Geometric ergodicity for families of homogeneous Markov chains

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Geometric ergodicity for families of homogeneous Markov chains

Leonid Galtchouk, Serguei Pergamenchtchikov

To cite this version:

Leonid Galtchouk, Serguei Pergamenchtchikov. Geometric ergodicity for families of homogeneous Markov chains. 2010. �hal-00455976v2�

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Geometric ergodicity for classes of homogeneous Markov chains

^∗

L. Galtchouk ^† S. Pergamenshchikov^‡ May 8, 2012

Abstract

The paper deals with non asymptotic computable bounds for the geometric convergence rate of homogeneous ergodic Markov processes.

Some sufficient conditions are stated for simultaneous geometric ergodicity of Markov chain classes. This property is applied to nonparametric estimating in ergodic diffusion processes.

MSC : 60F10, 60J05

Key words and phrases: Homogeneous Markov chain; Geometric ergodicity;

Convergence rate; Coupling renewal processes; Lyapunov function; Renewal theory; Non asymptotic exponential upper bound; Ergodic diffusion processes.

∗The second author is partially supported by the RFFI-Grant 09-01-00172-a.

†IRMA,Department of Mathematics, Strasbourg University, 7, rue R´en`e Descartes, 67084, Strasbourg, France, e-mail: leonid.galtchouk@math.unistra.fr

‡Laboratoire de Mathématiques Raphael Salem, Avenue de l’Université, BP. 12, Uni- versité de Rouen, F76801, Saint Etienne du Rouvray, Cedex France and Department of Mathematics and Mechanics,Tomsk State University, Lenin str. 36, 634041 Tomsk, Russia, e-mail: Serge.Pergamenchtchikov@univ-rouen.fr

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1 Introduction

In this paper we study a class of homogeneous Markov chains (Φ)_n≥0 with values in some measurable space (X, B(X)) defined by a parametrized family of transition probabilities

(P^ϑ_x)_x∈X_{, ϑ∈Θ}, (1.1)

where Θ is a parametric set for this family. For each ϑ ∈ Θ the sequence Φ = (Φ_n)_n≥0 is a homogeneous Markov chain defined on the measurable space (X ,B(X)) with a transition probability P^ϑ, i.e.

P(Φ₁ ∈Γ|Φ₀ =x) =P^ϑ_x(Γ).

Our main goal is to state geometric ergodicity for this class simultaneously over all values of the parameter ϑ∈Θ.

Geometric ergodicity is studied in a number of papers (see, for example, [1], [16]-[19]). We remind that a chain (Φ_n)_n≥0 on the space (X,B(X)) with a invariant measure π is calledgeometrically ergodicif there exist a X →[1,∞[ function V(x) and some constants R >0,κ >0 such that, for any n ≥1,

sup

x∈X

sup

0≤g≤V

1

V(x)|E_xg(Φ_n)−π(g)| ≤R e^−κn. (1.2) As we shall see later (see, Definition 6.1 below) the function V, providing the drift condition, is given by the Lyapunov functions (see, e.g. [13] in the case of diffusion processes and [7], [11], [12] for Markov chains). For this reason, in the sequel, we shall call such functions by the Lyapunov functions.

The property (1.2) is useful in applied problems related to the identifica- tion of stochastic systems, described by stochastic processes with dependent values, in particular, governed by stochastic difference or stochastic differential equations. Necessity of simultaneous geometric ergodicity appears in statistics, when one studies a minimax risk with respect to some family of distributions related to a statistical experiment. In particular, in this paper we shall apply simultaneous geometric ergodicity to nonparametric estimating the drift coefficient in the stochastic differential equation (see [2]):

dy_t=S(y_t) dt+σ(y_t)dW_t, 0≤t≤T , (1.3) whereSandσ are unknown functions andShas to be estimated from observations (y_t)_0≤t≤T. In studying minimax risks for kernel estimators we need to use geometric ergodicity for the process (1.3) simultaneously over all coefficients S and σ from some functional class (see (2.7) below). In this case, ϑ = (S, σ) is the class parameter.

It is clear that to apply the property (1.2) to some distribution family we have to find some explicit expressions for the parameters R > 0 and κ >

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0. It is a well-known problem in the Markov Chain Monte Carlo (MCMC) theory, when a stopping rule for simulations is based on the accuracy of n- step approximations. Therefore, we need to find computable bounds in (1.2).

Note that some explicit expressions for R and κ were calculated in [1], [17], [21] and [22] for ψ-irreducible homogeneous Markov chains. These results are not applicable in our case because it is not clear what does it mean ψ- irreducibility for a class of parametrized Markov chains. Note that in [1] and [17] the parameters R and κ were obtained by making use of the Kendall theorem. In [22] these parameters are obtained through the direct coupling method for the Markov chain. To this end the authors impose in [21]–[22]

some additional assumptions which are not satisfied in the case of the diffusion process (1.3) (see Remark 3.1 in Section 3). Moreover, it should be noted that the upper bound in (1.2) given in [21] is calculated under the assumption that the minorization condition holds on the whole state space. This assumption is never true for the model (1.3).

In the paper we apply the coupling method to the renewal process generated by entrance times of the process into the minorization set. Note, that Meyn and Tweedie use the same approach in order to obtain convergence results (see, [16], chapter 13). Their results imply the power convergence rate. To obtain the geometric rate we make use of the Lyapunov functions method for the related coupling process.

In order to explain the novelty of the method introduced in the paper, we give the scheme of proving the property (1.2). The first step consists in passing to a splitting chain, which yields a chain with an accessible atom.

Then, one makes use of the Regenerative Decomposition for splitting chains in order to evaluate the convergence rate (see [10], [16]). Let us remind that the principal term in this decomposition gives the deviation in the renewal theorem, which may be evaluated thanks to the Kendall renewal theorem that provides a geometric convergence rate. In our case the same convergence rate is obtained thanks to making use of the Lyapunov functions method for the coupling renewal process (see Theorem 4.1 in Section 4). This upper bound enables us to find the explicit non asymptotic exponential upper bound in the ergodic theorem for which we can find the supremum over the transition probability family in (1.1).

In this paper we find some sufficient conditions which provide simultaneous geometric ergodicity of the family (1.1) over all values of the parameterϑ. We check these conditions for the diffusion model (1.3). As corollary, we obtain explicit upper bounds for geometric convergence rate in the ergodic theorem for diffusion processes. These bounds may be used in the Monte Carlo technique to calculate some functionals of ergodic diffusion processes. In that case one can replace these functionals by the corresponding integrals with respect to the invariant density which has a simple explicit form. The accuracy of this

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approximation is given by the explicit non asymptotic bounds in the geometric convergence rate for diffusion processes.

The paper is organized as follows. In the next section the main results are stated. Section 3 provides the explicit formulas for parameters in the geometric convergence rate. Section 4 is devoted to related coupling renewal processes. In section 5 geometric ergodicity is proved for a parametrized class of homogeneous Markov chains. In section 6 we apply this property to stochastic differential equations. Some basic results on homogeneous Markov chains are given in the Appendix.

2 Main results

Assume now, that the family of transition probabilities (P^ϑ)_ϑ∈Θ satisfies the following conditions

H₁) There exist 0< δ <1, some set C ∈ B(X) and some probability measure ν on B(X) with ν(C) = 1 such that, for any A∈ B(X) with ν(A)>0,

x∈Cinf

ϑ∈Θinf P^ϑ(x, A)−δν(A)

> 0. (2.1)

For the sequel we denote η = inf

x∈C

ϑ∈Θinf P^ϑ(x, C)−δ

. (2.2)

H₂) There exist X → [1,∞) function V, some constants 0 < ρ < 1, D ≥ 1, and a set C from B(X) such that

V^∗ = sup

x∈C

V(x) <∞ and, for any x∈ X,

sup

ϑ∈Θ

E^ϑ_x(V(Φ₁)) ≤ (1−ρ)V(x) + D1_C(x). (2.3)

Here E^ϑ_x means the expectation with respect to the transition probability P^ϑ(x,·).

Remark 2.1. Condition H₂) is called the uniform drift condition and that of H₁) is the uniform minorization condition.

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Theorem 2.1. Assume the conditions H₁)–H₂) hold true with the same set C ∈ B(X). Then, for each θ ∈Θ, the chainΦadmits an invariant distribution π^ϑ on B(X). Moreover, for any n≥2,

sup

ϑ∈Θ

sup

x∈X

sup

0<f≤V

1 V(x)

E^ϑ_xf(Φ_n)− Z

X

f(z)π^ϑ(dz)

≤R^∗e^−κ^∗ⁿ, (2.4) where the parameters R^∗ = R^∗(ρ, δ, D, η, V^∗) and κ^∗ = κ^∗(ρ, δ, D, η₁, V^∗) are given in (3.5).

Apply now to the process (1.3). To this end we have to introduce some functional class of functions ϑ = (S, σ). First, for some x_∗ ≥ 1, M > 0 and L > β >0, we denote by V1 the class of functions S from C¹(R) such that

sup

|x|≤x_∗

|S(x)|+|S(x)˙ |

≤M and

−L≤ inf

|x|≥x_∗

S(x)˙ ≤ sup

|x|≥x_∗

S(x)˙ ≤ −β .

Second, for some fixed reals σ_min >0 and σ_max > σ_min, we denote by V2 the class of functions σ fromC²(R) such that, for allx∈R,

σ_min ≤ min (|σ(x)|, |σ(x)˙ |, |σ(x)¨ |)≤max (|σ(x)|,|σ(x)˙ |,|σ(x)¨ |)≤σ_max. Finally, we set

Θ = V1× V2. (2.5)

Note that (see, for example, [6]), for any function ϑ from Θ, the equation (1.3) admits a unique strong solution, which is an ergodic process having an invariant measure π_ϑ with the invariant density q_ϑ defined as

q_ϑ(x) = σ⁻²(x) exp{Rx

0 S₁(v)dv} R+∞

−∞ σ⁻²(z) exp{Rz

0 S₁(v)dv}dz , (2.6) where S₁(v) = 2S(v)/σ²(v).

Theorem 2.2. For any 0< ǫ≤1/2 and t >0,

sup

ϑ∈Θ

sup

x∈R

sup

0<g≤1

E^ϑ_xg(y_t)−R

Rg(x)q_ϑ(x)dx

(1 +x²)^ǫ ≤R_ǫe^−κ^ǫ^t, (2.7) where the parameters R_ǫ >0 and κ_ǫ >0 are given in (3.13).

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Remark 2.2. Note that the property (2.7) is called simultaneous geometric ergodicity. As is shown in Section 6, the function (1 +x²)^ǫ is a Lyapunov function. It should be said that in [20], for the process (1.3) with a constant diffusion coefficient (i.e. for σ = 1), an exponential bound for deviation (2.7) was obtained. The proof of that result was based on the coupling method applied directly to the diffusion process (1.3) provided the existence of a Lyapunov function. In contrast with [20], in our paper an explicit family of Lyapunov functions is given that is of help in applications.

Remark 2.3. It should be noted that the inequality (2.7) may be applied to Monte Carlo calculation of the expectation E^ϑ_xg(y_t). Indeed, the previous expectation can be replaced with the integral of g with respect to the invariant density (2.6). The precision of such approximation is given in (2.7).

3 Computable bounds for geometric conver- gence rate

In this section we introduce the parametersR^∗ andκ^∗ which make explicit the upper bound for geometric ergodicity in (2.4). For any 0 < γ <1, we denote

B^∗ = ˇU^∗

1 + Uˇ^∗V^∗ 1−(1−δη₁)^γ

, (3.1)

where η₁ =η/(1−δ) and Uˇ^∗ = max

1−ρ+D

(1−δ)(1−ρ)^1−γ(1−(1−ρ)^γ), V^∗ (1−ρ)^1−γ

.

We remind that the parameter η appears in the condition H₁). Moreover, we

put 









r_∗ = (1−γ)²|ln(1−ρ)| |ln(1−δη₁)| ln( ˇU^∗V^∗) +|ln(1−δη₁)| ; el = 2 +

ln(V^∗B^∗)

r_∗ +lneq(1−e^−r^∗)⁻¹ 2r_∗

,

(3.2)

where

e

q = 1−Bˇ₁^∗

2 and Bˇ^∗₁ = min

e^−r^∗, δ e^r^∗−1 +δη₁

. Here [a] means the integer part of a reel number a >0. Next,

Ae= V^∗B^∗+ 1

(1−e^−r^∗)(1−e^−γr^∗) and ̺e₁ = (1−γ)²er|ln(1−eǫ)|

lnAe+r_∗el+|ln(1−eǫ)|, (3.3)

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where er =|ln(1−eq)| and eǫ =δη₁(1−δ)^e^l−2. Now we introduce the following parameter

Ae₃ = 1−γ γ



3 + 2 e^r^∗Ae

1 +Aee ^r^∗^e^l (1−(1−eǫ)^γ) (e^r^∗−1)



 V^∗B^∗+e^e^κ (3.4)

and

e

κ= (1−γ)̺e₁r_∗ e

̺₁+ lnV^∗B^∗ .

We define the parameters R^∗ and κ^∗ in the Theorem 2.1 as











κ^∗ =κ^∗(ρ, δ, D, V^∗) = (1−γ)̺e₁r_∗ 2(̺e₁+ lnV^∗B^∗); R^∗ =R^∗(ρ, δ, D, V^∗) = 2(Ae₂ + 1)e^κ^∗+ 1

e^κ^∗−1 V^∗ (B^∗)² .

(3.5)

Further we define the upper bound in the ergodic Theorem 2.2. First of all, we take the set C in the minorization condition H₁) as the intervalC = [−K, K] for someK >0 and we chose the measure ν as the uniform distribution onC, i.e. for any measurable set A fromB(R),

ν(A) = 1

2K mes(A∩[−K , K]), (3.6) where mes(·) is the Lebesgue measure on R.

In order to define the threshold δ >0 we need the quadratic function

Ω_∗(z) =ω₁^∗z²+ω^∗₂z+ω₃^∗ (3.7) with ω^∗₁ = 4(Lσ_∗)² +Lσ_∗ + 1, ω₂^∗ = Lσ_∗σ_max +H₀^∗ and ω₃^∗ = H₁^∗ + 2(H₀^∗)², where

H₀^∗ = M +σ_∗

σ_min , H₁^∗ =M(1 +σ_∗) +L+ 2σ²_∗ and σ_∗ = σ_max σ_min .

We chose the parameter δ as

δ_K = 3Ke^−Ω^∗⁽^K)^e 2√

2πσ_max , (3.8)

where Ke =K/σ_min. Now we set K₀ =p

M₁+ 8σ_min+ 8ˇσ_∗+ 4q ˇ

σ_∗(M₁+M₂) + 16ˇσ_∗², (3.9)

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where ˇσ_∗ =σ²_max/(β(1−e^−β)),

M₁ = (M+βx_∗)²+σ_max² β

β² and M₂ =M₁(1−e^−β). We set

η_K = 1−δ_K− 4ˇσ_∗(K²+M₂)

(K²−M₁)² . (3.10)

Note, that for any K ≥K₀ δ_K ≤ 3Kee ⁻^K^e²

2√

2π ≤ 3

4√

eπ and η_K ≥ 3(√

eπ−1) 4√

eπ .

Therefore, Propositions A.7–A.8 imply the condition H₁) with the parameters δ_K andη_K defined in (3.8) and (3.10) for anyK ≥K₀. To check the condition H₂) we set

b^∗₀ =β(2 +x_∗)(1 +x_∗) + (M + 3)(3 +x_∗) and b^∗₁ = b^∗₀

2β . (3.11) Propositions 6.3–6.4 imply, that for any 0 < ǫ ≤ 1/2, the diffusion process (1.3) satisfies the drift condition H₂) withV(x) = (1 +x²)^ǫ,C = [−K_ǫ, K_ǫ],

ρ_ǫ = (1−ˇǫ) (1−e^−2ǫβ) and D_ǫ =V_ǫ^∗e^−2ǫβ+b^∗₁(1−e^−2ǫβ), (3.12) where

K_ǫ =K₀+ sb^∗₁

ˇ ǫ

1/ǫ

−1 and V_ǫ^∗ = (1 +K_ǫ²)^ǫ. Now we define

κ_ǫ =κ^∗(ρ_ǫ, δ_ǫ, D_ǫ, η_ǫ, V_ǫ^∗) and R_ǫ =R^∗(ρ_ǫ, δ_ǫ, D_ǫ, η_ǫ, V_ǫ^∗), (3.13) where the functions κ^∗ and R^∗ are defined in (3.5), δ_ǫ = δ_K

ǫ, η_ǫ = η_K

ǫ and D_ǫ =D_K

ǫ.

Remark 3.1. Note that we can not apply the bounds for geometric convergence rate from the paper [22] (Theorem 12) and [21] (Theorem 8) since there the bounds were obtained under the condition

(1 +K²)^ǫ > 2D_K

ρ_ǫ . (3.14)

This condition is not satisfied in our case for sufficiently small values of the parameter ǫ > 0. In fact, we apply the bounds (2.7) in [5] to point-wise estimating the drift coefficientS under observations of the process (1.3)at discrete times (t_k)_k≥1. The question of interest is the behavior of kernel estimators that are nonlinear functionals of observations. In order to study the non asymptotic estimating precision one needs of some concentration inequalities [4] based on the bounds (2.7) with the parameter ǫ >0 no matter how small.

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In order to illustrate the behavior of the geometric rate, we suppose that the parameters satisfy the following conditions:









lim_δ→0η= 1, lim_δ→0 1−η

δ = +∞, lim_ρ→1 D

V^∗ = 0 and lim_ρ→1 lnV^∗

|ln(1−ρ)| = 1.

(3.15)

It should remark that these conditions hold true for the process (1.3) with the parametric set (2.5) as L > β → ∞ and ˇǫ→0 for some fixed ǫ >0.

It should be noted, that the geometric rate in (1.2) obtained in [17] satisfies κ^∗ =O(δ⁸) as δ→0.

In [22] under the condition (3.14) this rate is “best“, i.e.

κ^∗ =O(δ) as δ →0.

Under the conditions (3.15) the coefficient κ^∗ defined in (3.5) κ^∗ = (c₁+o(1))δ^13/2+µ⁰^(γ)

|lnδ|² as δ →0, ρ→1, where c₁ >0 and µ₀(γ)→0 as γ →0.

Remark 3.2. Note that the condition L, β → ∞ on the drift function of the process (1.3) concerns the behavior of the function only outside of the interval [−x_∗, x_∗], i.e. outside of the informative part of the function S. Remind (see, [3]), that the class (2.5) is used to bound the function S on the interval [−x_∗, x_∗] and outside of the interval the conditions are imposed to preserve ergodicity.

Remark 3.3. It should remark that, unfortunately, the bounds from the papers [1], [17] are not applicable, in general case, to classes of Markov chains.

Indeed, irreducibility is one of conditions providing the geometrical rate in [17].

Therefore, in the case of a parametric Markov chain class, irreducibility measure should depend on a class parameter. It is not clear, what will going on this measure when one takes the supremum over the class parameter and how one needs to change the irreducibility condition in order to obtain uniform bounds over the class parameter ϑ∈Θ by using the proof in [17].

4 Coupling Renewal Method

In this Section we shall obtain a non asymptotic upper bound with explicit constants in the renewal theorem by making use of the coupling method. The notions used here can be found in [8], [11], [14].

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Let (Y_j)_j≥0 and (Y_j^′)_j≥0 be two independent sequences of random variables taking values in N. Assume that the initial random variables Y₀ and Y₀^′ have distributionsa= (a(k))_k≥0 and b= (b(k))_k≥0, respectively, i.e. for anyk ≥0,

P(Y₀ =k) =a(k) and P(Y₀^′ =k) =b(k).

The sequences (Y_j)_j≥1 and (Y_j^′)_j≥1 are supposed to be the i.i.d. sequences with the same distribution p= (p(k))_k≥0, i.e. for any k ≥0,

P(Y₁ =k) =P(Y₁^′ =k) =p(k).

We assume also that p(0) = P(Y₁ = 0) = 0, i.e. the sequences (Y_j)_j≥1 and (Y_j^′)_j≥1 take values in N^∗ =N\ {0}. Moreover, we suppose that the distributions a, b and psatisfy the following condition

C) There exists a real number r >0 such that ln max (Ee^rY⁰, Ee^rY¹)

≤υ_∗(r) and ln

Ee^rY⁰^′

≤υ_∗^′(r). (4.1) For any n≥0, we define the following stopping times

t_n = inf{k ≥0 : Xk

i=0

Y_i > n} and t^′_n= inf{k ≥0 : Xk

i=0

Y_i^′ > n}.

Further, we set

W_n =

t_n

X

j=0

Y_j−n and W_n^′ =

t^′_n

X

j=0

Y_j^′−n . (4.2) It is easy to see that the sequences (W_n)_n≥1 and (W_n^′)_n≥1 are homogeneous Markov chains taking values in N^∗ such that, for any n, k and l from N^∗,

P W_n =k|W_n−1 =l

=P

W_n^′ =k|W_n−1^′ =l

=p(k)1_{l=1}+1_{k=l−1}1_{l≥2}. (4.3) Firstly we study the entrance times (s_k)_k≥0 of the chain (W_n)_n≥0 to the state {1} which are defined ass₋₁ = 0 and for k ≥0

s_k= inf{l ≥s_k−1+ 1 : W_l= 1}. (4.4) One can check directly that the stopping times s_k, k ≥ 1, can be represented as

s_k =s₀+ Xk

j=1

ς_j. (4.5)

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One can check directly that in this case (ς_j)_j≥1 are i.i.d. random variables independent of s₀ and, for any l ∈N^∗,

P(ς₁ =l) =P(s₀ =l|W₀ = 1) =p(l), (4.6) i.e. the random variables (ς_j)_j≥1 have the same distribution as Y₁. Now we study the properties of the stopping time s₀.

Proposition 4.1. Assume that the condition C) holds. Then Ee^rs⁰ ≤3e^υ^∗^(r).

Proof. First of all, note that, for k≥2 and l ≥0, P(s₀ =l|W₀ =k) = 1_{l=k−1}. Moreover, taking into account that, for any k ≥1,

P(W₀ =k) =a(0)p(k) +a(k), (4.7) we obtain that

Ee^rs⁰ =P(W₀ = 1)Ee^rY¹+ X∞ k=2

e^r(k−1)P(W₀ =k)

≤ Ee^rY¹ +a(0)e^−rEe^rY¹ +e^−rEe^rY⁰ ≤3e^υ^∗^(r). Hence Proposition 4.1.

Now, we introduce the embedded Markov chain (Z_k)_k≥0 by Z_k =W_s^′

k (4.8)

and the corresponding entrance time to the state {1}:

̟ = inf{k ≥1 : Z_k= 1}. (4.9) In order to study the property of this stopping time, we need of the following notations

l_∗ =l_∗(r) = 2 +

"

ln e^2υ^∗^(r)(1−e^−r)⁻¹q(r) 2r

#

, (4.10)

where q(r) = (1− e^υ¹^(r))/2 and the parameter υ₁(r) < 0 will be specified below. Moreover, for any 0 < γ , ǫ_∗ <1, we set

A^∗₁(r) = A^∗(r)1 +A^∗(r)e^rl^∗

1−(1−ǫ_∗)^γ , (4.11) where

A^∗(r) = 1 +Q^∗(r)e^rl^∗^+υ^∗^(r)

(1−q(r))^1−γ (1−(1−q(r))^γ) and Q^∗(r) = e^υ^∗^(r) 1−e^−r .

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Proposition 4.2. Assume the condition C). Then, for any 0 < γ < 1, for any ǫ_∗ >0 and υ₁(r)≤ −r for which

0< ǫ_∗ ≤ min

1≤j≤l_∗−1 p(j) and lnEe^−rY¹ ≤υ₁(r), (4.12) one has

Ee^̺^∗^̟ ≤ Q^∗₁(r)A^∗₁(r), where Q^∗₁(r) =e^υ^∗^(r)+e^υ^∗^′^(r)+Q^∗(r) and

̺_∗ =̺_∗(r) = (1−γ)²|ln(1−q(r))| |ln(1−ǫ_∗)|

lnA^∗(r) +rl_∗+|ln(1−ǫ_∗)| . (4.13) Proof. Firstly we note that the sequence (4.8) is a homogeneous Markov chain with values in N^∗ such that, for any mand l fromN^∗ and for anyk ≥1,

P(Z_k=m|Z_k−1 =l) =P(W_ς^′

1−l =m|Y₀^′ = 0). (4.14) Note, thatW_k^′ =−k fork < 0 under the condition Y₀^′ = 0. Therefore, for any positive function V,

E[V(Z₁)|Z₀ =l] =EV(l−ς₁)1_{ς

1<l}+EQ(ς₁−l)1_{ς

1≥l}, (4.15) where Q(n) =E(V(W_n)|Y₀ = 0). Using the distribution (4.3) yields

Q(n)≤E V(W_n−1−1)|Y₀ = 0

+EV(Y₁)P W_n−1 = 1|Y₀ = 0 . Choosing now V(x) =e^rx one has

Q(n)≤e^−rQ(n−1) +EV(Y₁).

From the last inequality, taking into account that Q(0) = EV(Y₁), it follows that, for any n ≥1,

Q(n)≤e^−nrQ(0) +EV(Y₁) Xn

j=1

e^−(n−j)r ≤Q^∗(r), (4.16) where the upper bound Q^∗(r) is defined in (4.10). This implies that the last term in (4.15) can be estimated as

EQ(ς₁−l)1_{ς₁_≥l} ≤Q^∗(r)e^υ^∗^(r)−rl. Therefore,

E (V(Z₁)|Z₀ =l)

V(l) ≤e^υ¹^(r)+Q^∗(r)e^υ^∗^(r)−2rl.