i

ν ^(θ) (dx, dy)

= E

ν

h g(X T ) − P

g(X 0 )

g(X _T ^(θ) ) − P

g(X ₀ ^(θ) ) i .

Using that under the stationary regime, L ((X _t ) _t≥0 ) = L ((X _t ^(θ) ) _t≥0 ) = P _ν , we derive from Schwarz’s inequality that

Z g(x)g(y) − P

g(x)P

g(y)

ν ^(θ) (dx, dy) ≤ E _ν

(g(X _T ) − P

g(X ₀ )) ²

= ˆ σ _T ² (3.19) Owing to (3.17), this shows that for every θ ∈ [0, 1], (ˆ σ _T ^(θ) ) ² ≥ σ ˆ _T ² .

(ii) Set

U = g(X _T ) − P

g(X ₀ ) and V = g(X _T ^(θ) ) − P

g(X ₀ ^(θ) ).

From what preceeds and from the equality case in Schwarz’s inequality, we obtain that equality holds in (3.19) if and only if there exists λ ∈ R such that U = λV a.s., and thus if and only if U = V a.s. since U and V have the same distribution. As a consequence, equality holds in (3.19) if (X ₀ , X _T ) = (X ₀ ^(θ) , X _T ^(θ) ) a.s. So is the case if ν ^(θ) = ν _∆ .

3.5 Are invariant measures of duplicated diffusions always supported by the di- agonal?

This section is a summarized version of [15]. In particular, we refer to this paper for the proofs of the results.

We keep the notations of the previous part of the paper: denoting by (X _t ) _{t ≥ 0} the solution to (1.1) and by ( X ^(θ) _t ) _t≥0 the duplicated diffusion solution to (3.16), we consider the problems of existence and mostly of uniqueness of the invariant distribution for ( X ^(θ)

t ) _{t ≥ 0} .

In what follows we will always assume that the original diffusion X ^x has at least one invariant distri- bution denoted ν i.e. satisfying νP _t = ν for every t ∈ R ₊ .

Existence of an invariant distribution for (Q ^(θ) _t ) _{t ≥ 0} with marginals ν. Using that the family of probability measures on ( R ^d × R ^d , B or( R ^d ) ^⊗2 )

1 t

Z t

0

ν ^⊗ (dx ₀ , dx ^′ ₀ )Q ^(θ) _s (x ₀ , x ^′ ₀ , dy, dy ^′ )ds, t > 0

is tight since both its marginals on R ^d are equal to ν and that the semi-group (Q ^(θ) _t ) _{t ≥ 0} is Feller, one

easily shows that any limiting distribution ν ^(θ) as t → ∞ is an invariant distribution for (Q ^(θ) _t ) _t≥0

with both marginals equal to ν.

Uniqueness of the invariant distribution of Q ^(θ) _T (and thus for (Q ^(θ) _t ) t ≥ 0 ). It is clear that in full generality ( X ^(θ) _t ) t ≥ 0 may admit several invariant distributions even if X has only one such distribution.

So is the case in a fully degenerate setting: σ ≡ 0. Then, if the flow Φ(x, t) of the ODE x ˙ = b(x) has an invariant distribution ν, then both ν ^⊗2 and the image ν _∆ of ν on the diagonal of R ^d × R ^d are invariant for the duplicated o.d.e. and ν ^{⊗ 2} and the image ν ∆ on the diagonal of R ^d × R ^d are invariant and ν ^{⊗ 2} 6 = ν _∆ as soon as ν is not reduced to Dirac mass (think for instance to a 2-dimensional ODE with a limit cycle).

In the non-degenerate case (σ 6≡ 0), the situation is more involved and depends on the correlation θ between the two Brownian motions.

• θ ∈ [0, 1): if one writes W ^(θ) = θW + (1 − θ ² )

W f where f W is independent of W , one checks that for every t ≥ 0, the diffusion matrix, Σ _X

t

of the duplicated diffusion X ^(θ) = (X, X ^(θ) ) at time t satisfies

Σ _X

Σ ^∗

X

t

=

"

σσ ^∗ (X _t ) θσ(X _t )σ ^∗ (X _t ^(θ) ) θσ(X _t )σ ^∗ (X _t ^(θ) ) √

1 − θ ² σσ ^∗ (X _t ^(θ) )

# .

From this expression, it is straightforward than ellipticity or uniform ellipticity of σ (for X ^x which underlines q ≥ d) can be transferred to the couple (X, X ^(θ) ). Uniform ellipticity classically implies under standard regularity and growth/boundedness assumptions on the coefficients b, σ and their partial derivatives the existence of a (strictly) positive probability density p _t (x, y) for X. These conditions once again transfer to the coefficients of the diffusion (X, X ^(θ) ). Consequently, under these standard assumptions on b and σ which ensure uniqueness of the invariant distribution ν for X (and more precisely for P _T ), we get uniqueness for the “duplicated” diffusion process (X, X ^(θ) ) (and more precisely for Q ^(θ) _T ) as well.

In an hypoelliptic setting, it is also classical that if the diffusion coefficient of a diffusion satisfies the hypoelliptic H¨ormander assumption and if the deterministic system related to the Stochastic differential system (written in the Stratonovich sense) is controllable, then (the transition semi-group is strongly Feller and) the invariant distribution, if any, is unique. In fact, both properties can be transferred from the original SDE to the duplicated system so that the invariant distribution is unique (see [15] for details).

• θ = 1: this setting corresponds in our problem to the simulation of consistent Brownian in- crements for our Euler schemes. Owing to Proposition 3.3, in this case, uniqueness of the invariant distribution for Q ⁽¹⁾ _T implies that the variance in the CLT is equal to that of the original procedure.

However, the situation becomes significantly different since the resulting “duplicated stochastic sys- tem” becomes degenerate. It is clear that ν ∆ as defined above is an invariant distribution for the system, the question becomes: “is it the only one?”.

In what follows, we provide some answers: In Proposition 3.4, we state that under some asymptotic confluence assumptions, uniqueness holds for the invariant distribution of the “single” diffusion and is transferred to the duplicated diffusion. Then, when this assumption fails and if d ≥ 2, we show explicitely that it is possible to build some diffusions for which uniqueness holds for the “single” but not for the duplicated one.

Finally, we focus on the one-dimensional case where the property is essentially always true.

P ^ROPOSITION 3.4 (see [15]). Assume that the SDE is asymptotically confluent in the following

sense:

(AC) ≡ ∀ R > 0, ∃ δ _R > 0, such that ∀ x, y ∈ B _{| . |} (0; R), 0 < | x − y | ≤ δ _R = ⇒ h b(x) − b(y), x − y i + 1

2 Tr

(σ(x) − σ(y)(σ(x) − σ(y) ^∗

< 0, then ν and ν ∆ are respectively the unique invariant distributions for P

and Q ^(θ) _T .

R EMARK 3.5. • In [15], the reader will find other asymptotic confluence conditions which ensure the conclusion of Proposition 3.4.

• If b and σ are differentiable, Assumption (AC) is equivalent to

∀ R > 0, ∃ ε _R > 0 such that , | x | ≤ R = ⇒ J _b (x) + 1 2

X d

i=1

X q

j=1

X ^d

k=1

∂σ _ij

∂x _k (x) 2

< − ε _R

where J _b denotes the Jacobian d × d matrix of b.

As soon as d ≥ 2, there are examples of non asymptotically confluent diffusions having a unique invariant distribution ν but whose “duplicated system” (with the same Brownian motion) has several invariant distributions. One example is provided below.

A counterexample: We consider the 2-dimensional SDE with Lipschitz continuous coefficients defined by

b(x) =

x1 _{{ 0 ≤| x |≤ 1 }} − x

| x | 1 _{{| x |≥ 1 }}

(1 − | x | ) σ(x) = ϑDiag(b(x)) +

0 − cx ² cx ¹ 0

.

where ϑ, c ∈ (0, + ∞ ) are fixed parameters. Switching to polar coordinates X _t = (r _t cos ϕ _t , r _t sin ϕ _t ), t ∈ R ₊ show that this SDE also reads

dr _t = min(r _t , 1)(1 − r _t )(dt + ϑdW _t ¹ ), r ₀ ∈ R ₊ dϕ t = cdW _t ² , ϕ 0 ∈ [0, 2π).

where x ₀ = r ₀ (cos ϕ ₀ , sin ϕ ₀ ) and W = (W ¹ , W ² ) is a standard 2-dimensional Brownian motion.

Standard considerations about Feller classification (see [7], chapter 15.6, p. 226) show that, if x ₀ 6 = 0 (i.e. r ₀ > 0) and ϑ ∈ (0, √

2) then

r _t −→ 1 as t → ∞ . while it is classical background that

P a.s. ∀ ϕ ₀ ∈ R ₊ , 1 t

Z t

0

δ _e

s)

ds = ⇒ λ _S

as t → ∞

where S ¹ denotes the unit circle of R ² and λ _S

denotes the Lebesgue measure on S ¹ . Combining these two results straightforwardly yields

∀ x ∈ R ² \ { (0, 0) } , P -a.s. 1 t

Z _t

0

δ _X

ds ^{( R}

2

)

= ⇒ ν := λ _S

as t → ∞ .

On the other hand, given the form of ϕ _t , it is clear that if x = r ₀ e ^iϕ

and x ^′ = r ₀ ^′ e ^iϕ

, r ₀ , r ^′ ₀ 6 = 0 then

t lim →∞ | X _t ^x − X _t ^x

| = | e ^i(ϕ

^{− ϕ}

⁾ − 1 | . Now, taking ϕ ₀ 6 = ϕ ^′ ₀ and using that the weak limits of ( ¹ _t R t

0 Q _s (x, x ^′ , ., .)ds) _{t ≥ 0} always belong to the set of invariant distributions of (Q _t ) _t≥0 , one easily deduces that ν _∆ (defined by ν _∆ (h) = R

h(x, x)ν (dx)) can not be the only invariant distribution : otherwise, setting f _M (x, y) = | y − x | ∧ M (M > 0), one would have ¹ _t R t

0 Q _s f _M (x, x ^′ )ds → 0 for every M > 0).

In fact, one can also directly check (with the definition of the semi-group) that ν ⊗ ν is another invariant distribution for (Q _t ) _t≥0 (so that ν _∆ , ν ⊗ ν and all the convex combinations of these probability measures are invariant distributions for (Q _t ) _{t ≥ 0} ).

The one-dimensional case. Finally, one has a more unexpected result in one dimension d = q = 1.

T HEOREM 3.4 (see [15]). Assume that b and σ are locally Lipschitz continuous on R and that there exists a C ² -function V : R ^d → R ^∗ ₊ such that lim

|x|→+∞ V (x) = + ∞ and A V ≤ CV with C > 0. Assume that (P _t ) _{t ≥ 0} admits a unique invariant distribution. Then, ν _∆ = ν ◦ (ξ 7→ (ξ, ξ)) ⁻¹ is the unique invariant distribution of (Q ⁽¹⁾ _t ) _t≥0 .

R EMARK 3.6. When σ never vanishes, this result can be retrieved by introducing the scale function p of the diffusion defined by:

p(x) = Z x

x

dξe ⁻

R

2b σ2

(u)

, x ∈ R .

In fact, by a result of Has’minskii (see [6], Corollary A.2 p.274), when the diffusion is positive recurrent (see [15] for more detailed assumptions), then, for every x, y ∈ R ^d , p(X _t ^x ) − p(X _t ^y ) −−−−→ ^{t → + ∞} 0 a.s.

Denoting by µ an invariant distribution of (X _t ^x , X _t ^y ) t ≥ 0 , this implies that

∀ K ≥ 0, Z

| p(x) − p(y) | ∧ Kµ(dx, dy) ≤ lim sup 1 t

Z t

0

E _µ [ | p(X _s ^x ) − p(X _s ^y ) | ∧ K]ds = 0 As a consequence, p(x) = p(y) µ(dx, dy)-a.s. Since p is an increasing function, we can deduce that µ((x, x), x ∈ R ^d ) = 1 and thus that µ = ν _∆ .

The next three sections are devoted to the proofs of the main results (of Section 3). In Section 4, we begin with some first and second order expansions of the weak error related to the Euler scheme in a finite horizon in a non-constant step framework. These expansions are important tools for the sequel of the proof.

4 The weak error in a non constant-step framework

In the sequel, C denotes any non explicit positive constant. Likewise, for a function f with polynomial growth, we usually write | f (x) | ≤ C(1 + | x | ^r ) where the exponent r denotes a non explicit positive number.

Let T > 0. We denote by (ξ ^x,h _t ) _{t ∈ [0,T ]} the genuine Euler scheme with step sequence h := (h _k ) _{k ≥ 1} starting point x ∈ R ^d and by E

(g, x, h) the weak error between the Euler scheme and the diffusion, that is:

E

(g, x, h) = E [g(ξ ^x,h _T )] − E [g(X _T ^x )]. (4.20)

4.1 Preliminary lemmas

We denote by (t _k ) the sequence of discretization times:

t 0 = 0, t _k = X k

l=1

h _l , k ≥ 1, and we assume that there exists k

∈ N such that t _k

T

= T .

L EMMA 4.1. Let k ∈ N and let g : R ^d → R be such that the assumptions of Proposition 2.2 hold.

Then, (i) If m = 4,

|E

(g, x, h) | ≤ C _T (1 + | x | ^r )

k

X

k=1

(h _k ) ²

where r is a positive real number and C _T is a real constant only depending on T , b, σ, g (and their derivatives up to order 4) which is non-decreasing as a function of T .

(ii) If m = 5 or = 6, then,

E

(g, x, h) = 1 2

k

X

k=1

(h _k ) ² E [Φ _g (t _{k − 1} , ξ _t ^x,h

)] + R _T (x, h) where Φ _g is defined by (3.12) and

| R _T (x, h) | ≤ C _T (1 + | x | ^r )

( k h k ∞

if m = 5 k h k ² _∞ if m = 6

with r > 0, k h k ∞ = max ^k _k=1

h _k , and T 7→ C _T is a non-decreasing function only depending on T , b, σ, g and their partial derivatives up to order k.

Proof. (i) First, in order to alleviate the notations, we will write u instead of u _g . Then, by the definition of u, we have

E [g(ξ _T ^x,h )] − E [g(X _T ^x )] =

k

X

k=1

E [u(t _k , ξ _t ^x,h

) − u(t _k−1 , ξ _t ^x,h

)]. (4.21) By Proposition 2.2, t 7→ u(t, x) is a C ² -function and x 7→ u(t, x) is a C ⁴ function. Then, owing to the Itˆo formula, we have

u(t _k , ξ _t ^x,h

) − u(t _{k − 1} , ξ _t ^x,h

) = Z t

t

∂ _t u(s, ξ _s ^x,h ) + ¯ A u(s, ξ _s ^x,h , ξ ^x,h _s )

ds + M _t ⁽⁰⁾

− M _t ⁽⁰⁾

where (M _t ⁽⁰⁾ ) is a true martingale defined by M _t ⁽⁰⁾ = R t

0 h ∂ _x u(s, ξ s ^x,h ), σ(ξ ^x,h s )dW _s i and A ¯ u(s, x, x) = h ∂ _x u(s, x), b(x) i + 1

2 Tr

σ ^∗ (x)∂ _xx ² u(s, x)σ(x)

.