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Approximate hedging for non linear transaction costs on
the volume of traded assets
Romuald Elie, Emmanuel L´epinette
Abstract This paper is dedicated to the replication of a convex contingent claim h(S1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs rewrite as a non lin-ear functionG of the volume of traded assets, with G0(0)>0. For a stock with Black-Scholes mid-price dynamics, we exhibit an asymptotically convergent repli-cating portfolio, defined on a regular time grid withntrading dates. Up to a well chosen regularizationhn of the payoff function, we first introduce the frictionless replicating portfolio of hn(S1n), where Sn is a fictive stock with enlarged local volatility dynamics. In the market with frictions, a proper modification of this portfolio strategy provides a terminal wealth, which converges in probability to the claim of interesth(S1), as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G0(0) of the bid-ask spread. The main innovation of the paper is the introduction of a ’Leland type’ strategy for non-vanishing (non-linear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, that we pass through using an innovative approach based on the Malliavin calculus representation of the Greeks.
Key wordsLeland–Lott strategy, Delta hedging, Malliavin Calculus, transaction costs, order book.
Mathematics Subject Classification (2010)91G20 ; 60G44 ; 60H07 JEL ClassificationG11·G13
1 Introduction
The current high frequency of trading on the financial markets does not allow to neglect the frictions induced by market orders for buying or selling a given number of shares. Depending on the liquidity of the stock of interest, the marginal price of
CEREMADE, CNRS, UMR 7534, Universit´e Paris-Dauphine
any extra unit of stock can be significantly different. The shape of the order book and the size of the bid-ask spread determine the underlying cost induced by pass-ing an order on the market. Modelpass-ing order book dynamics and more importantly quantifying the impact of the trades on the underlying price have brought a lot of attention in the recent literature. Our concern in this paper is to look towards efficient alternatives in order to replicate options in the presence of transaction costs, related to the presence of order books.
This kind of induced cost rewrites as a function of the traded amount of shares instead of the more classical and less realistic traded amount of money. For simplic-ity here, the order book shape is supposed to be deterministic and has a stationary asymptotic behavior when the number of traded shares goes to zero. More pre-cisely, tradingγshares of stock at timetinduces a costG(t, γ) where the possibly non-linear functionGsatisfiesG(t, γ)∼ G0(0)|γ|+O(|γ|2), forγ small enough. We consider a financial market with one bond normalized to 1 and one stockS with Black–Scholes mid-price dynamics. Observe thatG0(0) interprets as the half size of the bid-ask spread. The order book induces frictions on any position taken on the stock and we investigate the replication of a European option with payoffh(S1), wherehis a convex function.
In the classical framework of proportional transaction costs on the amount of traded money, Leland [8] introduced an ingenious method in order to hedge efficiently call options on a discrete time grid. His idea relies on the use of the frictionless hedging strategy associated to a Black–Scholes stock with a suitably enlarged volatility, related to the chosen frequency of trading. As the number of trading dates goes to infinity, Lott [10] or Kabanov and Safarian [6] verified that the terminal value of the corresponding portfolio converges to the claim h(S1) of interest, under the additional condition that the transaction costs coefficient vanishes sufficiently fast as well. This unrealistic assumption has recently been relieved by L´epinette [9] via a proper modification of the replicating strategy.
The main motivation of the paper is the introduction of ’Leland-Lott’ ap-proximate hedging strategies in the realistic framework described above, where the amount of transaction costs is a non linear function of the number of traded shares of asset. This particular feature implies that the natural ’Leland-type’ en-larged volatility is associated to a local volatility model instead of a Black–Scholes one. Indeed, we consider the pricing function ˆCn and associated delta hedging strategy ˆCxn induced by a fictive asset with local volatility
ˆ σn : (t, x)7→ s |σx|2+σG0(0) r 8n π x , (1.1)
whereσis the Black–Scholes volatility of the stock and 1/nis the mesh size of the regular revision grid.
In the imperfect market of interest, we exhibit a portfolio starting with ini-tial wealth ˆCn(0, S0) and induced by a proper modification of the delta hedging strategy ( ˆCxn(t, St))0≤t≤T, in the spirit of [9]. The main result of the paper is the
interest h(S1), as the number of revision dates n tends to infinity. This conver-gence requires to consider payoff functionshwith bounded second derivatives. For derivatives with less regular payoff functions such as the classical call option, one simply needs to replacehby a well chosen more regular payoff functionhn, char-acterized in terms of number of trading datesnof the hedging strategy.
The approximate hedging strategy introduced in this paper allows therefore to replicate asymptotically a convex contingent claimh(S1) in a market with non vanishing transaction costs coefficient related to deterministic order books. The enhanced strategy only relies on the size 2G0(0) of the bid-ask spread and not on the global shape of the order book. The consideration of a fictive asset with local volatility dynamics of the form (1.1) induces lots of technicalities since the Lott–Kabanov methodology requires precise estimates on the sensitivities of the pricing function ˆCn in terms of the number n of trading dates. The rather com-putational obtention of these estimates relies on an innovative approach based on the Malliavin representation of the Greeks introduced in [4].
The paper is organized as follows: The next section presents the financial mar-ket with frictions and the replication problem of interest. Section 3 is dedicated to the main results of the paper: the construction of the modified volatility and corresponding fictive pricing and hedging functions, the Delta correction for the consideration of non-vanishing transaction costs coefficient, the payoff regulariza-tion and the convergence of the enhanced replicating strategy. Secregulariza-tion 4 details the proof of the convergence, whereas technical estimates on the derivatives of the fictive pricing function ˆCn are reported in Section 5.
Notations. For a function f from [0,1]×R to R, we denote byft, fx, ftx,fxx,
. . .the time and space partial derivatives. For a function f from R to R, the first and second derivatives are simply denoted∇f and∇2f. We denote byCa generic constant, which may vary from line to line. For possibly random constants, we use the notationCω.
2 Hedging under transaction costs on the traded volume of shares
In this section, we introduce the market model and formulate the financial deriva-tive replication problem under transaction costs induced by order book frictions.
2.1 The market model
We consider a financial market defined on a probability space (Ω, F ,Q), endowed with a 1-dimensional Brownian motionW. We denote by F = (Ft)t≥0the comple-tion of the filtracomple-tion generated byW.
Our model is the standard two-asset model with the time horizon T = 1 as-suming that it is specified under the unique martingale measure Q. The non-risky asset is thenum´eraire S0= 1, and the dynamics of the risky asset is given by the
stochastic equation
St=S0+ Z t
0
σSudWu, 0≤ t ≤1,
whereσ >0 is a constant. Up to considering discounted processes, all the results of the paper extend as usual to financial markets with non zero deterministic interest rates.
In a frictionless complete market of this form, the price at timetof a financial derivativeh(S1) is given byC(t, St) whereC is the unique solution of the PDE
(e0) =
Ct(t, x) + 12σ2x2Cxx(t, x) = 0, (t, x)∈[0,1)×(0, ∞)
C(1, x) =h(x), x ∈(0, ∞) . In presence of realistic transaction costs, where continuous hedging is not adequate anymore, this paper develops an asymptotic hedging strategy for the financial derivativeh(S1).
2.2 The order book frictions
We intend to take into account the frictions induced by the use of market orders in the financial market. When a portfolio manager buys or sells a given quantityγ 6= 0 of stockS, the presence of order books implies an additional cost, which is related to the volume γ of the order. We model these order book related costs via the introduction of a non linear continuous deterministic cost function G. Whenever an agent trades a (possibly negative) quantityγof stocksSon the financial market at timet, he shall pay an immediate costG(t, γ)>0.
We make the following stationary assumption on the asymptotic behavior of the cost functionGon the neighborhood ofγ= 0.
Condition (G): There exists a constantG0(0)>0 such that G(t, γ) =G0(0)|γ|+O(|γ|2), 0≤ t ≤1.
Remark 2.1 WhenSrepresents the mid-price dynamics of the risky financial asset, 2G0(0) interprets simply as the bid-ask spread of the asset in the order book of interest. We shall see in the following that for asymptotic replication purpose, only the size 2G0(0) of the bid-ask spread is relevant in our approach.
Remark 2.2 Of course, assuming that the order book is deterministic and that the bid-ask spread remains constant is unrealistic and hence restrictive. Nevertheless, we outline in this paper that this simple framework already raises interesting mathematical problems and leads to promising conclusions. The consideration of dynamic random order books, for which no unanimous model has emerged in the literature, shall be left for further research.
2.3 Portfolio dynamics and replication
Due to the presence of frictions on the market, inducing direct or indirect trans-action costs, we only consider portfolio strategies, where the manager changes his market position on a finite number nof revision dates (tni)0≤i≤n. For simplicity,
we assume in the paper that the revision dates (tni) define a uniform deterministic time grid, i.e.tni :=i/n, for 0≤ i ≤ n.
Remark 2.3 As observed in [2] or [13], the use of non uniform time grid, where the number of trading dates increases as the maturity is getting closer, allows to improve the convergence of the Leland type approximate hedging strategy. One can expect this property to remain satisfied in our context. A rigorous proof of this result requires very computational finer estimates, which go beyond the scope of this (already technical) paper. For the consideration of random time nets, we refer to the nice results of [5], which produces a robust asymptotic hedging strategy for vanishing linear transaction costs written in terms of the traded amount of money. A portfolio on the time interval [0,1] is given by an initial capitalx ∈R and an F-adapted piecewise-constant process (Hn)n∈N, whereH
n tn
i ∈ L
2(Ω) represents the number of shares of stock hold in the portfolio on the time interval [tni, tni+1), for any 0≤ i < n. Due to the order book frictions, the value of the portfolio process Vn associated to the piecewise-constant investment strategyHn is given by
Vtn=V0n+ Z t 0 HundSu− X tn i≤t G tni, Htnn i − H n tn i−1 , 0≤ t ≤1, n ∈N. (2.2) We aim at hedging the contingent claim with payoffh(S1), wherehis a convex function, for which precise regularity requirements are given in Section 3.3 below. We look towards a portfolio Vn, with terminal value converging toh(S1) as the number of trading datesntends to infinity.
3 Asymptotic hedging via volatility modification and payoff regularization In order to exhibit a portfolio strategy, whose asymptotic terminal value attains the claim of interesth(S1) despite the frictions, we formally explain in Section 3.1 the Leland methodology and consider a fictive asset with upgraded volatility. Since transaction costs rewrite in our framework as a function of the volume of traded asset, the fictive asset has non Lipschitz local volatility dynamics. After verifying in Section 3.2 that this stochastic differential equation has a unique solution, we introduce the corresponding pricing and hedging functions of the claimh(S1) for a frictionless market. Up to a proper strategy modification, we exhibit in Section 3.4 an asymptotic hedging strategy for the convex claim h(S1). For payoff functions with few regularity such as call option, a well chosen additional regularization method is exposed in Section 3.3.
3.1 Construction of the enlarged volatility function
In the frictionless Black–Scholes model, the price function of the convex claim h(S1) is the unique solutionC(., .) of the PDE (e0) and the exact self-financing
replication portfolio is given by
C(t, St) = Eh(S1) + Z t
0
Cx(u, Su)dSu, 0≤ t ≤1.
It exactly replicates the contingent claimh(S1) and is self-financing. In the pres-ence of transaction costs, Leland suggested in his famous paper [8] to substitute the volatilityσ by an artificially enlarged onebσn, related to the mesh 1/nof the trading replication grid. We briefly recall the main ideas behind this volatility enlargement and detail formally how it adapts to the framework of frictions con-sidered here.
For a sequence of volatility functions (σbn)n to be determined below, consider the following PDEs
ut(t, x) + 12bσ 2
n(x)x2uxx(t, x) = 0, (t, x)∈[0,1)×(0, ∞)
u(1, x) = h(x), x ∈(0, ∞) ,
forn ∈N. The solutionCn of this equation (if it exists) is the frictionless pricing function of a financial derivative with payoff function h, whenever the stock has b
σn local volatility dynamics.
We look towards a volatility function bσ
n allowing to take into account the
transaction costs induced on the n trading dates. More precisely, Ito’s formula implies that the formally supposed smooth functionCn verifies
Cn(t, St) =Cn(0, S0) + Z t 0 Cxn(u, Su)dSu+1 2 Z t 0 h σ2−bσ2n(Su) i Su2Cxxn (u, Su)du,
for 0≤ t ≤1 andn ∈N. Hence, the process (Cn(t, St))0≤t≤1can be approximately identified as a portfolio process with dynamics of the form (2.2) whenever the last term on the right hand side above corresponds to the transaction costs cumulative sum, i.e. equalizing the variations:
1 2 h
σ2−bσ2n(Su)
i
S2uCxxn (u, Su)∆u ' −G(u, Cxn(u+∆u, Su+∆u)− Cxn(u, Su)),
forn ∈N. A formal Taylor approximation gives
Cxn(u+∆u, Su+∆u)− Cxn(u, Su) =Cxtn(u, Su)∆u+Cxxn (u, Su) (Su+∆u− Su),
' Cxxn (u, Su) (Su+∆u− Su) ,
forn ∈N. Since his a convex function, we expectCxxn ≥0 and it follows formally
from Condition(G)together with the relationSu+∆u− Su' σSu(Wu+∆u− Wu)
that 1 2 h σ2−bσ 2 n(Su) i ∆u ' −G0(0)σ |Wu+∆u− Wu| 1 Su , n ∈N.
Taking the conditional expectation given Fu and plugging the classical estimate
E|Wu+∆u− Wu|=p2∆u/π, this leads to
1 2 h σ2−bσ2n(Su) i ∆u ' −G0(0) σ Su r 2∆u π , n ∈N.
For the regular trading grid considered here, ∆u = 1/n provides the following candidate for the upgraded volatility function:
b σn2 : x ∈(0, ∞)7→ σ2+G0(0)n1/2 r 8 π σ x, n ∈N. (3.3) Observe that this candidate upgraded local volatility function is degenerate at 0 and we prove in the next paragraph the well posed-ness of the corresponding local volatility fictive asset and associated pricing function.
3.2 The fictive asset dynamics
Let us consider a sequence of fictive assets, whose dynamics are given by the candidate upgraded volatility (bσn) defined in (3.3). We expect the fictive assets ( bSn)n to solve the following stochastic differential equation
b Stn=S0+ Z t 0 b γn( bSun)dWu, 0≤ t ≤ T , n ∈N, (3.4) where we introduced the notation
b γn:x 7→σbn(x)x= p σ2x2+σγn|x| , with γn := G0(0)n1/2 r 8 π, n ∈N.(3.5) Since the diffusion coefficients (bγn) are not Lipschitz, the existence of a unique process with such dynamics does not follow from the classical theorems. We puz-zle out this difficulty using the Engelbert & Schmidt criterion as detailed in the following lemma.
Lemma 3.1 Whatever initial condition(t, x)∈[0,1]×(0, ∞), the stochastic differen-tial equation (3.4) admits a unique strong solution( bSsn)t≤s≤1, starting from x at time
t. Furthermore, this solution remains non-negative.
Proof.We fixn ∈N and (t, x)∈[0,1]×(0, ∞). For anyz ∈R, observe that the diffusion coefficientbγn defined in (3.5) satisfies:
if Z ε
−ε
dy
|bγn(z+y)|2 =∞ , for anyε >0, then bγn(z) = 0. (3.6)
Indeed, forz 6= 0, takingε=|z|/2, we getRε
−ε dy |bγn(z+y)|
2 < ∞, so that the left hand
side condition of (3.6) implies z = 0, leading to bγn(z) = 0. Hence, the diffusion coefficientbγnsatisfies the Engelbert & Schmidt criterion, and, there exists a weak
solution to (3.4) with initial condition (t, x), see Theorem 5.4 in Section 5 of [7]. We now observe that the diffusion coefficient bγn also satisfies
|bγn(z)−bγn(y)|= p σ2z2+σγn|z| −pσ2y2+σγn|y| ≤ σ|z − y|+ p σ2y2+σγn|z| −p σ2y2+σγn|y| , (z, y)∈R 2 ,
since the derivative ofy 7→pσ2y2+σγn|z|is upper bounded by σ. We deduce |bγn(z)−bγn(y)| ≤ σ|z − y|+ √ σγn p ||z| − |y|| ≤ `(|z − y|), (z, y)∈R2, with ` :u 7→ σu+√σγnu. Since R0ε|`(duu)|2 = ∞, for any ε >0, we deduce from
Proposition 2.13 in Section 5 of [7] that pathwise uniqueness holds for the stochas-tic differential equation (3.1). Together with the existence of a weak solution ver-ified above, this implies the existence of a unique strong solution to (3.1) for any initial condition (t, x), see Corollary 3.23 in Section 5 of [7].
Finally, bSn remains non-negative, since it is continuous and Markovian, and the unique strong solution starting at 0 is the null one. 2
3.3 Payoff regularization and related pricing function
We now inquire the properties of the pricing functions associated to the fictive assets ( bSn)n and first discuss the regularity of the payoff function of interest.
We aim at hedging the contingent claim with payoff h(S1), where the payoff functionhis supposed to satisfy the following:
Condition (P): The convex functionh: [0, ∞)→R is affine outside the interval Condition (P):[1/K, K], with K >1.
Observe that most of the classical convex payoffs satisfy this condition. In par-ticular, under Condition (P), the map his Lipschitz and we denote byL >0 its smallest Lipschitz constant.
In the following, we shall sometimes require the payoff function to be continu-ously differentiable. Besides, in order to consider non-vanishing transaction costs, we need a control on the second order variations of the payoff function. In order to do so, we regularize the convex maph, as detailed in the following lemma. Lemma 3.2 There exists a sequence of convex maps (hn)n valued in C2([0, ∞),R) such that, for n large enough,
khn− hk∞≤ Lln(n) γ1n/6 , k∇hnk∞≤ L , k∇2hnk∞≤3L γ 1/6 n ln(n)1[1/2K,2K]. (3.7) Proof. We observe that h is affine on [0,1/K] and introduce the extension of h on R, which remains affine with the same slope on (−∞,0). For simplicity, this extended map is also denotedh. Forn ∈N, we introduce the convolution between hand the square kernel with support [−ln(n)/γn1/6,ln(n)/γn1/6]:
hn:x ∈[0, ∞)7→ 4 3 Z 1 −1 h x+yln(n) γ1n/6 ! (1− y2)dy . SincehisL-Lipschitz andR1
−1(1− y 2) dy= 3/4, we compute khn− hk∞≤ 4 3 Z 1 −1 Lln(n)y γ1n/6 (1− y2)dy = 2L 3 ln(n) γ1n/6 ≤ Lln(n) γ1n/6 , n ∈N.
Fix n ∈N. Observe that hn ∈ C2([0, ∞),
R) and, denoting abusively ∇h the right derivative ofh, we have
∇hn(x) = 4 3 Z 1 −1 ∇h x+yln(n) γn1/6 ! (1− y2)dy = 4 3 Z x+1 n x−1 n ∇h(z) 1− γ1n/6 ln(n)(z − x) 2 γn1/6 ln(n)dz , x ≥0. Sincek∇hk∞≤ L, we deduce thatk∇hnk∞≤ L.
Differentiating the second expression of ∇hn above, we deduce that ∇2hn(x) =4 3 Z x+n1 x−1 n ∇h(z)2γ 2/6 n ln(n)2(x − z) γn1/6 ln(n)dz= 8 3 Z 1 −1 −∇h x+yln(n) γn1/6 ! γn1/6 ln(n)ydy, forx ≥0. Using once again thatk∇hk∞≤ L, this yields
k∇2hnk∞≤ 8L 3 Z 1 −1 |y|γ 1/6 n ln(n)dy = 8L 3 γn1/6 ln(n) ≤ 3L γn1/6 ln(n) . (3.8) Besides, sincehis affine on [K, ∞), we deduce that
hn(x) = 4 3 Z 1 −1 ∇h(K) x − K+yln(n) γn1/6 ! (1− y2)dy = ∇h(K)(x − K) = h(x),
for any x ≥ K + ln(n)/γn1/6. The exact same reasoning applies for x ≤ 1/K −
ln(n)/γn1/6. Hence, forn large enough such thatγn1/6/ln(n)≥ K, hn is affine and
therefore ∇2hn = 0 outside the interval [1/2K,2K]. Combined with (3.8), this
completes the proof. 2
Remark 3.4 Wheneverhis valued inC2([0, ∞),R), the regularization procedure is not necessary since (3.7) is satisfied as soon as nis large enough. Hence one can simply usehinstead of (hn)n.
The sequence of regularized approximating payoff functions (hn)n in hand, we
can now introduce the associated valuation PDEs, given by: (en) = b Cnt(t, x) + 12bσ 2 n(x)x2Cbxxn (t, x) = 0, (t, x)∈[0,1)×(0, ∞), b Cn(1, x) =hn(x), x ∈(0, ∞). ,
forn ∈N. The existence of a unique strong solution for this PDE is given in Propo-sition 3.3 below. For sake of completeness and since the corresponding differential operator is not uniformly parabolic on [0,1)×(0, ∞), the proof of this proposition is reported in Appendix. As expected, the solution of the PDE interprets as the valuation function of the option with payoffhn on the terminal value of the fictive asset bS1n, introduced in the previous section.
Proposition 3.3 For any n ∈ N, the PDE (en)has a unique solution denoted Cb
n
, which moreover satisfies
b Cn(t, x) = Et,x h hn( bS1n) i , (t, x)∈[0,1]×(0, ∞), n ∈N. (3.9)
3.4 Delta correction and asymptotic hedging for non vanishing transaction costs coefficient
Even in a frictionless complete setting, a contingent claim can never be perfectly replicated in practice, since continuous time hedging is not feasible. As detailed in Section 2.3, we consider portfolios where the position in the assets changes on the regular discrete time grid (tni)i≤n. In this framework, we claim that the upgrade
(bσn)n of volatility and the regularization (h
n)
n of the payoff detailed in Section
3.2 and Section 3.3 allows to counterbalance asymptotically the frictions due to order book related transaction costs. This claim is the content of the next theorem, which is the main result of the paper.
Theorem 3.4 Consider the sequence of portfolios(Vn)nassociated to the initial
con-ditions( bCn(0, S0))n and the investment strategies(Hn)n defined by
Htn:= bCxn(tni, Stn i)− X j≤i b Cxn(tnj, Stn j−1)−Cb n x(tnj−1, Stn j−1) ,
for t ∈[tni, tni+1)and0≤ i < n. Then, the sequence of portfolio values rewrite Vtn= bCn(0, S0) + Z t 0 HundSu− X tn i≤t Gtni, Htnn i − H n tn i−1 , 0≤ t ≤1, n ∈N, (3.10) and(V1n)n converges in probability to the payoff h(S1)as n goes to ∞.
The proof of this theorem is presented in Section 4 below, and it requires sharp estimates on the derivatives of ( bCn)n, whose proofs are postponed to Section 5.
Remark 3.5 Observe that the hedging strategy does not simply consist in consider-ing theDeltaassociated to the fictive asset ( bSn)n. Indeed, as observed in [6, 11] for
the classical framework of transaction costs proportional to the amount of money, this original Leland replicating strategy does not converge to the claim of interest, unless the transaction costs vanish fast enough as the number of trading datesn increases. As in [9], the extra term in the definition of (Hn)n allows to consider
non vanishing transaction costs. In particular, observe that the change of position at timetni, fori ≤ n, in the portfolioVn is given by bCxn(tni, Stn
i)−Cb
n x(tni, Stn
i−1).
Remark 3.6 Our main result also allows to quantify the effects of a volume based trading taxation, on the cost of hedging strategies for convex derivatives. Indeed, in order to render most of the high frequency trading arbitrage opportunities ir-relevant, the regulator is still looking towards the best way to create a tax on trading orders. Nevertheless, the exact consequences of such a regulation on asset management strategies or more generally risk management strategies is not yet completely understood. Simple questions on this subject still lack fully satisfying answers: Should the regulator create a tax on the volume of traded asset or the quantity of traded money? Should he use a linear tax? What are the consequences of using a different shape of tax function? In our simplifying Black–Scholes frame-work, our conclusions are that the global shape of the taxation does not really matters from a hedging perspective since only the asymptotic behavior around 0 is relevant. Besides, Theorem 3.4 exhibits the volatility change related to a volume based taxation instead of a more classical amount based one.
4 Proof of the main result
Due to the consideration of volume related non linear transaction costs, the exhib-ited trading strategy is based on a pricing function of a stock model with non linear dynamics. Hence, classical estimates are not available for the sensitivities of the price function in terms of the volatility parameter. But, we require to understand precisely the dependence of the price sensitivities with respect to the number of trading datesnwhich affects the modified volatility parameter. We overcome this difficulty, using Malliavin derivative type representation of the Greeks, as detailed in the next subsection. This leads to sharp estimates, which allow to derive the convergence of the approximating replicating portfolio to the claim of interest at maturity.
4.1 Representation and estimates for the modified price function sensitivities Recall that the price function ˆCn is given by
ˆ Cn : (t, x)7→Et,x h hn( bS1n) i . (4.11)
A well chosen probability change leads classically to a nice representation of theDelta of the option presented below.
Lemma 4.1 For n ∈Nand any initial condition(t, x)∈[0,1]×(0, ∞), the s.d.e. dSe n u =bγn( eS n u)dWu+γbn∇bγn( eS n u)du (4.12)
has a unique solutionSen, which moreover remains strictly positive. Besides, we have b Cxn(t, x) = Et,x h ∇hn( eSn1) i , (t, x)∈[0,1]×(0, ∞), n ∈N. (4.13)
Proof.Fixn ∈N. The existence of a unique solution to (4.12) follows from similar arguments as the one presented in Lemma 3.1. Besides, since R1
0 ρn(u)du = ∞ where ρn:u 7→exp Z 1 u 2σ2y+σγn σ2y2+σγnydy = σ 2+σγ n σ2u2+σγnu,
Theorem 2.16 and 2.17, [1], ensure that eSn remains strictly positive for a given positive initial condition.
The mappings y 7→ bσn(ey) and y 7→ |bσn(ey)|2 admit locally Lipschitz first derivatives because their second derivatives are locally bounded. Let denoteSn:= ln bSn. By virtue of Theorem 39 (V.7) and Theorem 38 (V.7)[12], we deduce that there exists a version of the mappingy 7→ Snt,y, which is continuously differentiable
and so isx 7→Sbnt,xon (0, ∞), for anyt ∈(0,1). Precisely, for a given initial condition
(t, x)∈[0,1]×(0, ∞), the tangent process∇Sb
n is given by ∇Sb n u= 1 + Z u t ∇bγn( bSns)∇Sb n sdWs, t ≤ s ≤ T .
Besides, differentiating expression (4.12) provides bCxn(t, x) = Et,x[∇hn( bS1n)∇Sbn1]. Assume for the moment that∇Sb
n is a positive martingale and introduce the new
equivalent probability Pn defined bydPn=∇Sb1ndQ, so that b Cxn(t, x) = EP n t,x h ∇hn( bS1n) i . (4.14)
Girsanov theorem asserts that the processWn given bydWun=dWu− ∇bγ
n
( bSun)du
is a standard Brownian motion under Pn. Hence, the dynamics of bSnunder Pnare given by dSb n u =bγ n ( bSun)dWun+bγ n ∇bγ n ( bSun)du .
Therefore, the law of bSn under Pnis identical to the one of eSnunder Q and (4.14) rewrites as (4.13).
The rest of the proof is dedicated to the verification that ∇Sbn is indeed a positive martingale.
For anyp ∈N, let us introduce the stopping time τp:= inf{s ≤1 : bSns ≤ x/(1 +p)} ,
with the convention that inf∅ =∞. Applying Gronwall’s lemma, we verify that supt≤s≤1∇Sb
n
s∧τn is square integrable, hence∇Sb
n
.∧τn is a martingale. Let us define
the change of measuredQp:=∇Sb1∧τpdQ. Then,
E[∇Sb
n
1]≥E[∇Sb
n
1∧τp1τp=∞] = Qp(τp=∞), p ∈N. (4.15)
As (τp)p, let us define the sequence (˜τp)p associated to the process eSn given by
(4.12). By construction, observe thatτphas the same law under Qpthanτepunder Q, for anyp ∈N. It follows that Qp(τp=∞) = Q(˜τp=∞)→Q(˜τ∞=∞) where ˜
τ∞ is the first time when eSn hits zero. But eSn remains strictly positive, so that (4.15) implies that E[∇Sb1n]≥1. Since∇Sbnis a supermartingale, we then conclude. 2 We now provide an expression for the second derivative of the price function ˆ
Cn, in the spirit of the Malliavin representation of the Greeks presented in [4]. Lemma 4.2 For any n ∈N, we have
b Cxxn (t, x) = Et,x ∇hn( eSn1) Z 1 t πundWu , (t, x)∈[0,1]×(0, ∞), (4.16) where πn is defined by πnu := ∇Seun (1− t)bγ n( eSn u) , 0≤ t ≤ u ≤1. (4.17)
Proof.Fix any initial condition (t, x)∈[0,1]×(0, ∞) andn ∈N. Differentiating (4.13) with respect tox, we directly compute
b Cxxn (t, x) = Et,x h ∇2hn( eS1n)∇Se n 1 i = Et,x 1 1− t Z 1 t ∇2hn( eS1n)DsSe n 1 ∇Se n s ˆ γ( eSsn) ds .
Recall that the Malliavin derivative and the tangent process only differ by their initial conditions. Hence, recalling the definition (4.17) ofπn, the integration by parts formula yields
b Cxxn (t, x) = Et,x Z 1 t Ds[∇hn( eS1n)]πnsds = Et,x ∇hn( eS1n) Z 1 t πnsdWs . 2 Similarly, the third derivative of the price function also has such type of repre-sentation in expectation, where we emphasize that the stochastic integrals consid-ered below are of Skorokhod type, since the integrand is not necessarilyF-adapted. Lemma 4.3 For any n ∈N, we have
b Cxxxn (t, x) = Et,x ∇hn( eSn1) Z 1 t ¯ πundWu , (t, x)∈[0,1]×(0, ∞), (4.18) whereπ¯n is defined by ¯ πnu :=∇xπnu+πnu Z 1 t πsndWs , 0≤ t ≤ u ≤1. (4.19) Proof.Fix any initial condition (t, x)∈[0,1]×(0, ∞) andn ∈N. Differentiating (4.16) with respect toxand following a similar reasoning as above yields
b Cxxxn (t, x) = Et,x ∇hn( eS1n) Z 1 t ∇xπnsdWs +∇2hn( eS1n)∇Se n 1 Z 1 t πnudWu = Et,x ∇h( eS1n) Z 1 t ∇xπnsdWs + Z 1 t Ds[∇hn( eS1n)]πns Z 1 t πundWu ds . Hence, the Malliavin integration by parts formula provides
b Cxxxn (t, x) = Et,x ∇h( eS1n) Z 1 t ∇xπsndWs +∇hn( eS1n) Z 1 t πsn Z 1 t πnudWu dWs , and the definition (4.19) concludes the proof. 2
The exact same line of arguments provides a similar representation for the fourth derivative of the pricing function.
Lemma 4.4 For any n ∈N, we have
b Cxxxxn (t, x) = Et,x ∇hn( eS1n) Z 1 t ˆ πnudWu , (t, x)∈[0,1]×(0, ∞), (4.20) whereπˆn is defined by ˆ πnu :=∇x¯πnu+πnu Z 1 t ¯ πsndWs , 0≤ t ≤ u ≤1. (4.21) These representations allow to derive estimates on the dependance of the derivatives of the pricing function ˆCn, in terms of the parameter n. The rather computational obtention of these estimates is reported in Section 5 below.
Proposition 4.5 There exist a constant C and a continuous function f on (0, ∞) which do not depend on n ∈N, such that
|Cb n x(t, x)| ≤ C , (4.22) 0 ≤ Cb n xx(t, x)≤ C p(1− t)γn x−1/2 (4.23) |Cb n xxx(t, x)| ≤ C γn(1− t)x −1 +p C γn(1− t) x−3/2, (4.24) |Cb n xxxx(t, x)| ≤ f(x) √ 1− tγn + f(x) (1− t)γn + f(x) (1− t)5/4γ5/4 n + f(x) (1− t)3/2γ3/2 n ,(4.25) |Cb n xt(t, x)| ≤ f(x) (1− t)4/3ln(n), (4.26)
for any(t, x)∈[0,1]×(0, ∞)and n ∈N.
Remark 4.7 Observe that (4.23) also indicates that the price function bCnis convex with respect to the space variable. Indeed, the pricing function inherits the con-vexity of the payoff. This observation is crucial in order to ensure that a volatility upgrade allows to compensate the transaction costs.
4.2 Asymptotics of the hedging error
The subsection is dedicated to the proof of Theorem 3.4, the main result of the paper. We verify below that the sequence (V1n)nof terminal values for approximate
replicating portfolios converges toh(S1), as the number of trading dates ntends to infinity.
For anyn ∈N, we rewrite the hedging strategy (Htn)0≤t≤1asHn= ˆHn+Knwith ˆ Htn:= bCxn(tni, Stn i) and K n t := X j≤i b Cxn(tnj−1, Stn j−1)−Cb n x(tnj, Stn j−1), (4.27)
for t ∈[tni, tni+1) and 1≤ i ≤ n. We also denote∆Hˆ n
t := ˆHtn+−Hˆt−n and∆ Ktn:=
Ktn+− Kt−n . Therefore the terminal value of the candidate replicating portfolioV1n rewrites V1n= bCn(0, S0) + Z 1 0 HundSu− X i<n G tni, ∆Hˆtnn i +∆ K n tn i , n ∈N. (4.28)
Besides, the dynamics of bCn and the definition (3.3) of ˆσn yields hn(S1) = bCn(0, S0) + Z 1 0 b Cxn(u, Su)dSu+1 2 Z 1 0 σγnSuCb n xx(u, Su)du , n ∈N. Plugging the two expressions above together directly leads to the following tractable decomposition of the hedging error
V1n− h(S1) =F0n+F n 1 +F n 2 +F n 3 +F n 4 ,
for anyn ∈N, where F0n:=h n (S1)− h(S1) + Z 1 tn n−1 ( ˆHtn−Cb n x(t, St))dSt− G 1, ∆Hˆ1n+∆ K n 1 , F1n:= Z tnn−1 0 ( ˆHtn−Cb n x(t, St))dSt, F2n:= Z 1 0 KtndSt F3n:= n−1 X i=1 G0(0)|∆Hˆtnn i +∆ K n tn i| − n−1 X i=1 G tni, ∆Hˆtnn i +∆ K n tn i , F4n:= 1 2 Z 1 0 σγnStCb n xx(t, St)dt − n−1 X i=1 G0(0)|∆Hˆtnn i +∆ K n tn i| .
We now prove that each sequence of random variables (Fjn)n for j= 0, . . . ,4
goes to zero in probability, asngoes to infinity.
Proposition 4.6 The sequences(F0n),(F1n), (F2n)and(F3n)converge to0 in proba-bility as n goes to ∞.
Proof.We prove the convergence of each sequence separately. Step 0. Convergence of (F0n)n.
By construction of (hn), (3.7) implies that the first termhn(S1)− h(S1) tends to 0 as hn → h. The second one converges to 0 because ( bCxn(., S.))n is bounded
according to (4.22). As for the last term, observe from (4.13) that |∆Hˆ1n|=|Cb n x(1, S1)−Cb n x(tnn−1, Stn n−1)|= ∇h n (S1)−E h ∇hn( ˜S1n) ˜S n tn n−1=St n n−1 i ≤ k∇2hnk∞E h |S1−S˜n1| ˜S n tn n−1=Stnn−1 i , n ∈N. As E|S1− Stn
n−1| ≤ Cp1/n, we deduce from (3.7) that
|∆Hˆ1n| ≤ C γn1/6 √ nlnn +C γn1/6 lnnE|S˜ n 1 −S˜tnn n−1| , n ∈N.
From the dynamics (4.12) of ˜Sn, we compute directly E|S˜1n−S˜tnn n−1| ≤ C
p γn/n
so that|∆Hˆ1n|goes to 0 asngoes to infinity. Very similarly, we show that|∆ K1n| converges also to 0 and Condition(G)provides the convergence ofF0n to 0.
Step 1. Convergence of (F1n)n.
Applying the Ito formula, we directly compute that ˆ Htn−Cb n x(t, St) =Mtn− Mtnn i +A n t − Antn i , t n i ≤ t < tni+1, i < n −1, (4.29) where the sequence of processes (Mn)n and (An)n are given by
Mn:= Z . 0 σSuCb n xx(u, Su)dWu and An:= Z . 0 b Cxtn(u, Su) +1 2σ 2 Su2Cb n xxx(u, Su) du ,
for anyn ∈N. SinceS has bounded moments, (4.23) together with the Cauchy– Schwartz inequality yield
E(Mtn− Mtnn i) 4≤ C γ2 n Z tni+1 tn i du 1− u !2 ≤ C γ2 n , tni ≤ t < t n i+1, i < n −1. Besides, (4.24) together with (4.26) indicate that
E|Ant − Antn i| 4≤ C Z t tn i ln(n)−1du (1− u)4/3 + γ−n1du (1− u)+ γn−1/2du √ 1− u !4 ≤ C ln(n) −1 n(1− t)4/3 + γn−1/2 n(1− t) !4 , tni ≤ t < tni+1, i < n −1.
Plugging the last two estimates in (4.29) leads directly to
E|F1n|2 ≤ C γn + C n2 Z tnn−1 0 |ln(n)|−2 (1− t)8/3 + γ−n1 (1− t)2 dt ≤ C γn + C|ln(n)|−2 n1/3 + C nγn ,
for anyn ∈N, so that E|F1n|2 goes to 0 asngoes to infinity.
Step 2. Convergence of (F2n)n.
From the definition ofKn given in (4.27), we directly compute
F2n=− X i≤n−1 Z tni tn i−1 b Cxtn(u, Stn i−1)(Stni − S1)du
Combining the Cauchy–Schwartz inequality together with (4.26) yields
E|F2n| ≤ C lnn X i≤n−1 E[|Stn i − S1| 2]1/2 Z tni tn i−1 du (1− u)4/3 ≤ C nlnn X i≤n−1 1 (1− tn i)5/6 ≤ C lnn →0. Step 3. Convergence of (F3n)n.
For any 0≤ i ≤ n, observe that ∆Hˆtnn i +∆ K n tn i = bC n x(tni, Stn i)−Cb n x(tni, Stn i−1) = bC n xx(tni, Stn i)(Stni − Stni−1),
where the random variableStn
i is betweenStni−1 andStni. Hence, (4.23) together
with Condition(G)yield|F3n| ≤ Cωχn3 where
χn3 := X i≤n−1 1 γn(1− tni)(St n i − Stni−1) 2 ,
for anyn ∈N. But Eχn3 ≤ C(γn)−1lnn, henceF3n→0 as ngoes to∞. 2
Proof.For anyn ∈N, we writeF4n= P4
i=1L n
i with the summands
Ln1 := 1 2 Z 1 0 σγnStCb n xx(t, St)dt − 1 2n n−1 X i=1 σγnStn i−1Cb n xx(tni−1, Stn i−1), Ln2 := n−1 X i=1 b Cxxn (tni−1, Stn i−1) σγn 2n − G 0(0) σ |∆Wtn i| , Ln3 :=σG0(0) n−1 X i=1 Stn i−1Cb n xx(tni−1, Stn i−1)|∆Wtni| − Z tni tn i−1 σSuCb n xx(u, Su)dWu ! , Ln4 :=G0(0) n−1 X i=1 Z tni tn i−1 σSuCb n xx(u, Su)dWu − G0(0) n−1 X i=1 |∆Htnn i +∆K n tn i| .
Observe that the previous decomposition uses the convexity of the price function given in (4.23), see Remark 4.7.
It now suffices to show thatLni →0 fori= 1, . . .4 as detailed in the steps below. Step 1. Convergence of (Ln1)n.
We have|Ln1| ≤ Cω(|Ln11|+|Ln12|) where, by virtue of (4.23),
Ln11 :=γn n−1 X i=1 Z tni tn i−1 St− Stn i−1 p γn(1− t) dt , Ln12 :=γn n−1 X i=1 Z tni tn i−1 Stn i−1 b Cxxn (t, St)−Cb n xx(tni−1, Stn i−1) dt . We have E|Ln11| ≤ C p
γn/n →0. For the second term, we use the Taylor expansion
b Cxxn (t, St)−Cb n xx(tni−1, Stn i−1) = bCxxx(¯t n i,S¯tn i)(St− Stni−1) + bCxxt(¯t n i,S¯tn i)(t − t n i−1), for some random variables ¯tni and ¯Stn
i, for t
n
i−1 ≤ t < t n
i. Besides, differentiating
the dynamics of bCn, we observe that b
Cxttn =−2σ2Cb
n
xx−(2σ2x+σγnx) bCxxxn −(σ2x2+σγnx) bCxxxxn , (4.30)
for anyx ∈(0, ∞) andn ∈N. Hence, combining (4.23), (4.24) and (4.25), we get
Ln12 ≤ Cωγn n−1 X i=1 Z tni tn i−1 St− Stn i−1 γn(1− t)dt +Cω γn2 n Z tnn−1 0 dt p γn(1− t) + dt γn(1− t) + dt γn5/4(1− t)5/4 + dt γn3/2(1− t)3/2 ! , for anyn ∈ N. Hence the Cauchy–Schwartz inequality and a direct computation yield ELn12≤ C ln(n) √ n + 1 n1/4+ 1 n3/8 → 0. Step 2. Convergence of (Ln2)n.
We use the equalityE|∆Wtn
i|=p2/πnfrom which we deduce
E σγn 2n − G 0(0) σ |∆Wtn i| 2 =V arG0(0)σ |∆Wtn i| = σ 2 G0(0)2 n ,
for anyi ≤ n. The independence of the increments of the Brownian motion together with (4.23) yield E(Ln2)2≤ C n X i≤n−1 1 γn(1− tni) ≤Cln(n) n√n →0. Step 3. Convergence of (Ln3)n.
We use the inequality ||a| − |b|| ≤ |a − b|. Therefore, the Cauchy-Schwarz in-equality and the Ito isometry give us
E|Ln3| ≤ C n−1 X i=1 Z tni tn i−1 E Stni−1Cb n xx(tni−1, Stn i−1)− SuCb n xx(u, Su) 2 du !1/2 .
By the Ito formula, we getd[StCbxxn (t, St)] =ftndWt+gntdtwhere
ftn :=σStCb n xx(t, St) +σSt2Cb n xxx(t, St), gnt :=StCb n xxt(t, St) +1 2σ 2 St3Cb n xxxx(t, St) +σ2St2Cb n xxx(t, St),
for 0≤ t ≤1 andn ∈N. Hence, we derive
E|Ln3| ≤ C n−1 X i=1 1 n Z tni tn i−1 E|fsn|2ds+ 2 n2 Z tni tn i−1 E|gns|2ds !1/2 . (4.31) Estimates (4.23) and (4.24) provide
Z tni tn i−1 E|fun|2du ≤ C n 1 p(1− tn i)γn + 1 (1− tn i)γn !2 , i ≤ n . Besides, combining (4.30) together with (4.23), (4.24) and (4.25), we get
Z tni tn i−1 E|gnu|2du ≤ Cγn n γn−1/2 p(1− tn i) + γ −1 n (1− tn i) + γ −5/4 n (1− tn i)5/4 + γ −3/2 n (1− tn i)3/2 ! . Plugging these last two estimates in (4.31), similar computations as in Step 1 yield to the convergence of E|Ln3|to zero.
Step 4. Convergence of (Ln4)n.
We first verify that we may replace∆Kn by∆K˜n where ∆K˜tnn i :=− Z tni tn i−1 b Cxtn(u, Su)du , i ≤ n .
To do so, it suffices to show thatχn→0 where χn:= X i≤n−1 Z tni tn i−1 b Cxtn(u, Su)−Cb n xt(u, Stn i−1) du.
Using a Taylor expansion, we compute b Cxtn(u, Su)−Cb n xt(u, Stn i−1) = bC n xxt(u,S¯tn i−1) Su− Stn i−1 , for some random variable ¯Stn
i−1 between Su and Stni−1, for any 0≤ t
n
i ≤ u ≤1.
Hence (4.30) together with (4.23), (4.24) and (4.25) imply thatχn≤ Cωχ¯n where
¯ χn:=γn n−1 X i=1 Z tni tn i−1 Su− Stn i−1 √ 1− tγ1n/2 +Su− St n i−1 (1− t)γn + Su− Stn i−1 (1− t)5/4γn5/4+ Su− Stn i−1 (1− t)3/2γn3/2 ! du, for n ∈N. As E Su− Stni−1 ≤ Cn −1/2 for
tni ≤ u ≤ tni+1, we easily conclude that E ¯χn →0. At last, replacing ∆Ktnn
i by ∆
˜ Ktnn
i and using the inequality ||a| − |b|| ≤
|a − b|, we deduce from Ito’s formula together with (4.24) that |Ln4| ≤ Cω Z tnn−1 0 b Cxxx(u, Su)du ≤ cω Z tnn−1 0 du p(1− u)γn + du (1− u)γn ! →0. 2
5 Price sensitivities estimation
This section is dedicated to the obtention of the estimates presented in Proposition 4.5 above, which allow to upper bound the sensitivities of the price function bCnin terms of the number of trading datesn. The control of each sensitivity is presented separately. These estimates, namely (4.22), (4.23), (4.24), (4.25) and (4.26), are obtained using the Malliavin representation of the Greeks detailed in Section 4.1. This particular feature is new in the classical scheme of proof for the obtention of Leland type convergence theorems.
In all the section, we fix (t, x)∈[0,1]×(0, ∞) and omit the subscript{t, x}in order to alleviate the notations.
5.1 Estimates (4.22) and (4.23) on the first and second derivatives
First observe that estimate (4.22) directly follows from the representation (4.13), since (∇hn)n is bounded. The rest of this subsection is dedicated to the obtention
of (4.23).
We fix (t, x)∈[0,1]×(0, ∞). Using (4.16) together with the Cauchy–Schwartz inequality, we derive |Cˆxxn (t, x)| ≤ k∇hnk∞ Z 1 t E π n u 2 du 1/2 , n ∈N. (5.32) We now focus more closely on the dynamics of the processes (πn)n defined by
(4.17). First, according to the dynamics of eSn, the tangent process∇Sen satisfies d∇Se n u =∇bγn( eS n u)∇Se n udWu+ |∇bγn( eSun)|2+bγn( eS n u)∇2bγn( eS n u) ∇Se n udu ,
forn ∈N. Besides, Ito’s formula implies that 1/bγn( eSn) has the following dynamics d 1 b γn( eSun) =−∇bγn( eS n u) |bγn( eS n u)|2 dSe n u+ 2|∇bγn( eS n u)|2−bγn( eS n u)∇2γbn( eS n u) 2bγn( eS n u) du =−∇bγn( eS n u) |bγn( eSun)|2 dWu− ∇2 b γn( eSnu) 2 du , n ∈N.
A direct application of the integration by parts formula hence implies
dπnu = ∇2 b γn( eSun)∇Se n u 2(1− t) du = − σ2γn2 8|bγn( eSun)|2 πundu , n ∈N. (5.33) Therefore, we deduce that
πnu=πtnexp − Z u t σ2γn2 8|bγn( eS n s)|2 ds ≤ 1 (1− t)bγn(x) , 0≤ u ≤1, n ∈N. (5.34)
Plugging this expression together withbγn(x)≥ √
σγnxin (5.32) provides (4.23).
Indeed (5.34) also indicates that πn and hence ∇S˜n are non-negative, so that b Cxxn (t, x) = Et,x h ∇2hn( eS1n)∇Sen1 i ≥0.
5.2 Estimate (4.24) on the third derivative
This subsection is dedicated to the obtention of (4.24) and divides in 3 steps. Step 1. Estimate decomposition
Using (4.18), we derive |Cb n xxx(t, x)| ≤ k∇hnk∞E ¯Z n 1 where ¯Z n 1 := Z 1 t ¯ πundWu, n ∈N. (5.35) Let us introduce the sequence of processes (Zn)n given by
Zsn:=
Z s
t
πnudWu, n ∈N. (5.36) By the definition of (¯πn) given in (4.19), we compute
¯ Z1n = Z 1 t ∇xπnudWu+ Z 1 t πunZ1ndWu = Z 1 t ∇xπnudWu+ Z1n 2 − Z 1 t πnuDuZ1ndu = Z 1 t ∇xπnudWu+ Z1n 2 − Z 1 t |πnu|2du − Z 1 t πun Z 1 u DuπsndWs du = Z n 1 2 − Z 1 t |πun|2du+ Z 1 t ∇xπnu− Z u t πnsDsπnuds dWu, n ∈N.
Plugging this expression in (5.35) and using Ito’s formula, we deduce |Cb n xxx(t, x)| ≤ C A1n/2+Bn , n ∈N, (5.37)
where (An) and (Bn) are respectively defined by
An := E Z 1 t πunZundWu 2 and Bn:= E Z 1 t ∇xπnu− Z u t πnsDsπunds dWu , forn ∈N. We now fixn ∈N and intend to control the termsAnandBnseparately.
Step 2. Control of (An)n
Recall from (5.34) that |πn| ≤ 1/(1− t)bγn(x). Hence, we get from a direct
application of Ito’s formula that
An= E Z 1 t πnu Z u t πsndWs dWu 2 = Z 1 t E πun Z u t πnsdWs 2 du .
We recall from (5.34) that |πn| ≤1/(1− t)bγn(x) and deduce from the previous expression An = Z 1 t E πun Z u t πnsdWs 2 du ≤ 1 (1− t)2| b γn(x)|2 Z 1 t Z u t E π n s 2 dsdu . (5.38)
Using once again the same relation together with|ˆγn(x)|2≥ σγnxyields
A1n/2 ≤ 1 2(1− t)2| b γn(x)|4 ≤ 1 √ 2σ(1− t)γnx . (5.39) Step 3. Control of (Bn)n
We now turn to the more intricate termBn. Let us introduce the notation
bn := ∇xπn− Z . t [Dsπnu]πsnds , so that Bn= E Z 1 t bnudWu . (5.40) By virtue of the martingale moment inequalities, there existsC >0 such that
Bn ≤ CE Z 1 t |bnu|2du 1/2 ≤ C√1− tE sup t≤u≤1 b n u , n ∈N. (5.41) In order to control the last term on the r. h. s. , we look towards the dynamics of (bn)n. Differentiating the dynamics of (πn) given in (5.33), we compute separately
d∇xπun=− σ2γ2n 8|bγn( eSun)|2 ∇xπundu+ σ2γn2∇bγn( eS n u) 4|bγn( eSnu)|3 ∇Se n uπnudu , (5.42) dDsπun=− σ2γ2n 8|bγn( eS n u)|2 Dsπnudu+ σ2γn2∇bγn( eS n u) 4|bγn( eS n u)|3 DsSe n uπnudu , t ≤ s , (5.43)
SinceDsSern=∇Sernbγn( eS n s)/∇Sesn=∇Sern/{(1− t)πsn}for t ≤ s ≤ r ≤1, we deduce Z u t πnsDsπunds=− Z u t Z u s σ2γn2πns 8|bγn( eS n r)|2 Dsπrndrds+ Z u t Z u s σ2γn2∇bγn( eS n r) 4|bγn( eS n r)|3 DsSe n rπrnπnsdrds =− Z u t Z r t πnsDsπrnds σ2γn2dr 8|bγn( eSrn)|2 + Z u t r − t 1− t σ2γ2n∇bγn( eS n r) 4|bγn( eSrn)|3 ∇Se n rπnrdr,
fort ≤ u ≤1. Combining this expression with (5.42), we get
dbnu=− σ2γn2 8|bγn( eSun)|2 bnudu+ σ2γn2∇bγn( eS n u) 4|bγn( eSun)|2 |πun|2(1− u)du . (5.44) Notice thatbnt =−∇bγn(x)/(1−t)|bγn(x)|
2<0. From the dynamics ofbn, we observe
thatbnincreases as long asbnis negative. Once it becomes positive, it must remain non negative, since the negative part of the drift disappears as soon asbn reaches 0. Indeed,bn=Lnπn/πnt where Ln :=bnt + Z . t σ2γn2∇bγn( eS n r) 4|bγn( eS n r)|2 πnrπtn(1− r)dr (5.45)
is strictly increasing. From there, we deduce that bn andLn have the same sign. Hencebn is always non negative on [τn,1] where τn:= inf{s ∈[t,1], bns = 0} ∧1.
Therefore, we get |bnu| ≤ −bnt1{bu≤0}+b n u1{u≥τn} ≤ −bnt − 1{u≥τn} Z u τn σ2γ2nbnr 8|bγn( eSrn)|2 dr+1{u≥τn} Z u τn σ2γn2∇bγn( eS n r) 4|bγn( eSnr)|2 |πrn|2(1− r)dr,
for anyt ≤ u ≤1, which directly leads to |bnu| ≤ |bnt|+Γun, with Γn:=−
Z .
t
∇bγn( eSrn)(1− r)2πnrdπrn. (5.46)
Since∇bγn is non-negative andπ
n is decreasing, we deduce that
E sup t≤u≤1 b n u ≤ ∇bγn(x) (1− t)|bγn(x)|2 + EΓ n 1 . (5.47)
We now focus on the last term of this expression and observe from a direct appli-cation of the integration by parts formula that
Γun= (1− t)∇bγn(x)|π n t|2−(1− u)∇bγn( eS n u)|πun|2 + Z u t |πrn|2(1− r)d∇bγn( eS n r)− Z u t |πrn|2∇bγn( eS n r)dr , t ≤ u ≤1. (5.48) We compute ∇bγn(x) = 2 σ2x+σγn 2pσ2x2+σγnx, ∇ 2 b γn(x) =− σ2γ2n 4bγ 3 n(x) , ∇3bγn(x) =3 σ2γn2 4 ∇bγn(x) b γn4(x) ,
and deduce from the application of Ito’s formula that d∇bγn( eSun) = −σ2γn2 4bγ 2 n( eSun) dWu−∇b γn( eSun) 2 du = −σ 2γ2 n 4bγ 2 n( eSun) dWu− ∇γ( eSun) dπun πn u .(5.49)
Plugging this expression in (5.48) directly leads to
Γun≤(1− t)∇bγn(x)|π n t|2+Nun+ 1 2Γ n u , t ≤ u ≤1, where Nn :=−R. t|π n r|2(1− r) σ2γ2 n 4bγ 2 n( eSnr)
dWr.Since Γn ≥0, it follows that (Nun)u≥t
is a supermartingale whenceEN1n≤0. We deduce an upper bound onEΓ1n which plugged in (5.47) provides E sup t≤u≤1 b n u ≤ ∇bγn(x) (1− t)|bγn(x)| 2 + 2(1− t)∇bγn(x)|π n t|2 = 3∇bγn(x) (1− t)|bγn(x)| 2. (5.50) Together with (5.41) and the expression∇bγn(x)/bγn(x)≤ C/x, we get
Bn≤ C xbγn(x) √ 1− t ≤ C p γn(1− t) x−3/2,
which, combined with (5.37) and (5.39), provides (4.24).
5.3 Estimate (4.25) on the fourth derivative
This subsection is dedicated to the obtention of (4.25). Fixn ∈N. The represen-tation (4.20) directly provides
|Cˆxxxx(t, x)| ≤ k∇hnk∞E Z 1 t ˆ πundWu , (5.51)
and we now intend to control the term E R1 t πˆ n udWu in several steps. Step 1. A tractable Decomposition forE
R1 t πˆ n udWu . Let introduce the notation
¯ Zun := Z u t (bns+ 2πnsZsn)dWs, t ≤ u ≤1, so that Z¯1n= Z 1 t ¯ πsndWs,
where (bn)n is defined above and given bybn:=∇πn−Rt.πrn(Drπn)dr. The
defi-nition of ˆπn given in (4.21) implies Z 1 t ˆ πnudWu= Z 1 t ∇π¯undWu+ Z 1 t πunZ¯1ndWu = ∇Z¯1n+ Z 1 t πunZ¯1ndWu. (5.52)
Using integration by parts formulae, observe thatR1 t π n uZ¯1ndWu rewrites Z1nZ¯1n− Z 1 t πunDu[ ¯Z1n]du =Z1nZ¯1n− Z 1 t πun(bnu+ 2πunZun)du − Z 1 t πun Z 1 u (Dubns+ 2Du[πnsZsn])dWs du = Z 1 t ZundZ¯un+ Z 1 t ¯ ZundZun− Z 1 t Z s t πunDubnsdu −2Zsn Z s t πnuDuπnsdu dWs −2 Z 1 t πsn Z s t |πun|2du dWs−2 Z 1 t πns Z s t Z r t πnuDuπrndu dWr dWs.
Plugging this expression together with∇Z¯1n= R1
t(∇b n
s+ 2Zsn∇πsn+ 2πns∇Zsn)dWs
and the definition ofbn in (5.52), we obtain Z 1 t ˆ πundWu = Z 1 t cnsdWs+ Z 1 t ZundZ¯un+ Z 1 t ¯ ZundZun + 2 Z 1 t Zsnbns +πns Z s t bnrdWr − πsn Z s t |πun|2du dWs, wherecn:=∇bn−R. tπ n
r(Drbn)dr. Introducing the dynamics ofZn and ¯Zn in the
previous expression, we get Z 1 t ˆ πnudWu= Z 1 t cnsdWs+ 3 Z 1 t Zsnbns+πns Z s t bnrdWr dWs + 2 Z 1 t πns |Zsn|2+ Z s t πrnZrndWr− Z s t |πnr|2dr dWs.
Using Ito’s formula together with the definition of ¯Zn, we deduce E Z 1 t ˆ πnudWu ≤3C1n+ 3C n 2 +C n 3 , (5.53) where we set C1n:= E Z 1 t πns Z s t ¯ ZrndWr dWs , C2n:= E Z 1 t ZsnbnsdWs , C3n:= E Z 1 t cnsdWs . We now require to control these three terms separately.
Step 2. Control of (C1n)
Using twice the martingale moment inequality, we compute
C1n ≤ C πntE s Z 1 t ( ¯Zn u) 2 du ≤ c πnt √ 1− tE sup t≤u≤1 |Z¯un| ≤ C πnt √ 1− t √1− tE sup t≤u≤t |bnu|+ 2πtn √ 1− tE sup t≤u≤1 Z n u ! ≤ C πnt(1− t) E sup t≤u≤t |bnu|+ 2c|πnt|2 √ 1− t ! .
Plugging (5.50) in this expression, it follows that C1n ≤ C b γn(x)(1− t) ∇bγn(x) |bγn(x)|2 +√ 1 1− t|bγn(x)| 2 . Since|bγn(x)|2≥ σγnxand|∇bγn(x)|/|bγn(x)| ≤3/2x, we deduce that
C1n≤ C (1− t)γnx 1 x+ 1 p(1− t)γnx ! . (5.54) Step 3. Control of (C2n)
Applying the martingale moment inequality together with the relation (5.46), we deduce C2n ≤ C √ 1− tE sup t≤u≤1 |bnuZun| ≤ C √ 1− t |bnt|E sup t≤u≤1 |Zun|+ E sup t≤u≤1 Γun|Zun| ! , where Γn defined in (5.46) is non negative and increasing. Using once again the martingale moment inequality, we derive
C2n ≤ C(1− t)|bnt|πnt +C
√
1− tE sup
t≤u≤1
Γun|Zun| . (5.55)
Observe that the integration by parts formula yields dΓunZun=−∇bγn( eS
n
u)(1− u)Zun2πundπnu+ΓunπundWu.
The Jensen inequality applied to the concave functionx 7→√xyields the inequality (R
f(u)udu)2≤(R
f(u)du)(R
f(u)u2du). Sinceπn is decreasing, we deduce that 2 sup t≤u≤1 |ΓunZun| ≤ sup t≤u≤1 Z u t ΓrnπrdWr + Γ n 1Γ n 1 1/2 , (5.56)
where, using (5.49), we have Γnu :=− Z u t ∇bγn( eS n r)(1− r)|Zrn|22πrndπnr = Z u t |πnr|4∇bγn( eS n r)(1− r)dr − Z u t |πnr|2|Zrn|2∇bγn( eS n r)dr − Z u t |πnr|2|Znr|2(1− r)∇bγn( eS n r) dπnr πrn + 2 Z u t (πnr)3(1− r)d D |Zn|2, ∇bγn( eS n )E r+N n u ,
for t ≤ u ≤1, with Nn a local martingale. Hence, we deduce that 1 2Γ n u ≤ Nun+ χn1+χn2 where χn1 := Z 1 t (πrn)4∇bγn( eS n r)(1− r)dr ≥0, χn2 :=−4 Z 1 t (1− r)|Zrn||πrn|2dπnr ≥0.
Applying Ito’s formula to (|πnr|4∇bγn( eS
n
r)(1− r)2)t≤r≤1together with the relation (5.49) yields χn1 =|πnt|4∇bγn(x)(1− t) 2 + 4 Z 1 t (1− r)2∇bγn( eSnr)(πnr)3dπnr − Z 1 t (1− r)2(πnr)4∇bγn( eS n r) dπrn πn r + N11,n,
whereN1,nis a lower bounded local martingale, so that E|χn1| ≤ |πnt|4∇bγn(x)(1− t)
2
.
From the martingale inequality together with Ito’s formula, we get
E|χn2| ≤4E sup r |Zrn| Z 1 t −(1− r)|πnr|2dπrn ≤ 4 √ 1− tπnt (1− t) |πtn|3 3 , where the last inequality follows from the monotonicity ofπntogether with Doob’s inequality.
We deduce that E|χn1+χn2| < ∞, so that 0≤12Γ n
u≤ Nun+E[|χn1+χn2|Fu], which
implies thatNn is a supermartingale. Therefore EN1n≤0 and EΓ1n≤2E[χn1+χn2]. Hence, the two previous inequalities together with (5.56) lead to
2E sup u |ΓunZun| ≤E sup u Z u t ΓrnπnrdWr +pEΓ1n s |πn t|4 (1− t)−2∇bγn(x) + 4 3 (πtn)4 (1− t)−3/2 . The martingale moment inequality and the monotonicity ofΓn andπn ensure
E sup t≤u≤1 Z u t ΓrnπrndWr ≤ CE s Z u t |Γn rπrn|2dr ≤ Cπtn √ 1− tEΓ1n. Plugging EΓ1n ≤2(1− t)∇bγn(x)|π n
t|2 observed in (5.50) together with the
defini-tions ofπtn andbnt in the previous expressions and (5.55) leads to
C2n ≤ C √ 1− t ∇bγn(x) b γn(x)3 √ 1− t+ ∇bγn(x) b γn(x)2 + ∇bγn(x) b γn(x)4(1− t)+ 1 b γn(x)4(1− t)3/2 Since∇bγn(x)/bγn(x)≤3/2xandbγn(x) 2≥ σγ nx, we compute C2n≤ f(x) (1− t)γn + f(x) (1− t)5/4γ5/4 n , (5.57)
for some continuous functionf. Step 4. Control of C3n
We now turn to the last term Cn3 and observe from the martingale moment inequality that C3n= E Z 1 t cnsdWs ≤ CE s Z 1 t |cn s|2ds ≤ C √ 1− tE sup t≤s≤1 |cns| . (5.58)
In order to control this last term, we compute the dynamics of cn defined as ∇bn−R.
tπ n
sDsbnds. We deduce from the dynamics ofbn given in (5.44) that
d∇bnu =− γn2 8|bγn( eSun)|2 ∇bnudu+ γn2 4 ∇bγn |bγn| 3( eS n u)∇Se n ubnudu + γ 2 n 2 ∇bγn |bγn| 2( eS n u)πnu∇πnu(1− u)du+ γn2 4 ∇ ∇bγn |bγn| 2 ( eSun)∇Se n u|πun|2(1− u)du .
Similarly, we compute dDsbnu =− γ2n 8|bγn( eS n u)|2 Dsbnudu+ γn2 4 ∇bγn |bγn|3( eS n u)DsSe n ubnudu + γ 2 n 2 ∇bγn |bγn|2 ( eSun)πnuDsπnu(1− u)du+ γ2n 4 ∇ ∇bγn |bγn|2 ( eSun)DsSe n u|πnu|2(1− u)du ,
for t ≤ s ≤ u ≤1. Since D.Seun=∇Se.n/{(1− t)πns}, we deduce following the same
line of arguments as in Step 3 of the previous section that dcnu=− γn2 8|bγn( eS n u)|2 cnudu+ γ2n 4 ∇bγn |bγn| 3( eS n u)bnu 1− u 1− t ∇Se n udu + γ 2 n 2 ∇bγn |bγn|2( eS n u)πnu(1− u)bnudu+ γ2n 4 ∇ ∇bγn |bγn|2 ( eSun)∇Se n u|πun|2 (1− u)2 1− t du . Therefore, Ito’s formula together with the definition ofπn leads to
cnu πun = c n t πnt + 6 Z u t ∇bγn( eSrn)(1− r) bnr πnr γn2 8 πnrdr |bγn( eSrn)|2 + Z u t γn2 4 ∇2bγn b γn −2|∇bγn| 2 |bγn| 2 ( eSrn)|πrn|2(1− r)2dr , t ≤ u ≤1. Sinceπn and∇2 b
γn are decreasing, this relation combined with (5.58) implies
C3n ≤ C √ 1− t(|cnt|+ EX1n+ EY1n) , (5.59) with Xn:= Z . t −∇bγn( eS n r)(1− r) bnr πn r 2πnrdπnr ≥0, Yn:= Z . t −2|∇bγn( eSrn)|2− ∇2bγn( eS n r)bγn( eS n r) |πnr|2(1− r)2dπnr ≥0.
We first focus on the processYn and, sinceπn is decreasing, observe that 0 ≤ Yn ≤ |πnt|2 Z . t −2|∇bγn( eSrn)|2− ∇2bγn( eS n r)bγn( eS n r) (1− r)2dπrn. (5.60)
Applying Ito’s formula to the process(1− u)2πun/bγn( eS
n u)2 t≤u≤1, we get (1− u)2πun |bγn( eSnu)|2 −(1− t) 2πn t |bγn(x)| 2 = Z u t 3|∇bγn( eS n r)|2 b γn( eSrn)4 −∇ 2 b γn( eSrn) b γn( eSnr)3 πnr|bγn( eS n r)|2 (1− r)−2 dr − Z u t 2(1− r)π n r |bγn( eSnr)|2 dr+ Z u t (1− r)2 |bγn( eSrn)|2 dπnr−2 Z u t πnr (1− r)−2 |∇bγn( eSrn)|2 |bγn( eSnr)|2 dr − NuY ,
whereNY is a local martingale given byNY :=R.
t2(1−r) 2πn r∇bγn( eS n r)/bγn( eS n r)2dWr. Plugging 2dπrn=πnr∇2bγn( eS n r)bγn( eS n
r)drin the previous equality provides
βun:= 1 2 Z u t (1− r)2πrn 2|∇bγn( eS n r)|2 |bγn( eSrn)|2 −∇ 2 b γn( eSrn) b γn( eSrn) dr (5.61) = (1− u) 2πn u |bγn( eS n u)|2 −(1− t) 2πn t |bγn(x)|2 + Z u t 2(1− r)π n r |bγn( eS n r)|2 dr+NuY , t ≤ u ≤1. (5.62)
Let pickv ∈[t,1] and define for r ∈[v,1],Nr∗:=
Z r
v
NuYdu. By virtue of Theorem
65,IV-6, [12], (Nr∗)r∈[v,1] is a local martingale. Moreover, (5.62) implies that Z r v βundu ≤ Z r v (1− u)2πun [bγn( eS n u)|2 du+ Z r v Z u t 2(1− r)π n r |bγn( eSrn)|2 drdu+Nr∗ ≤3(1− v) Z 1 t (1− u)πnu |bγn( eS n u)|2 du+Nr∗ (5.63)
Besides, observe that Z 1 t (1− r)πrn |bγn( eSrn)|2 dr = 8 γ2 n Z 1 t −(1− r)dπrn ≤ 8 γ2 n (1− t)πtn. (5.64)
This estimate together with (5.63) and βn ≥ 0 imply that (Nr∗)r∈[v,1] is a su-permartingale, as a local martingale bounded from below. Therefore, since βn is increasing, we deduce from (5.63) and (5.64) that
Eβvn≤ 1 1− vE Z 1 v βundu ≤ 24 γ2 n (1− t)πnt , t ≤ v <1.
Asv →1, using the Fatou lemma sinceβn≥0, we derive EY1n ≤
γn2
4 |π
n
t|2Eβ1n ≤ 6(1− t)|πnt|3. (5.65)
We now focus on the termX1n and observe from (5.44) that
d bnr πn r = σ 2γ2 n∇bγn( eS n r) 4|bγn( eSrn)|2 πrndr = − ∇bγn( eS n r) 2 dπ n r ,
so thatbn/πn is increasing and therefore 2d|bnr/πnr| ≤ −∇bγn( eS
n
r)dπrn. Hence, Ito’s
formula implies directly Xun ≤ ∇bγn(x) |bnt| πtn(1− t)|π n t|2− Z u t ∇bγn( eS n r) |bnr| πnr |πnr|2dr − Z u t |∇bγn( eS n r)|2|πrn|2(1− r)2dπrn+ Z u t |bnr| πrn (1− r)|πrn|2d∇bγn( eS n r),
fort ≤ u ≤1. Plugging (5.49) in this expression, we deduce 0 ≤ 1 2X n u ≤ ∇bγn(x)(1− t)|b n t|πtn+Y1n+NuX, t ≤ u ≤1, (5.66)
whereNX is a local martingale. Since EY1n< ∞, we deduce thatNX is a super-martingale so that EN1X ≤0. Hence, combining (5.59) together with (5.65) and (5.66) provides C3n ≤ C √ 1− t|cnt|+∇bγn(x)(1− t)|b n t|πtn+ (1− t)|πnt|3 = √C 1− t |∇2bγn(x)| |bγn(x)| 2 + 3 |∇bγn(x)| 2 |bγn(x)| 3 + 1 (1− t)|bγn(x)| 3 . Since|∇2 b γn|bγn(x)≤ Cγn/x,∇bγn/bγn(x)≤ C/xand 1/bγn(x)≤ C √ γnx, this yields C3n≤ C γn √ 1− t 1 x5/2+ 1 γnx3/2 . (5.67)
5.4 Estimate (4.26) on the crossed derivative
This subsection is dedicated to the obtention of (4.26). This finer estimate is nec-essary in order to consider transaction costs coefficients which do not vanish as the number of trading datesngoes to infinity. It requires the obtention of stronger estimates on ( bCnxx) and ( bCxxxn ) which are made possible via the control (3.7) on
the sequence of payoff functions (hn)n.
We recall that the initial condition (t, x) is fixed and Et,xdenotes E[. |Sent =x].
Let us first derive some a priori estimates on ( eSn)n and (∇Sen)n.
Lemma 5.1 There exist a constant C and a continuous function f on (0, ∞) which do not depend on n such that
Et,x∇Se n u ≤ C , t ≤ u ≤1, (5.68) Et,xSe n u ≤ Cf(x), t ≤ u ≤1, (5.69) Et,x ∇Se n u 2 ≤ Cf(x), t ≤ u ≤1, (5.70) Et,x|Se n u|3/2 ≤ Cf(x), t ≤ u ≤1, (5.71) Et,x|Se n u|2 ≤ C √ γnf(x), t ≤ u ≤1, (5.72)
Proof.We fixn ∈N andu ∈[t,1] in order to verify each estimate separately.2 Proof of(5.68). Recall that∇Se n satisfies d ∇Se n u =∇bγn( eS n u)∇Se n udWu+σ2∇Se n udu.
Using the dynamic of eSn and the Ito formula, we verify easily that eSn has finite moments of all orders. As∇Se
n u =πun(1− t)bγn( eS n u) , we deduce that∇Se n u has also
finite moments of all orders. We also know that the process ∇Sen is positive and R.
t∇bγn( eS
n
u)∇SenudWuis a local martingale which turns out to be a martingale once
stopped by a sequence of stopping timesτk,n→ ∞a.s. ask → ∞. By the Fatou Lemma, we deduce that
E∇Se n u ≤1 + lim inf k E Z τk,n t σ2∇Se n udu ≤ 1 + E Z u t σ2∇Se n rdr .,
Using the Gronwall lemma, we conclude about (5.68).2 Proof of(5.69).
By virtue of (5.68), we have 0 ≤ ∇xEt,xSeun = Et,x∇Seun ≤ C. Hence, a Taylor
expansion directly leads to Et,xSe n u = Et,xSe n u−Et,0Se n u ≤ Cx . 2 Proof of(5.70)
From the s.d.e. satisfied by eSnu, we deduce that there is a constant C such
thatE|Se
n
u|2≤ Cγng(x) for some continuous functiong. To do so, it suffices to use
inequality (5.69) and apply the Gronwall lemma. Recall that ∇Se n= πn(1− t)bγn( eSn) =πn(1− t) q σ2| e Sn|2+σγn e Sn. Asπn≤ πtn, we conclude about (5.70).2 Proof of(5.71). We have∇xEt,x|Se n u|3/2= (3/2)Et,x|Se n u|1/2∇Se n
u. Using the Cauchy-Schwarz
in-equality and Inequalities (5.69) and (5.70), we deduce that 0≤ ∇xEt,x|Seun|3/2 ≤
Cg(x), for some continuous function g. Hence (5.69) follows from a Taylor expan-sion.2
Proof of(5.72).
We have ∇xEt,x|Seun|2 = 2Et,xSeun∇Seun. We then use the Cauchy–Schwarz
in-equality with Inin-equality (5.70) and the inin-equality Et,x|Se
n
u|2≤ Cγng(x). The
con-clusion follows as previously. 2
We now provide finer estimates on ( bCxxn ) and ( bCxxxn ).
Lemma 5.2 There exists a continuous function f such that |Cb n xx(t, x)| ≤ f(x) (1− t)4/3γ nln(n) , (t, x)∈[0,1]×(0, ∞), n ∈N. Proof. Fixn ∈N. From (4.17) and (5.34), we compute
b Cxxn (t, x) = E[∇2hn( eS1n)∇Se n 1] = E ∇2hn( eS1n) b γn( eS1n)π1n (1− t)−1 ≤ C √ γnk∇2hnk∞ (1− t)−1 E [π n 1], since∇2hn vanishes outside a compact subset of (0, ∞) which does not depend of nand hence∇2hn( e Sn1)bγn( eS n 1) is bounded byC √ γnk∇2hnk∞.
We now look towards a sharp estimate of E[π1n]. The expression ofπn given in (5.34) together with Jensen inequality yield
E [πn1]≤ πnt 1− t Z 1 t E e− (1−t)σ2 γ2n 8|γn( eb Snu )|2du ≤ π n tγn−2 (1− t)2 Z 1 t E " |bγn( eSun)|2e − (1−t)σ2 γ2n 16|bγn( eSnu )|2 # du,(5.73) where we used the boundxe−x ≤ C, x ≥0, for the last inequality. We split the expectation of the r.h.s. in the expression above in two parts. The first one is bounded fornlarge enough as follows, by virtue of (5.69) and (5.72):
E " |bγn( eSun)|2e −(1−t)σ2 γ2n 16γ2 ( eSnu )1 {Seun≤ √ γn} # ≤ γne− (1−t)σ2 γ1/2n 16 f(x), (5.74)
where f is a continuous function which may change from line to line. Observe that the Cauchy-Schwarz inequality and (5.71) yields E[ eSun1Sen
u≥
√ γn]≤ γ
−1/6 n f(x).
Therefore, the second term is bounded by
E " |bγn( eS n u)|2e −(1−t)σ2 γ2n 16γ2 ( eSnu )1 {Senu≥ √ γn} # ≤(σ2√γn+γ5n/6)f(x) ≤ γn5/6f(x). (5.75)
Together withx1/3e−x≤ C,x ≥0, plugging (5.74) and (5.75) in (5.73) yields E [π1n]≤ πtnγn−2 (1− t) γ5n/6f(x) (1− t)1/3 ≤ f(x) γn5/3(1− t)7/3 . (5.76)
Together with (3.7), plugging this estimate in the first inequality of this proof
concludes the proof. 2
Lemma 5.3 Fix n ∈N. There exists a continuous function f such that |Cb
n
xxx(t, x)| ≤
f(x)
(1− t)4/3γnln(n), (t, x)∈[0,1]×(0, ∞), n ∈N.
Proof. Fix n ∈N. As observed in Section 5.2, we have |Cb n xxx(t, x)| ≤ k∇hnk∞An1/2+ ¯Bn, where ¯Bn:= E ∇hn( eS1n) Z 1 t bnudWu (5.77)
and (An)n and (bn)n are respectively given in (5.37) and (5.40). As already
ob-served in (5.38), we have An ≤ (1− t) −2 |bγn(x)| 2 Z 1 t Z u t E π n s 2 dsdu ≤ (1− t) −2|πn t|2 |bγn(x)| 2 Z 1 t Z u t Z s t Ee− (s−t)σ2 γ2n 4|bγn ( eSnr )|2 s − t drdsdu. Using the boundx1/2e−x≤ C for x ≥0, we deduce
An≤ C(1 − t)−2|πn t|2 |bγn(x)| 2γn Z 1 t Z u t 1 (s − t)3/2 Z s t E " b γn( eSnr)e −(s−t)σ2 γ2n 8|bγn( eSnr )|2 # drdsdu.
Since the exponential on the r.h.s is smaller than 1, we directly deduce from (5.69) that A1n/2 ≤ C (1− t)−1|πnt| |bγn(x)|γn1/2 γn1/4f(x) (1− t)−3/4 ≤ f(x) (1− t)5/4γn5/4 ≤ f(x) (1− t)4/3γ nln(n) .(5.78)
We now focus on the second term on the r.h.s. of (5.77) and rewrite ¯ Bn= E Z 1 t bnu∇2hn( eSn1)DuSe n 1du= E Z 1 t bnu∇2hn( eS1n) ∇Se n 1 (1− t)πnu du , t ≤ u ≤1. Observe from (5.45) that the processbn is given by
bnu= πnu πn t bnt +πun Z u t σ2γn2∇bγn( eS n r) 4bγ2 n( eSnr) πrn(1− r)dr.
Moreover, recall that∇Se1n=π1n(1−t)bγn( eS1n) and∇2hnvanishes outside a compact subset independent ofn. Plugging these estimates in the expression of ¯Bn, we get
¯ Bn ≤ C√γnk∇2hnk∞ |bnt| πnt (1− t)E[π n 1] + E π1n Z 1 t σ2γ2n∇bγn( eS n r) 4bγ 2 n( eSrn) πnr(1− r)2dr .
Recalling the process βn defined in (5.61), observe that the expression ofbnt to-gether with|∇bγn| ≤1 +|∇bγn| 2 and (5.34) lead to ¯ Bn ≤ C√γnk∇2hnk∞ ∇bγn(x) b γn(x) (1− t)E[π1n] + E π1n Z 1 t −(1− r)2dπnr + E [πn1β n 1] ≤ C√γnk∇2hnk∞ 1− t x E[π n 1] + (1− t)2π n tE [πn1] + E [π n 1β n 1] ≤ k∇ 2 hnk∞f(x) (1− t)4/3γ7n/6 + C√γnk∇2hnk∞E [πn1β1n], (5.79)
where the last inequality follows from (5.76).
The rest of the proof is dedicated to the control of E [πn1β1n]. We follow the notations of the previous section and observe from the monotonicity ofβntogether with (5.63) that Eπ1nβn1 ≤ lim v→1Eπ n 1 1 1− v Z 1 v βundu ≤lim v→1Eπ n 1 3 Z 1 t (1− r)πnr b γ2 n( eSrn) dr+ 1 1− v Z 1 v NrYdr .
Since the first term in the parenthesis is bounded byCπnt(1− t)γ−n2, (5.76) yields
Eπn1β1n≤ Cπtn(1− t)γn−2 (1− t)4/3γ7/6 n f(x) + lim v→1Eπ n 1 1 1− v Z 1 v NrYdr . (5.80)
Regarding the last term, we first observe from (5.63) that Z u v NrYdr ≥ −3(1− v) Z 1 t (1− r)πnr γ2( eSn r) dr ≥ −C(1− v)πntγn−2, v ≤ u ≤1.
This provides an upper bound for (R.
vN Y
r dr)−and the integration by parts formula
yields πun Z u v NrYdr ≤ −C(1− v)πtnγn−2 Z u v dπnr+ Z u v πnrNrYdr , v ≤ u ≤1.
Moreover, the last term on the r.h.s is a supermartingale as a bounded from below local martingale. Hence, by virtue of the Lebesgue theorem, we finally deduce that
lim v→1Eπ n 1 1 1− v Z 1 v NrYdr ≤ C(1− v)πntγ−n2lim v→1E Z 1 v d πrn πtn = 0.
Combining this estimate with (5.77), (5.78), (5.79) and (5.80) and (3.7) concludes
the proof. 2
Proof of(4.26).
In order to derive the upper bound (4.26), it suffices to derive the expression of bCxtn(t, x) from bCxxn (t, x) and bCnxxx(t, x) by differentiating the p.d.e. (en) and to