Limit Theorem for a Modified Leland Hedging Strategy under Constant Transaction Costs rate

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Limit Theorem for a Modified Leland Hedging Strategy under Constant Transaction Costs rate

Sebastien Darses, Emmanuel Denis

To cite this version:

Sebastien Darses, Emmanuel Denis. Limit Theorem for a Modified Leland Hedging Strategy under

Constant Transaction Costs rate. 2010. �hal-00467704�

STRATEGY UNDER CONSTANT TRANSACTION COSTS RATE

SEBASTIEN DARSESANDEMMANUEL DENIS

Abstract. We study the Leland model for hedging portfolios in the presence of a constant proportional transaction costs coefficient. The modified Leland’s strategy defined in [2], contrarily to the classical one, ensures the asymptotic replication of a large class of payoff. In this setting, we prove a limit theorem for the deviation between the real portfolio and the payoff. As Pergamenshchikov did in the framework of the usual Leland’s strategy [11], we identify the rate of convergence and the associated limit distribution. This rate turns out to be improved using the modified strategy and non periodic revision dates.

Keywords: Asymptotic hedging – Leland-Lott strategy – Transaction costs – Martingale limit theorem.

1. Introduction

The present paper is concerned with the study of asymptotic hedging in the pres- ence of transaction costs. The asymptotic replication of a given payoff is performed via a modified Leland’s strategy recently introduced in [2].

Let us briefly recall the history and the main known results about Leland’s strategy. In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. His main idea was to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodically revised portfolio whose terminal value approximates the payoff. The intuition behind this practical method is to compensate for transaction cost by increasing the volatility in the following way:

(1.1) bσ_t²=σ²+σ√ nkn

p8/πp f^′(t),

where n is the number of the portfolio revision dates and kn =k0n^−α, α∈[0,¹₂] is the transaction costs coefficient generally depending ofn;f is an increasing and smooth function whose inverse g := f⁻¹ defines the revision dates tⁿ_i := g(_nⁱ), 1≤i≤n.

The principal results on convergence for models with transaction costs can be described as follows. First consider the case of approximate hedging of the European call option using the strategy with periodic portfolio revisions (i.e g(t) =t). We know the following results withT = 1:

(a) For α = ¹₂, Lott gave the first rigorous result on the convergence of the approximating portfolio value V₁ⁿ to the payoff V1 = (S1 −K)+. The sequence V₁ⁿ−V1 tends to zero in probability [10], and a stronger result holds: nE(V₁ⁿ−V1)²converges to a constantA1>0[5];

1

(b) Forα∈(0,¹₂), the sequenceV₁ⁿ−V1tends to zero in probability (see [8]), and it is shown in [1] thatn^pαE (V₁ⁿ−V1)²→0as n→ ∞, withpα< α.

(c) Forα= 0, the terminal values of portfolios do not converge to the European call as shown by Kabanov and Safarian [8]. Namely, there is a negative σ{S1}-measurable random variableξsuch thatV₁ⁿ−V1→ξin probability.

Pergamenshchikov [11] then analyzed the rate of convergence and proved a limit theorem: the sequencen¹⁴(V₁ⁿ−V1−ξ)converges in law to a mixture of Gaussian distributions [11]. He noticed that one can increase the modified volatility to obtain the asymptotic replication. To do so, he utilizes the explicit form of the systematic hedging error for the European call option.

For related results see also [6] and [12].

For models including uniform and non-uniform revision intervals one needs to im- pose certain conditions on the scale transform g. Generalizations of some of the above results to this more technical case as well as extensions to contingent claims of the form h(S1) can be found in [12, 5, 1]. In particular, n^1/2+αE (V₁ⁿ−V1)² converges to a constant in the case α >0. Moreover, forα= ¹₂, the distributions of the processY_tⁿ :=n¹²(V_tⁿ−Vbt)^∗in the Skorohod spaceD[0,1]converges weakly to the distribution of a two-dimensional Markov diffusion process component (see [4]). Notice that the asymptotic replication still does not hold for α = 0 in this more general setting. For more details we refer to [1, 3, 4] and references therein.

We solve the caseα= 0for a large class of payoff and with specific non uniform revision dates by means of the modified strategy introduced in [2]. This one makes the portfolio’s terminal value converge to the contingent claim asntends to infinity, that is the approximation error vanishes. The analysis we performed here suggests that it might be difficult to obtain a better convergence rate regarding uniform revision dates. In the framework of the non uniform grid we use, concentrating the revision dates near the maturityT = 1 accelerates the convergence rate. We leave the issue concerning periodic revision dates as an open problem.

The asymptotic behavior of the hedging error is a practical important issue.

Since traders obviously prefer gains than losses, measuring theL²-norm of hedging errors is strongly criticized. Of course, the limiting distribution of the hedging error is much more informative. Our present work also aims at tackling this issue: we prove that

n^14+p(V₁ⁿ−h(S1))−−−−→_n ^d

→∞ Z,

where the law ofZ is explicitly identified andp >0depends on the choosen grid.

The paper is organized as follows. In Section2, we introduce the basic notations, models and assumptions of our study; In particular we recall the modified Leland’s strategy defined in [2]. In Section 3, we state our main result: a limit theorem for the renormalized asymptotic hedging error. In Section 4, we establish two lemmas concerning, on one hand, random variables constructed from the geometric Brownian motion, and on the other hand, some change of variables for the revision dates. These auxiliary results will be used repeatedly throughout the paper. In

∗Note thatYtⁿcorresponds to the deviation (up to a multiplicative constant) between the “real world” portfolio and the theoretical Leland’s portfolioVbt=C(t, Sb t)whereCbis the modified heat equation solution suggested by Leland whose terminal value is the payoff function.

Section 5, we prove the main result. An appendix finally recalls all the known technical results we need for the various proofs.

2. Notations and Models

2.1. Black–Scholes model and hedging strategy. We are given a filtered prob- ability space(Ω,F,(Ft)[0,1],P)on which a standard one-dimensional(Ft)-adapted Brownian motionW is defined. As usual, we denote byL²(Ω)the space of square in- tegrableF1-measurable random variables endowed with its normkXk2:=√

E X². We consider the classical Black–Scholes model composed of two assets without transaction costs, i.e. k0 = 0and bσ=σ. The first one is riskless (bond) with the interest rater= 0and the second asset isS= (St),t∈[0,1], a geometric Brownian motion that is

St = S0e^{σWt−12σ2t}. (2.2)

It satisfies the SDE

dSt = σStdWt,

with positive constants S0, σ. It means that the risky asset is seen under the martingale measure.

The well-known Black and Scholes problem without transaction costs is to hedge a payoff h(S1), h being a continuous function of polynomial growth. The pricing function solves the terminal valued Cauchy problem

(2.3)







Ct(t, x) +σ_t²

2 x²Cxx(t, x) = 0, t∈[0,1], x >0, C(1, x) = h(x).

Its solution can be written as C(t, x) =

Z _∞

−∞

h

xe^ρty−ρ

2 t 2

ϕ(y)dy (2.4)

whereρ²_t = (1−t)σ² andϕis the standard Gaussian density.

Without transaction costs (σ=bσ) the self–financed portfolio process reads Vt = C(0, S0) +

Z t 0

Cx(u, Su)dSu. (2.5)

In the Itô formula for C(t, St) the integral over dt vanishes and, therefore, Vt = C(t, St) for allt ∈ [0,1]. In particular, V1 =h(S1): At maturity the portfolioV replicates the terminal payoff of the option. Modeling assumptions of the above formulation include frictionless market and continuous trading for instance.

However, an investor revises the portfolio at a finite set of dates Tⁿ ={ti∈[0,1], i= 0,· · · , n}

and keeps Cx(ti, St_i) units of the stock until the next revision date ti+1. It is well known that this discretized model converges to the Black–Scholes one in the sense that the corresponding portfolio terminal value converges to the payoff as the number of revision dates tends to infinity.

2.2. Reminder about Leland’s strategy. We are now concerned with transac- tion costs. We directly work in a discrete time setting.

Leland suggested to replaceσin the Cauchy problem above by a suitable modi- fied volatilitybσ. In the case whereσbdoes not depend ont, the solutionCb satisfies

C(t, x) =b C(t, x,bσ), (2.6)

i.e. practitioners do not need to rectify their algorithms to compute the strategy.

Leland obtained an explicit expression of bσby equalizing the transaction costs of the portfolio and the drift term generated by the additional term bσ−σ² > 0 in the Ito expansion of the payoff h(S1) =C(1, Sb 1). In the general case, the pricing function can be written as

C(t, x) =b Z _∞

−∞

h

xe^{ρnty−(ρnt)2/2} ϕ(y)dy (2.7)

where

(ρⁿ_t)² :=

Z 1 t bσ_s²ds, (2.8)

b

σ²_t := σ²+σ√ nkn

p8/πp f^′(t), (2.9)

ϕis the Gaussian density andg=f⁻¹ is the revision date function.

2.3. A possible modification of Leland’s strategy. In the model with propor- tional transaction costs and a finite number of revision dates the current value of the portfolio process at timetis described as

(2.10) V_tⁿ:=V₀ⁿ+ Z t

0

D_uⁿdSu−X

ti<t

k0Sti|Dⁿ_i+1−Dⁿ_i|

where Dⁿ is a piecewise-constant process with Dⁿ =D_iⁿ on the interval (ti−1, ti], ti =tⁿ_i, i≤n, are the revision dates, andDⁿ_i areF^ti−1-measurable random vari- ables.

Recall that the transaction costs coefficient is a constantk0>0(that isα= 0in the Leland model) and the datesti are defined by a functiong, namelyti =g(_nⁱ).

Let us denote byf the inverse ofg. Set for alli0< n J_iⁿ₀(t) = {i≥i0, ti ≤t, ti∈ Tⁿ} and let us define the dates

t⁻_n(t) = t(n−1)∧maxJ₀ⁿ(t)

t⁺_n(t) = t1+(n−1)∧maxJ₀ⁿ(t).

The “enlarged volatility”, depending onn, is given by the formula (2.9).

We modify the usual Leland strategy (see [2]) by considering the process Dⁿ with

Dⁿ_i := Cbx(ti−1, Sti−1)−

i−1

X

j=1

Z tj

t_j−1

Cbxt(u, Stj−1)du.

(2.11)

Moreover, let us define

K_tⁿ:= X

i∈J₁ⁿ(t)

∆K_tⁿⁿ

i

where∆K_tⁿⁿ

0 := 0and fori≥1,

∆K_tⁿⁿ

i :=−

Z t_i ti−1

Cbxt(u, Su)du.

(2.12)

In the same way, we set

Lⁿ_t := X

i∈J₁ⁿ(t)

∆Lⁿ_tⁿ

i

where∆Lⁿ_tn

0 := 0and fori≥1,

∆Lⁿ_tⁿ

i :=−

Z ti

ti−1

Cbxt(u, Sti−1)du.

(2.13)

2.4. Assumptions and notational conventions. Throughout the paper, we adopt the following rules:

(i) we will often omit the indexes n and the variablet (especially in the ap- pendix) when there is no ambiguity;

(ii) the constants C appearing in the various inequalities is independent of n and may change from one line to the next;

(iii) we use the classical Landau notations O and o. These quantities will be always deterministic.

We use the abbreviations

δt := Cbx(t, St), (2.14)

γt := Cbxx(t, St).

(2.15)

We denote by(δⁿ_t)t the process equal toδtⁿ_i on the interval[tⁿ_i, tⁿ_i+1)and (γ_tⁿ)t is defined similarly. For an arbitrary processH, we set ∆Ht_i =Ht_i−Ht_i−1.

We will work under the following assumptions:

(A1) The functiong has the following form:

(2.16) g(t) = 1−(1−t)^µ, µ∈ 1,3 +√ 57 8

! .

(A2) his a convex and continuous function on[0,∞)which is twice differentiable except the pointsK1<· · ·< Kpwhereh^′andh^′′admit right and left limits;

|h^′′(x)| ≤M x^−β forx≥Kp whereβ≥3/2.

Assumption (A1) is not too restrictive. A trader can in particular choose µ sufficiently close to1to balance its portfolio quasi periodically. However, as we will see, it is more preferable to increaseµto obtain a better rate of convergence.

Note thatf(t) = 1−(1−t)^1/µ, hence the derivativef^′ forµ >1explodes at the maturity date and so does the enlarged volatility. We define the increasing function

p:=p(µ) := µ−1 4(1 +µ). (2.17)

Under Assumption (A1), we have0< p <1/16.

In the sequel, the quantity

Q(µ) = µ^1/2−2p(1 +µ)^4p 2^4pp

8/π4p+1

(2.18)

will frequently appear.

0 i-1 i i+1 i+2 i+3 1

0 t(i-1) t(i) t(i+1) t(i+2) t(i+3) 1

Revision dates

f(t, 1/mu)

Figure 1. Revision dates withf(t) = 1−(1−t)^1µ, µ= 1.5 andn= 10.

3. Main Result

In [2], it is proven thatV₁ⁿ converges in probability toh(S1). Our main result here provides the rate of convergence and identifies the associated limit distribution of the deviation:

Theorem 3.1. Consider the portfolioVⁿ defined by (2.10) and (2.11) under As- sumptions (A1) and (A2). The following convergence then holds:

n^14+p(V₁ⁿ−h(S1))−−−−→_n ^d

→∞ Z, (3.19)

where the law ofZ is given by the characteristic functionφZ(s) =Ee^−s

2 2η² with η² := Q(µ)(k0σ)^1−4pS₁²×

Z _∞

0

x^4p

(Z _∞

x

J(y, S1)dy 2

+

1− 2 π

Je(x, S1)² )

dx, and

J(x, S1) := 1 2x

Z ∞

−∞

h^′(Sue^√xy+x/2)(−y²−√

xy+ 1)ϕ(y)dy (3.20)

J(x, Se 1) := 1

√x Z ∞

−∞

h^′(Sue^√xy+x/2)yϕ(y)dy.

(3.21) Moreover

n^12+2pE (V₁ⁿ−h(S1))²−−−−→_n

→∞ E η².

4. Auxiliary results

4.1. Geometric Brownian motion and related quantities. In the sequel, we shall use the decomposition given by Ito formula

(4.22) Cbx(t, St) =Cbx(0, S0) +Mc_tⁿ+Abⁿ_t where

Mc_tⁿ :=

Z t 0

σuSuCbxx(u, Su)dWu, Abⁿ_t :=

Z t 0

Cbxt(u, Su) +1

2σ²_uS_u²Cbxxx(u, Su)

du.

The processMcⁿ is a square integrable martingale on[0,1]by virtue of [2].

We set foru < v

Eu^v= Sv

Su −1, and

[Eu^v]_c = E (|Eu^v|)− |Eu^v|. {Eu^v}²s := (Eu^v)²sgnEu^v.

In the sequel, we will use several times the following basic results.

Lemma 4.1. For alli the following inequalities and expansions hold:

E (Eu^v)^2m ≤ Cm(v−u)^m, u≤v (4.23)

E Et^ti−ⁱ 1

2

= σ²∆ti(1 +o(1)) (4.24)

E h Et^ti−1ⁱ

i2

c =

1−2

π

σ²∆ti(1 +o(1)) (4.25)

E h Et^ti−ⁱ 1

i2

csgnEu^v =

1−2 π

σ²(∆ti)³²(1 +o(1)) (4.26)

E {Et^ti−1ⁱ }²s = k(∆ti)^3/2

1 +o(n^−1/4) . (4.27)

Proof. We refer to [1] or [4]. For the sake of completeness we recall the proof of the last one. Let us notice the equality in law

{Et^ti−ⁱ 1}²s

=d expn

σp

∆tjξ−σ²∆tj/2o

−12 1_ξ≥σ√

∆tj/2−1_ξ≤σ√

∆tj/2

,

whereξ is the standard Gaussian variable. Sinceξ and−ξ has the same law, this yields

E{Et^ti−1ⁱ }²s = E

e^uξ−u2/2−12

−

e^{−uξ−u2/2}−12 1ξ≥u/2

−E

e^−uξ−u/2−12

1_|ξ|≤u/2, whereu=σp

∆tj. Moreover, we have the inequality E

e^{−uξ−u2/2}−12

1_|ξ|≤u/2≤u⁴.

From [5], we recall that E

e^uξ−u2/2−12

−

e^{−uξ−u2/2}−12

1ξ≥u/2= 2

√2πu³+O(u⁴).

We then conclude.

4.2. Basic results concerning the revision dates. The function ρt decreases fromρ0 to0. The following useful bounds are obvious:

ρ²_t ≥ (σ²+cn¹²)(1−t) (4.28)

ρ²_t ≤ σ²(1−t) +σk0n¹²p

8/π(1−t)¹²(1−f(t))¹². (4.29)

Moreover, it is straightforward that ρ²_t ≥ cn¹²p

f^′(t)(1−t), (4.30)

provided thatf^′ is no decreasing.

Note that there is a constantC independent ofnsuch that for alli≤n−1, 1−ti−1

1−ti ≤ C.

(4.31)

From there we deduce

∆ti

1−ti ≤ C.

(4.32)

We shall often use the inequality

nX−1 i=1

∆ti

1−ti ≤C log(n) whereC is a constant independent ofn.

Lemma 4.2. Fix x > 0 and t := t(n, x) such that x = ρ²_t. Set xi−1 = ρ²_ti−1 and xi = ρ²_ti where ti−1, ti are such that t ∈[ti−1, ti). Then, x∈ (xi, xi−1] with

|xi−1−xi| ≤c n^−1/2,c is a constant. There exists a constantC >0 such that

∆tin^1/2+2p

xi−1−xi ≤ C(x+ 1).

(4.33)

Moreover, for a givenx,(1−t)≤cn^−1/2x→0asn→ ∞and

∆tin^1/2+2p xi−1−xi −−−−→_n

→∞

Q(µ)x^4p (σk0)^4p+1. (4.34)

Proof. Let us write

∆tin^1/2+2p xi−1−xi

= n^2p

σ²n^−1/2+σk0

p8/π_∆t¹

i

Rti

ti−1

pf^′(u)du

n→∞∼

n^2p σk0

p8/πp f^′(˜ti) where˜ti∈[ti−1, ti]. Moreover

x=ρ²_t = σ²(1−t) +σk0

p8/πn^1/2 Z 1

t

pf^′(u)du

= σ²(1−t) +σk0

p8/πn^1/22µ^1/2

1 +µ(1−t)^1+µ2µ

and

1−t= x−σ²(1−t) σk0

p8/πn^1/2 1 +µ 2µ^1/2

!_1+µ^2µ .

Note thatx≥cn^1/2(1−t)so that(1−t)≤cn^−1/2x→0. In a similar way, we have

xi−1−xi=ρ²_ti−1−ρ²_ti =σ²∆ti+cn^1/2 q

f^′(ˇti)∆ti

whereˇti∈[ti−1, ti]. We deduce that xi−1−xi=σ²∆ti+cn^1/2

q

f^′(ˇti)g^′(θi)n⁻¹ whereθi∈[(i−1)/n, i/n]. Moreover,

q

f^′(ˇti)g^′(θi) = g^′(θi) pg^′(f(ˇti))

is bounded since f(ˇti) ∈ [(i−1)/n, i/n], i ≤n−1. Hence there is a constant c satisfyingxi−1−xi≤cn^−1/2. Sinceρis decreasing,x∈[xi, xi−1].

Eventually,˜ti∈[ti−1, ti] is such thatx˜i =ρ²_˜t

i∈[xi, xi−1]andx˜i→x. Similarly we have

1−t˜i= x˜i−σ²(1−t˜i) σk0

p8/πn^1/2 1 +µ 2µ^1/2

!_1+µ^2µ (4.35)

which yields q

f^′(˜ti) =µ^−1/2 x˜i−σ²(1−˜ti) σk0

p8/πn^1/2 1 +µ 2µ^1/2

!¹_1+µ^−µ (4.36)

and

∆tin^1/2+2p xi−1−xi _n∼

→∞

n^2p σk0

p8/πp f^′(˜ti)

n→∞∼

n^2p σk0

p8/πµ^1/2 σk0

p8/πn^1/2

˜

xi−σ²(1−˜ti) 2µ^1/2 1 +µ

!^1−µ_1+µ

n→∞∼

1 σk0

p8/πµ^1/2 σk0

p8/π

˜

xi−σ²(1−˜ti) 2µ^1/2 1 +µ

!^1−µ_1+µ . Sincex˜i→xandt˜i→0, we deduce that

∆tin^1/2+2p xi−1−xi −−−−→_n

→∞

1 σk0

p8/πµ^1/2 σk0

p8/π x

2µ^1/2 1 +µ

!^1−µ_1+µ . Since0<(µ−1)/(1 +µ)<1, we also find a constantcsuch that

∆tin^1/2+2p

xi−1−xi ≤c˜xi−σ²(1−t˜i)^µ−1+µ1 ≤c(x+ 1),

which concludes the proof.

We now stress an important remark regarding a slight abuse of notation repeat- edly used along the paper.

Remark 4.3. Throughout the sequel, we shall often use the change of variable x=ρ²_t withdx=−bσ²_tdt. For ease of notation, we will use the abuse of notation t instead oft(n, x) := (ρ²)⁻¹(x)when applying this change of variable in an integral.

Similarly, a direct computation yields the following lemma.

Lemma 4.4. Sety >0 andv:=v(n, y)such thaty=ρ²_v. There exists a constant C >0 such that

(1−v)n^1/2+2p

y ≤C y.

Moreover, for a giveny,(1−v)≤cn^−1/2y →0as n→ ∞and (1−v)n^1/2+2p

y −−−−→_n

→∞

µ^−1/2−2p(1 +µ)^4p+1y^4p 2^4p

σk0

p8/π4p+1 .

5. Proof of the limit theorem

The proof is divided into three parts. In Step1we split the hedging error into a martingale partM and a residual partε. In Step2we show that the residual terms tend to0inL²(Ω) with raten^14+pas ntends to infinity. We identify in Step3the asymptotic distribution of the martingale n^14+pMⁿ and we conclude the proof of the main result.

5.1. Step 1: Splitting of the hedging error. Comparing Expression (2.10) with the Ito expansion of h(S1) = C(1, Sb 1) yields the following decompositions.

The hedging error reads

V₁ⁿ−h(S1) =M₁ⁿ+εⁿ₁ (5.37)

where for alln∈N,Mⁿ is a martingale of terminal value M₁ⁿ := k0

X

i≤n−1

γti−1S_t²_i−1h Et^ti−ⁱ 1

i

c+ Z 1

0

K_uⁿdSu. (5.38)

The residual term can be splited as

εⁿ_t =Rⁿ₀(t) +Rⁿ₁(t) +Rⁿ₂(t) +Rⁿ₃(t) (5.39)

where

Rⁿ₀(t) := k0

X

i∈J₁ⁿ(t)

γti−1S²_ti−1 σ r2

π

pnf^′(ti−1)∆ti−E Et^ti−ⁱ 1

! (5.40)

Rⁿ₁(t) :=

Z t 0

(δⁿ_u−δu)dSu, (5.41)

Rⁿ₂(t) := k0

X

i∈J₁ⁿ(t)

|∆δ_tⁿ_i+ ∆K_tⁿ_i| − |∆δ_tⁿ_i+ ∆Lⁿ_ti| St_i, (5.42)

Rⁿ₃(t) :=

Z t 0

(Lⁿ_u−K_uⁿ)dSu. (5.43)

5.2. Step2: The mean square residue tends to 0 with rate n^12+2p.

The most technical part of this paper is the following. The deviation of the approximating portfolio from the payoff has been written in an integral form by virtue of the Ito formula. The “real world” portfolio may be interpreted as a discrete- time approximation of the theoretical portfolioC(t, Sb t)yielding the residual terms above. Consequently, the following analysis is mainly based on Taylor approxima- tions involving the successive derivatives of Cb and so heavily utilizes estimates of the appendix. Standard tools from stochastic calculus are also frequently used.

Theorem 5.1. The following convergence holds:

n^12+2pE(εⁿ_t)²−−−−→

n→∞ 0.

(5.44)

To prove this theorem, we show the suitable convergence to 0 concerning the Rj, 0≤j≤3.

Lemma 5.2.

n^12+2pE (Rⁿ₀)²−−−−→_n

→∞ 0.

(5.45)

Proof. We have E Et^t_i−1ⁱ

= 4Φ

σ√

∆ti

2

−2 = σ r2

π

p∆ti+ (∆ti)o(1),

σ r2

πn¹²p

f^′(ti−1)∆ti = σ r2

π

p∆tiεi

whereεi=n¹²√

∆ti

pf^′(ti−1)verifies

|εi−1| ≤ c∆ti

1−ti

by virtue of Lemma 6.12. Hence, there is a constantC >0 such that:

sup

t |R₀ⁿ(t)| ≤ Ck0 nX−1

i=1

γti−1S_t²_i−1(∆ti)³² 1−ti

.

From Corollary 6.5 and Inequalities (4.28–4.32), we deduce the following n^14+p

s E

sup

t |Rⁿ₀(t)| 2

≤ Cn^18+p

nX−1 i=1

(∆ti)³² (1−ti)^5/4 (5.46)

≤ Cn^18+p

n¹⁴ logn−−−−→_n

→∞ 0.

(5.47)

A Taylor formula suggests to write the following splitting:

Rⁿ₁ = σ(Rⁿ₁₀−Rⁿ₁₁−Rⁿ₁₂−Rⁿ₁₃+ 2Rⁿ₁₄), (5.48)

where

Rⁿ₁₀(t) := X

i≤n

γti−1S_t²_i−1 Z ti∧t

ti−1∧tEt^u_i−1

Su

Sti−1

dWu

Rⁿ₁₁(t) :=

nX−1 i=1

Z ti∧t t_i−1∧t

Cbxt(ti−1, Sti−1)(u−ti−1)SudWu,

Rⁿ₁₂(t) := 1 2

n−1

X

i=1

S_t³_i−1 Z ti∧t

t_i−1∧t

Cbxxx(eti−1,Seti−1) Et^ui−1

2 Su

St_i−1dWu, Rⁿ₁₃(t) := 1

2

n−1

X

i=1

Sti−1

Z ti∧t ti−1∧t

Cbxtt(eti−1,Seti−1)(u−ti−1)² Su

St_i−1dWu, Rⁿ₁₄(t) := −1

2

nX−1 i=1

S²_ti−1 Z ti∧t

ti−1∧t

Cbxxt(eti−1,Seti−1)Et^ui−1(u−ti−1) Su

St_i−1dWu. Lemma 5.3.

n^32+2pE sup

t∈[0,1]

Rⁿ₁₀(t)

!2

−−−−→

n→∞ 0.

(5.49)

Proof. The Doob inequality yields n^32+2pE sup

t (Rⁿ₁₀(t))²≤4n^32+2pE(Rⁿ₁₀(1))²

where the r.h.s tends to 0 as shown below. Indeed, by the independence of the increments of the Wiener process, we write:

E (Rⁿ₁₀(1))²=σ² Xn i=1

Λti−1

Z ti

ti−1

E Et^ui−1

2 S_u² S_t²_i−1du where

Λt:=ECb_xx² (t, St)S_t⁴. (5.50)

It is easy to check the following asymptotic

E Et^ui−1

2 S_u²

S_t²_i−1 = σ²(u−ti−1) + (u−ti−1)O(n⁻¹).

Therefore

E(R₁₀ⁿ(1))²=σ⁴ 2

X

i≤n

Λti−1(∆ti)²(1 +O(n⁻¹)) where∆ti=g^′(θi)/n withθi∈[(i−1)/n, i/n].We then deduce

n^32+2pE(Rⁿ₁₀(1))²= σ⁴(1 +O(n⁻¹)) 2

X

i≤n

Λti−1(∆tin)∆tin^12+2p xi−1−xi

(xi−1−xi)

wherexi=ρ²_ti. So, we have:

n^32+2pE(Rⁿ₁₀(1))²= σ⁴(1 +O(n⁻¹)) 2

Z ρ²₀ 0

fn(x)dx where

fn(x) = Xn i=1

Λti−1(∆tin)∆tin^12+2p xi−1−xi

1(xi,xi−1](x).

Let us remark the abuse of notations ρ²₀ = ρ²_0,n and ti = t(n, xi) as previously mentioned.

First, let us show thatfn satisfies the dominated convergence bound condition.

Ifx∈(xi, xi−1]then from Corollary 6.5, we have 0≤Λti−1 ≤ C

√xi−1e^−xi−1/4≤ C

√xe^−x/4. Thus, from (4.33) we obtain

fn(x)≤ C

√xe^−x/4(1 +x).

(5.51)

Regarding the pointwise convergence offn, for a givenx∈(xi, xi−1], there exists u∈[ti−1, ti)such that x=ρ²_u ≥cn¹²(1−u). It follows that not onlyu→1 but also ti, ti−1 → 1. Recall that ∆ti = g^′(θi)n⁻¹ where θi ∈ [(i−1)/n, i/n]. Thus g(θi)→1andθi→1sincef is continuous. Therefore∆tin→g^′(1) = 0. Moreover, note that

Λti−1 = 1 xi−1

Z ∞

−∞

e^{2σ√ti−1z−σ2ti−1}Υi(z)ϕ(z)dz

where

Υi(z) = Z _∞

−∞

h^′

e^{σ√ti−1z−σ}

2ti−1

2 +√xi−1y+^xi−₂¹

yϕ(y)dy 2

. Applying the Lebesgue theorem, we deduce that Λti−1 converges to

Λ(x) := 1 x

Z _∞

−∞

e^2σz−σ2 Z _∞

−∞

h^′ e^σz−σ

2

2 +√xy+^x₂

yϕ(y)dy 2

ϕ(z)dz.

Finally, together with (4.34),fn−−−−→

n→∞ 0a.e. We then apply the Lebesgue theorem to conclude the following limit

σ⁴(1 +O(n⁻¹)) 2

Z ρ²₀ 0

fn(x)dx−−−−→_n

→∞ 0.

Lemma 5.4. n^12+2pE(sup_tRⁿ₁₁(t))²−−−−→

n→∞ 0.

Proof. Using the Doob inequality, we obtain thatE(sup_tRⁿ₁₁(t))²≤4E(Rⁿ₁₁(1))². By independence of the increments of the Wiener process, we deduce that

n^12+2pE(R₁₁ⁿ(1))²=n^12+2p

n−1

X

i=1

ECb_xt²(ti−1, Sti−1)S_t²_i−1 Z ti

ti−1

(u−ti−1)²E Su

St_i−1

2

du.

It follows that

n^12+2pE (Rⁿ₁₁(1))²≤cn^12+2p

n−1

X

i=1

ECb_xt²(ti−1, Sti−1)S_t²_i−1(∆ti)³≤cn^−14+2plogn, since Corollary 6.14 gives

ECb_xt²(ti−1, Sti−1)S_t²_i−1 ≤cn¹⁴f^′(ti−1) (1−ti−1)³²

wherenf^′(ti−1)∆ti is bounded. We then conclude.

Lemma 5.5. n^12+2pE(sup_tRⁿ₁₂(t))²−−−−→_n

→∞ 0.

Proof. As previously, we have the Doob inequalityE(sup_tRⁿ₁₂(t))²≤4E(Rⁿ₁₂(1))² and the equality

4E (Rⁿ₁₂(1))²=

nX−1 i=1

Z ti

ti−1

E Cb_xxx² (eti−1,Seti−1)S_t⁶_i−1

1− St

Sti−1

4

S_t² S_t²_i−1

! dt.

From (6.125), there exists a constantC such that:

E Cb_xxx⁴ (eti−1,Seti−1) ≤ C ρ⁸_ti. (5.52)

Using the Cauchy-Schwarz inequality and (4.23) withm= 8, we deduce that n^12+2pE(Rⁿ₁₂(1))² ≤ Cn^12+2p

nX−1 i=1

(∆ti)³ n(1−ti)²

≤ Cn^2plogn n³²

which proves the desired convergence to0.

Lemma 5.6. n^12+2pE(sup_tRⁿ₁₃(t))²−−−−→

n→∞ 0.

Proof. We still consider the Doob inequalityE(sup_tRⁿ₁₃(t))²≤4E (Rⁿ₁₃(1))² and 4E(Rⁿ₁₃(1))²≤

n−1

X

i=1

Z ti

ti−1

E

Cb_xtt² (eti−1,Seti−1)(t−ti−1)⁴S_t² dt.

Moreover, using Lemma 6.19 and the Cauchy-Schwarz inequality, we deduce that E

Cb_xtt² (eti−1,Seti−1)S_t²

≤ c (1−ti)⁴. Then, we obtain

n^12+2pE(Rⁿ₁₃(1))²≤Cn^12+2p

n−1

X

i=1

(∆ti)⁵

(1−ti)⁴ ≤C n^−12+2plogn.

The conclusion follows.

Lemma 5.7. n^12+2pE(sup_tRⁿ₁₄(t))²−−−−→

n→∞ 0.

Proof. We use the Doob inequalityE(sup_tRⁿ₁₄(t))²≤4E(Rⁿ₁₄(1))²and the equal- ity

4E(Rⁿ₁₄(1))²=

nX−1 i=1

Z ti

ti−1

E S_t⁴_i−1Cb_xxt² (eti−1,Seti−1)

1− St

Sti−1

2

(t−ti−1)² S_t² S²_ti−1

! dt.

From (6.126), we deduce that E S⁴_ti−1Cb_xxt² (eti−1,Set_i−1)

1− St

Sti−1

2

S²_t S_t²_i−1

!

≤ct−ti−1

(1−ti)³. Then,

n^12+2pE (Rⁿ₁₄(1))²≤cn^12+2p

n−1

X

i=1

(ti−ti−1)⁴

(1−ti)³ ≤c n^−12+2plogn.

Thus, we can conclude.

Let us now study the residual term Rⁿ₂. Again, a Taylor formula suggests to write the following splitting:

Rⁿ₂ =Rⁿ₂₀+· · ·+Rⁿ₂₄, (5.53)