A Scaling Limit for Utility Indifference Prices in the Discretized Bachelier Model
aa r X i v : . [ m a t h . P R ] F e b A SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THEDISCRETIZED BACHELIER MODEL
ASAF COHEN AND YAN DOLINSKY
Abstract.
We consider the discretized Bachelier model where hedging is done on an equidis-tant set of times. Exponential utility indifference prices are studied for path-dependent Eu-ropean options and we compute their non-trivial scaling limit for a large number of tradingtimes n and when risk aversion is scaled like nℓ for some constant ℓ >
0. Our analysis ispurely probabilistic. We first use a duality argument to transform the problem into an optimaldrift control problem with a penalty term. We further use martingale techniques and stronginvariance principles and get that the limiting problem takes the form of a volatility controlproblem.
AMS Classification:
Primary: 91G10, 60F15
Keywords:
Utility indifference, strong approximations, path-dependent SDEs, asymptoticanalysis Introduction
Taking into account market frictions is an important challenge in financial modeling. In thispaper, we focus on the friction that the rebalancing of the portfolio strategy is limited to occurdiscretely. In such a realistic situation, a general future payoff cannot be hedged perfectly evenin complete market models such as the Bachelier model or the Black–Scholes model.We consider hedging of a path-dependent European contingent claim in the Bachelier modelfor the setup where the investor can hedge on an equidistant set of times. Our main result isproviding the asymptotic behavior of the exponential utility indifference prices where the riskaversion goes to infinity linearly in the number of trading times. Namely, we establish a nontrivial scaling limit for indifference prices where the friction goes to zero and the risk aversiongoes to infinity.This type of scaling limits goes back to the seminal work of Barles and Soner [2], whichdetermines the scaling limit of utility indifference prices of vanilla options for small proportionaltransaction costs and high risk aversion. Another work in this direction is the recent article [1]which deals with scaling limits of utility indifference prices of vanilla options for hedging withvanishing delay H ↓ A/H for some constant A . In general,the common ground between the above two works and the present paper it that all of them startwith complete markets and consider small frictions, which make the markets incomplete and thederivative securities cannot be perfectly hedged with a reasonable initial capital. Then, insteadof considering perfect hedging, these papers study utility indifference prices with exponentialutilities and with large risk aversion. In contrast to the previous two papers which treated onlyvanilla options (path-independent), in this paper we are able to provide the limit theorem forpath-dependent options. Date : February 25, 2021.A. Cohen acknowledges the financial support of the National Science Foundation (DMS-2006305). Y. Dolinskyis supported in part by the GIF Grant 1489-304.6/2019 and the ISF grant 160/17.
Although the topic of discrete time hedging in the Brownian setting was largely studied, seefor instance, [3, 17, 13, 18, 14, 15, 10, 16, 11, 4], these papers studied the optimal discretizationof given hedging strategies such as the delta hedging strategy. In the present study, insteadof tracking a given hedging strategy we follow the well known approach of utility indifferencepricing which is commonly used in the setup of incomplete markets (see [5] and the referencestherein). In other words, this approach says that the price of a given contingent claim should beequal to the minimal amount of money that an investor has to be offered so that she becomesindifferent (in terms of utility) between the situation where she has sold the claim and the onewhere she has not.We now put our contribution in the context of asymptotic analysis of risk-sensitive controlproblems. Such problems model situations where a decision maker aims to minimize smallprobability events with significant impacts. Typically, the small probability event emerges froma state process with a volatility that vanishes with the scaling parameter. The limiting behaviorsof such risk-sensitive control problems are governed by deterministic differential games, see e.g.,[7] and the references therein. In our case, the volatility of the state process does not vanish, butis rather of order O (1), and the small probability event emerges from the discrete approximationof the stochastic integral on the grid. As a result, the structure of the limiting problem is quitedifferent: rather than a deterministic (drift control) differential game, we obtain a stochastic(volatility) control problem. The connection between the indifference price and the differencebetween two values of two (non asymptotically) risk-sensitive problems is well-known, see e.g.,[19]. It stems from the fact that the utility indifference pricing is a normalized version ofthe certainty equivalent criterion. Our study is concerned with the scaling limit of the utilityindifference prices when the market friction goes to zero and the risk-aversion goes to infinity.Let us outline the key steps in establishing the asymptotic result. Our approach is purelyprobabilistic and allows to consider European contingent claims with path-dependent payoffs.The first step in establishing the main result goes through a dual representation of the valuefunction. This representation is closer in nature to the form of the limiting stochastic volatilitycontrol problem. In the dual problem, there is only one player: a maximizing (adverse) playerthat controls the drift; the investor’s role is translated into a martingale condition. The control’scost is small, hence allowing the maximizer to choose controls with high values. The second andthe main step is to analyze the limit behavior of the dual representation. This is done via upperand lower bounds.There are two main challenges in the proof of the upper bound. The first one is comparingbetween the dual and the limiting penalty terms. The second one is due to the fact that theconsistent price systems in the dual representation are not necessary tight. To handle the firstchallenge, we work on the discrete-time grid, and for any level of penalty in the limiting problemwe are able to construct an optimal penalty term in the prelimit problem (Lemma 4.1). Thestructure of this best penalty term also serves us in the proof of the lower bound. We overcomethe second difficulty by applying a strong invariance principle (Lemma 4.2) and not the widelyused weak convergence approach which is not helpful here. Specifically, for any control in theprelimit dual problem we construct a process in the form of the limiting problem, which is closein probability to the original process and for which the penalization term is bounded in thedesired direction.The proof of the lower bound is made by an explicit construction driven by Lemma 4.1. Akey ingredient in the proof is a construction of a path-dependent stochastic differential equation(SDE), which translates the consistent price systems given via the drift control into the limitingvolatility control problem. SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 3
The rest of the paper is organized as follows. In Section 2 we provide the model and themain result. Section 3 provides the duality result. Finally, the proof of the main result is givenin Section 4. 2.
The model and the main results
Let (Ω , F , P ) be a complete probability space carrying a one-dimensional Wiener process( W t ) t ∈ [0 ,T ] with natural augmented filtration ( F Wt ) t ∈ [0 ,T ] and time horizon T ∈ (0 , ∞ ). Weconsider a simple financial market with a riskless savings account bearing zero interest (forsimplicity) and with a risky asset X with Bachelier price dynamics X t = X + σW t + µt, t ∈ [0 , T ] , (2.1)where X ∈ R is the initial asset price, σ > µ ∈ R is the constantdrift. These parameters are fixed throughout the paper.Fix n ∈ N and consider an investor who can trade the risky asset only at times from the grid { , T /n, T /n, ..., T } . For technical reasons, in addition to the risky asset ( X t ) t ∈ [0 ,T ] we assumethat the financial market contains short time horizon options with payoffs in the spirit of poweroptions. Formally, for any k = 0 , , ..., n , at time kT /n the investor can buy but not sell Europeanoptions that can be exercised at the time ( k +1) T /n with the payoff | X ( k +1) T/n − X kT/n | . Denoteby h ( n ) the price of the above option. For simplicity, we will assume that the price does notdepend on k . Moreover, we will assume the following scaling:lim n →∞ n / h ( n ) = ∞ (2.2)and lim n →∞ nh ( n ) = 0 . (2.3)This investment opportunity can be viewed as an insurance against high values of the stockfluctuations. Roughly speaking, the term | X ( k +1) T/n − X kT/n | is of order O ( n − / ). Howeversince the payoff of this option is quite extreme we expect that the corresponding price will bemore expensive than O ( n − / ). The scaling which is given by (2.2)–(2.3) says that the optionprice is more expansive than O ( n − / ) but cheaper than O (1 /n ). Remark 2.1.
It is possible to replace the payoff | X ( k +1) T/n − X kT/n | with the payoff | X ( k +1) T/n − X kT/n | ǫ for ǫ > and assume that the new option price ˆ h ( n ) satisfies lim n →∞ n ˆ h ( n ) = 0 and lim n →∞ n ǫ/ ˆ h ( n ) = ∞ . However, for simplicity we work with power = 3 . In line with the above, the set A n of trading strategies for the n -step model consists of pairs( γ, δ ) = { ( γ k , δ k ) } ≤ k ≤ n − such that for any k , γ k , δ k are F WkT/n -measurable and in addition δ k ≥ V γ,δ kTn := k − X i =0 γ i (cid:0) X ( i +1) T/n − X iT/n (cid:1) + k − X i =0 δ i (cid:0) | X ( i +1) T/n − X iT/n | − h ( n ) (cid:1) , k = 0 , , ..., n. (2.4)Next, let C [0 , T ] be the space of all continuous functions z : [0 , T ] → R equipped with theuniform norm || z || := sup ≤ t ≤ T | z t | and let f : C [0 , T ] → [0 , ∞ ) be a Lipschitz continuous functionwith respect to the uniform norm. For the n -step model we consider a European option withthe payoff f n ( X ) := f ( p n ( X )) where p n ( z ) returns the linear interpolation of (( kT /n, z kT/n ) : A. COHEN AND Y. DOLINSKY k = 0 , . . . , n ) for any function z : [0 , T ] → R . It is immediate from the Lipschitz continuity of f that it has a linear growth, and consequently that E P [exp ( αf n ( X ))] < ∞ , ∀ α ∈ R . (2.5)The investor will assess the quality of a hedge by the resulting expected utility. Assumingexponential utility with constant absolute risk aversion λ >
0, the utility indifference price ofthe claim f n ( X ) does not depend on the investor’s initial wealth and takes the well-known form π ( n, λ ) := 1 λ log inf ( γ,δ ) ∈A n E P h exp (cid:16) − λ (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i inf ( γ,δ ) ∈A n E P h exp (cid:16) − λV γ,δT (cid:17)i . (2.6)The following scaling limit is the main result of the paper. Theorem 2.1.
For n → ∞ and re-scaled high risk-aversion nℓ with ℓ > fixed, the utilityindifference price of f n ( X ) has the scaling limit lim n →∞ π ( n, nℓ ) = π ( ℓ ) := sup ν ∈V E P h f (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt i , (2.7) where X ( ν ) t := X + Z t √ ν u dW u , t ∈ [0 , T ] ,g ( y ) := y − ln y − , y > . and V is the class of all bounded, non negative ( F Wt ) -predictable processes ν . Let us finish this section with the following remark.
Remark 2.2.
Observe that if we take ℓ to infinity then we have lim ℓ →∞ π ( ℓ ) = sup ν ∈V E P h f (cid:16) X ( ν ) (cid:17)i . The above right-hand side can be viewed as the model-free option price, see [12] . This correspondsto the case where the investor wants to super-replicate the payoff f ( X ) without any assumptionson the volatility. For the case where ℓ → it is straightforward to check that lim ℓ → π ( ℓ ) = E P h f (cid:16) X ( σ ) (cid:17)i where X ( σ ) t = X + σW t , t ∈ [0 , T ] . Namely, we converge to the unique price of the continuous time complete market given by (2.1). A duality result
Set n ∈ N . Denote by Q n the set of all probability measures Q ∼ P with finite entropy E Q (cid:20) log (cid:18) d Q d P (cid:19)(cid:21) < ∞ relative to P and such that the processes { X kT/n } ≤ k ≤ n and ( k − X i =0 | X ( i +1) T/n − X iT/n | − kh ( n ) ) ≤ k ≤ n SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 5 are, respectively, Q -martingale and a Q -supermartingale, with respect to the filtration {F WkT/n } ≤ k ≤ n .Denote by ˆ Q the unique probability measure such that ˆ Q ∼ P and { X t } t ∈ [0 ,T ] is ˆ Q -martingale.From (2.2) it follows that for sufficiently large n , we have ˆ Q ∈ Q n and so Q n = ∅ .We arrive at the dual representation for the certainty equivalent of f n ( X ) which is definedby 1 λ log (cid:18) inf ( γ,δ ) ∈A n E P h exp (cid:16) − λ (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i(cid:19) where λ > Proposition 3.1.
Let n ∈ N be sufficiently large such that Q n = ∅ and fix an arbitrary λ > .Then, λ log (cid:18) inf ( γ,δ ) ∈A n E P h exp (cid:16) − λ (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i(cid:19) = sup Q ∈Q n E Q (cid:20) f n ( X ) − λ log (cid:18) d Q d P (cid:19)(cid:21) . (3.1) Proof.
Since for any ( γ, δ ) ∈ A n and λ > V λγ,λδkT/n = λV γ,δkT/n for all k = 0 , , ..., n , thenwithout loss of generality we take λ = 1. As usual, the proof rests on the classical Legendre–Fenchel duality inequality xy ≤ e x + y (log y − , x ∈ R , y > , with ‘=’ iff y = e x . (3.2)We start with proving the inequality ‘ ≥ ’ in (3.1).Let ( γ, δ ) ∈ A n that satisfies E P h exp (cid:16) − (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i < ∞ . Choose an arbitrary Q ∈ Q n . From the Cauchy–Schwarz inequality and (2.5) it follows that E P [ e − V γ,δT / ] < ∞ . This together with (3.2) for x = max (cid:16) , − V γ,δT / (cid:17) and y = d Q d P , gives that E Q h max (cid:16) , − V γ,δT (cid:17)i = 2 E P (cid:20) d Q d P max (cid:16) , − V γ,δT / (cid:17)(cid:21) < ∞ . From (2.4) it follows that the portfolio value process { V γ,δkT/n } ≤ k ≤ n is a local Q -supermartingale,and so from [8, Proposition 9.6] and the inequality E Q (cid:2) max(0 , − V γ,δT ) (cid:3) < ∞ we get E Q [ V γ,δT ] ≤ z >
0, the followingestimates hold E P h exp (cid:16) − (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i ≥ E P (cid:20)(cid:18) e − V γ,δT + V γ,δT ze − f n ( X ) d Q d P (cid:19) e f n ( X ) (cid:21) ≥ E P (cid:20)(cid:18) − ze − f n ( X ) d Q d P (cid:18) log (cid:18) ze − f n ( X ) d Q d P (cid:19) − (cid:19)(cid:19) e f n ( X ) (cid:21) = − z (log z −
1) + z E Q (cid:20) f n ( X ) − log d Q d P (cid:21) . Indeed, the first inequality follows since E Q [ V γ,δT ] ≤ x = − V γ,δT and y = ze − f n ( X ) d Q /d P . Finally, taking supermum over z in thelast expression, one obtains from (3.2) that the supremum is exp (cid:16) E Q h f n ( X ) − log d Q d P i(cid:17) . Thisimplies the inequality ‘ ≥ ’ in (3.1). A. COHEN AND Y. DOLINSKY
Next, we prove the reversed inequality ‘ ≤ ’ in (3.1). Define the probability measure ˆ P by d ˆ P d P = e f n ( X ) E P [ e f n ( X ) ] . Without loss of generality we can assume that the right-hand side of (3.1) is finite. Thus forany Q ∈ Q n E Q (cid:20) log d Q d ˆ P (cid:21) = E Q (cid:20) log d Q d P − f n ( X ) (cid:21) + log (cid:16) E P [ e f n ( X ) ] (cid:17) . Next, we show that the supremum in (3.1) is attained. We use the well-known Komlos-argument,see e.g., [8, Lemma 1.69], to obtain a maximizing sequence { Q m } m ∈ N ⊂ Q n , for which Z m := d Q m /d ˆ P converges almost surely as m → ∞ ; in fact, convergence holds also in L (ˆ P ), because,without loss of generality, { H ( Q m | ˆ P ) } m ∈ N can be assumed to be bounded, where H ( Q m | ˆ P ) := E ˆ P [ Z m log Z m ] = E Q m (cid:20) log d Q m d ˆ P (cid:21) . Hence, Z := lim m →∞ Z m yields the density (Radon-Nikodym derivative with respect to ˆ P ) ofa probability measure Q , which by Fatou’s lemma satisfies H ( Q | ˆ P ) ≤ lim m →∞ H ( Q m | ˆ P ) and E Q (cid:2) | X ( k +1) T/n − X kT/n | (cid:12)(cid:12) F kT/n (cid:3) ≤ lim inf m →∞ E Q m (cid:2) | X ( k +1) T/n − X kT/n | (cid:12)(cid:12) F kT/n (cid:3) ≤ h ( n ) , ∀ k ≤ n. So, attainment of the supremum in (3.1) is proven if we can argue that Q is a martingalemeasure for { X kT/n } ≤ k ≤ n . To this end, it is sufficient to argue that for any k the measures (cid:0) Q m ◦ ( X kT/n ) − (cid:1) m ∈ N are uniformly integrable in the sense thatlim M →∞ sup m ∈ N E Q m h | X kT/n | {| X kT/n | >M } i = 0where · equals 1 on the event · and 0 otherwise. From H¨older’s inequality and (2.5) it followsthat for any k , E ˆ P (cid:2) exp( | X kT/n | / ) (cid:3) < ∞ . Hence, from (3.2)sup m ∈ N E Q m h | X kT/n | / i = sup m ∈ N E ˆ P h Z m | X kT/n | / i ≤ E ˆ P h exp( | X kT/n | / ) i + sup m ∈ N H ( Q m | ˆ P ) < ∞ and the uniform integrability follows.Now, we arrive at the final step of the proof. We follow the approach in [9]. Indeed, theperturbation argument for the proof of Theorem 2.3 there shows that the entropy minimizingmeasure’s density is of the form Z = e − ξ E ˆ P [ e − ξ ]for some random variable ξ with E Q [ ξ ] = 0 and E Q [ ξ ] ≤ Q ∈ Q n . The separationargument for [9, Theorem 2.4] shows that ξ is contained in the L ( Q )-closure of { V γ,δT : ( γ, δ ) ∈A n } − L ∞ + for any Q ∈ Q n , where L ∞ + is the set of all non negative random variables, whichare uniformly bounded. Since Q n = ∅ , Lemma 3.1 in [23] yields that ξ must be of the sameform ξ = V γ ,δ T − R for some ( γ , δ ) ∈ A n and some R ≥
0. As a result, we may bound theleft-hand side in (3.1) as followslog (cid:18) inf ( γ,δ ) ∈A n E P h exp (cid:16) − (cid:16) V γ,δT − f n ( X ) (cid:17)(cid:17)i(cid:19) SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 7 ≤ log (cid:16) E P [ e f n ( X ) ] (cid:17) + log (cid:16) E ˆ P [exp( − V γ ,δ T )] (cid:17) ≤ log (cid:16) E P [ e f n ( X ) ] (cid:17) + log (cid:0) E ˆ P [exp( − ξ )] (cid:1) = log (cid:16) E P [ e f n ( X ) ] (cid:17) + E Q (cid:2) ξ + log (cid:0) E ˆ P [exp( − ξ )] (cid:1)(cid:3) = log (cid:16) E P [ e f n ( X ) ] (cid:17) − E Q [log Z ]= sup Q ∈Q n E Q (cid:20) f n ( X ) − λ log d Q d P (cid:21) . The first inequality follows by the definition of ˆ P ; the second inequality uses R ≥
0; the firstequality is derived by E Q [ ξ ] = 0; the second equality follows by the definition of ξ ; and finally,the last equality is deduced by our choice of Q as a measure that attains the supremum over Q n . (cid:3) An equivalent form for the dual problem.
We now rewrite the right-hand side of (3.1)by using the explicit expression for the Radon–Nykodim derivative, using Girsanov’s kernels.For any probability measure Q ∼ P , let ψ Q be an ( F Wt )-progressively measurable process suchthat d Q d P |F Wt = exp (cid:16) Z t ψ Q s dW s − Z t | ψ Q s | ds (cid:17) , t ∈ [0 , T ] . We refer to ψ Q as the Girsanov kernel associated with Q . The dynamics of X can be writtenequivalently as X t = X + σW Q t + σ Z t ψ Q s ds + µt, t ∈ [0 , T ] , where W Q t := W t − R t ψ Q s ds is a Wiener process under Q . We obtain that for any n ∈ N and λ > Q ∈Q n E Q (cid:20) f n ( X ) − λ log (cid:18) d Q d P (cid:19)(cid:21) = sup Q ∈Q n E Q (cid:20) f n ( X ) − λ Z T | ψ Q s | ds (cid:21) . (3.3)4. Proof of Theorem 2.1
The proof relies on two bounds, which are provided in two separate subsections.4.1.
Upper bound.
This section is devoted to the proof of the inequality ‘ ≤ ’ in (2.7). We startwith the following lemma which provides an intuition for the penalty term g ( y ) := y − ln y − y > Lemma 4.1.
Consider a filtered probability space ( ¯Ω , ¯ F , { ¯ F t } t ∈ [0 , , ¯ P ) which supports an ( ¯ F t ) -Wiener process ¯ W . As usual we assume that the filtration { ¯ F t } t ∈ [0 ,T ] is right-continuous andcontains the null sets. Then, for any ≤ t < t ≤ T and any ( ¯ F t ) -progressively measurableprocess ψ , satisfying ¯ E [ R t t ψ u du | F t ] = 0 , one has, ¯ E h Z t t ψ u du (cid:12)(cid:12) F t i ≥ g (cid:16) t − t ¯ E h(cid:16) ¯ W t − ¯ W t + Z t t ψ u du (cid:17) (cid:12)(cid:12) F t i(cid:17) . (4.1) Moreover, set the parameterized (by β ) processes ( θ βt ) t ∈ [ t ,t ] and ( ϑ βt ) t ∈ [ t ,t ] by θ βt := Z tt ( β ( t − t ) − ( t − s )) − d ¯ W s , β > A. COHEN AND Y. DOLINSKY ϑ βt := − Z tt ( β ( t − t ) + ( t − s )) − d ¯ W s , β > . (4.3) Then (4.1) holds with equality for ψ t , t ∈ [ t , t ] , given by ψ t = θ ¯ βt n E h(cid:16) ¯ W t − ¯ W t + R t t ψ u du (cid:17) (cid:12)(cid:12) F t i >t − t o + ϑ ¯ βt n E h(cid:16) ¯ W t − ¯ W t + R t t ψ u du (cid:17) (cid:12)(cid:12) F t i By the independent increments and the scaling property of Wiener process, we assumewithout loss of generality that t = 0, t = 1 and F is the trivial σ -algebra.Obviously, (4.1) holds trivially if ¯ E [ R ψ t dt ] = ∞ , hence, for the rest of the proof we alsoassume that ¯ E [ R ψ t dt ] < ∞ . This in turn implies that ¯ E [ ψ t ] < ∞ for almost every t ∈ [0 , E [ ψ t ] =0 for any t ∈ [0 , ψ t := ψ t − ¯ E [ ψ t ] , t ∈ [0 , ψ , then¯ E h Z ψ t dt i ≥ ¯ E h Z ¯ ψ u dt i ≥ g (cid:16) ¯ E h(cid:16) ¯ W + Z ¯ ψ t dt (cid:17) i(cid:17) = g (cid:16) ¯ E h(cid:16) ¯ W + Z ψ t dt (cid:17) i(cid:17) , where the second inequality follows from (4.1) applied to ¯ ψ and the last equality follows since R ¯ E [ ψ t ] dt = 0.Next, by applying standard density arguments in L ( dt ⊗ ¯ P ) we can assume that ψ is a simpleprocess (see Section 3.2 in [21]) in the sense that ψ is bounded and there exists a deterministicpartition 0 = t < t < ... < t m = 1 such that ψ is a (random) constant on each interval( t i , t i +1 ]. Hence, for the rest of the proof we fix a simple process ψ which satisfies ¯ E [ ψ t ] = 0, forall t ∈ [0 , E [( ¯ W + R ψ t dt ) ] > E [( ¯ W + R ψ t dt ) ] < 1. Whenthe expected value equals 1, (4.1) follows immediately since g (1) = 0. Case I: ¯ E [( ¯ W + R ψ t dt ) ] > 1. Let ¯ β be given by (4.5). Observe that ¯ E [( ¯ W + R ψ t dt ) ] =¯ β/ ( ¯ β − E h(cid:16) ¯ W + Z ψ t dt (cid:17) − ¯ β Z ψ t dt i ≤ ¯ β ¯ β − − ¯ βg (cid:16) ¯ β ¯ β − (cid:17) = ¯ β log (cid:16) ¯ β ¯ β − (cid:17) . (4.6)Let ( F ¯ Wt ) t ∈ [0 , be the augmented filtration generated by the Brownian motion ( ¯ W t ) t ∈ [0 , andlet ( u t ) t ∈ [0 , be the optional projection of ψ on ( F ¯ Wt ) t ∈ [0 , (exists since ψ is bounded). Set, v t := ψ t − u t , t ∈ [0 , T ]. Observe that u t = ¯ E [ ψ t |F ¯ W ] for all t . This follows from the fact that¯ W [ t, − ¯ W t is independent of ψ t and ¯ W [0 ,t ] , where ¯ W [ a,b ] is the restriction of ¯ W to the interval SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 9 [ a, b ]. Thus,¯ E "(cid:18) ¯ W + Z ψ t dt (cid:19) − ¯ β Z ψ t dt = ¯ E "(cid:18) ¯ W + Z u t dt (cid:19) − ¯ β Z u t dt + ¯ E "(cid:18)Z v t dt (cid:19) − ¯ β Z v t dt ≤ ¯ E "(cid:18) ¯ W + Z u t dt (cid:19) − ¯ β Z u t dt (4.7)where the inequality follows from the Jensen inequality and the fact that ¯ β > ψ is simple it follows thatthere exists a (jointly) measurable map κ : [0 , × Ω → R such that κ t,s is F ¯ Wt ∧ s measurable forall t, s ∈ [0 , 1] and u t = Z t κ t,s d ¯ W s , dt ⊗ ¯ P a.s.For each t ∈ [0 , T ], we are applying the martingale representation theorem for the randomvariable u t . Utilizing the fact that ψ is piecewise constant, we obtain that κ is jointly measurablein s and t . This is essential in the sequel when we apply Fubini’s theorem.Define the processes ζ s := Z s κ t,s dt, η s := Z s κ t,s dt, s ∈ [0 , . Recall the process θ β given by (4.2). We get,¯ E h(cid:16) ¯ W + Z u t dt (cid:17) − ¯ β Z u t dt i = ¯ E h Z (cid:0) (1 + ζ s ) − ¯ βη s (cid:1) ds i ≤ ¯ E h Z (cid:16) (1 + ζ s ) − ¯ βζ s − s (cid:17) ds i ≤ ¯ E (cid:20)Z (cid:18)(cid:16) − s ) θ ¯ βs (cid:17) − ¯ β (1 − s ) | θ ¯ βs | (cid:19) ds (cid:21) = ¯ β log (cid:16) ¯ β ¯ β − (cid:17) . Indeed, the first equality follows from the stochastic Fubini theorem and the Itˆo-isometry (seeChapter IV in [24]) and the first inequality follows from the Cauchy–Schwarz inequality. Thistogether with (4.7) completes the proof of (4.6). Observe that for κ t,s := θ ¯ βs , t, s ∈ [0 , 1] theabove two inequalities are in fact equalities. Moreover, it easy to check that¯ E h(cid:16) ¯ W + Z u t dt (cid:17) i = Z (cid:16) − s ) θ ¯ βs (cid:17) ds = ¯ β ¯ β − ψ = θ ¯ β we have an equality in (4.1). Case II: ¯ E [( ¯ W + R ψ t dt ) ] < 1. Let ¯ β be given by (4.5). Observe that ¯ E [( ¯ W + R ψ t dt ) ] =¯ β/ ( ¯ β + 1) . In order to prove the inequality (4.1) it sufficient to show that¯ E h(cid:16) ¯ W + Z ψ t dt (cid:17) + ¯ β Z ψ t dt i ≥ ¯ β/ ( ¯ β + 1) + ¯ βg (cid:16) ¯ β ¯ β + 1 (cid:17) = ¯ β log (cid:16) ¯ β + 1¯ β (cid:17) . (4.8) Let u, v, ζ, η defined as in Case I. Recall the process ϑ β given by (4.3). Then, by using similararguments as in Case I we obtain¯ E "(cid:18) ¯ W + Z ψ t dt (cid:19) + ¯ β Z ψ t dt ≥ ¯ E "(cid:18) ¯ W + Z u t dt (cid:19) + ¯ β Z u t dt = ¯ E h Z (cid:0) (1 + ζ s ) + ¯ βη s (cid:1) ds i ≥ ¯ E h Z (cid:0) (1 + ζ s ) + ¯ βζ s / (1 − s ) (cid:1) ds i ≥ ¯ E (cid:20)Z (cid:18)(cid:16) − s ) ϑ ¯ βs (cid:17) + β (1 − s ) | ϑ ¯ βs | (cid:19) ds (cid:21) = ¯ β log (cid:16) ¯ β + 1¯ β (cid:17) and (4.8) follows. Finally, we notice that for κ t,s := ϑ ¯ βs , t, s ∈ [0 , 1] the above two inequalitiesare in fact equalities. In addition it is easy to check that¯ E h(cid:16) ¯ W + Z u t dt (cid:17) i = Z (cid:16) − s ) ϑ ¯ βs (cid:17) ds = ¯ β ¯ β + 1and so for ψ = ϑ ¯ β we have an equality in (4.1). (cid:3) Next, fix n ∈ N . The next lemma provides a bound for an expected payoff calculated withrespect to a given discrete-time martingale M , which later on will stand for { X kT/n } ≤ k ≤ n . Theidea is to construct a continuous-time martingale that is close in distribution to the process M on the discrete set of times and whose volatility is piecewise constant between two consecutivepoints on the discrete-time set. Lemma 4.2. Let { M k } ≤ k ≤ n be a martingale defined on some probability space, with M = X and that satisfies for any k = 0 , . . . , n − , ˆ E (cid:2) | M k +1 − M k | | M , . . . , M k (cid:3) ≤ h ( n ) (4.9) where ˆ E is the expectation with respect to the given probability space. Assume that for some K > , ˆ E h f n ( M ) − nℓ n − X k =0 g (cid:16) nσ T ˆ E (cid:2) | M k +1 − M k | | M , . . . , M k (cid:3) (cid:17)i ≥ − K (4.10) where f n ( M ) = f ( p n ( M )) and with abuse of notations p n ( M ) is the linear interpolation of (( kT /n, M k ) : k = 0 , . . . , n ) ( p n ( M ) is a random element in C [0 , T ] ). Then, there exists aconstant C > (that depends only on ℓ , K , and f , through the Lipschitzity and the lineargrowth), which is independent of n , such that, ˆ E h f n ( M ) − nℓ n − X k =0 g (cid:16) nσ T ˆ E (cid:2) | M k +1 − M k | | M , . . . , M k (cid:3) (cid:17)i ≤ C ( h ( n ) n ) / + sup ν ∈V E P h f n (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt i . (4.11) Proof. The Lipschitz continuity of f implies that it has a linear growth. This together with theDoob inequality for the martingale M , the simple bound g ( y ) ≥ y/ − SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 11 there exists a constant ˆ C > ℓ , K , and f , through the Lipschitzity andthe linear growth), which is independent of n , such that,ˆ E (cid:20) max ≤ k ≤ n M k (cid:21) ≤ ˆ C. (4.12)From Lemma 3.2 in [6] and (4.9) it follows that we can redefine the martingale { M k } ≤ k ≤ n ona new probability space that supports a sequence of independent and identically distributedrandom variables { Y k } ≤ k ≤ n , each has the standard normal distribution, such that for each k = 0 , , ..., n − Y k +1 is independent of { M i } ≤ i ≤ k , andˆ P (cid:18) max k =0 ,...,n | M k − ˆ X k | > ( h ( n ) n ) / (cid:19) < ¯ C ( h ( n ) n ) / , (4.13)for some universal constant ¯ C > 0, which is independent of the parameters in the model, withˆ X k := X + k − X i =0 Y i +1 ˆ E [( M i +1 − M i ) | M , . . . , M i ] , k = 0 , . . . , n. We abuse notation and use ˆ P for the probability measure under the original space and the newspace and keep using the notation M for the martingale under the new probability space.Next, we use this representation to embed the law of { ˆ X k } ≤ k ≤ n into the original Brownianprobability space from section 2: (Ω , F , {F Wt } t ∈ [0 ,T ] , P ). By applying Theorem 1 in [25] weobtain that there exist measurable functions χ k : R k +2 → R , k = 0 , , ..., n − 1, such that forany k , L ( Y , . . . , Y k +1 , M , . . . , M k , M k +1 )= L ( Y , . . . , Y k +1 , M , . . . , M k , χ k +1 ( Y , . . . , Y k +1 , M , . . . , M k , ξ ))where ξ has the standard normal distribution and is independent of { Y i } ≤ i ≤ k +1 and { M i } ≤ i ≤ k .Define on the probability space(Ω , F , {F Wt } t ∈ [0 ,T ] , P ) inductively the process { ¯ M i } ≤ i ≤ n , { ¯ Y i } ≤ i ≤ n and { ¯ ξ i } ≤ i ≤ n as follows. Set ¯ M = X . Define:¯ Y = p n/T W T/n , ¯ ξ = 3 W T/ (3 n ) − W T/ (2 n ) q Var (cid:0) W T/ (3 n ) − W T/ (2 n ) (cid:1) , ¯ M = χ ( ¯ Y , ¯ M , ¯ ξ ) , and for k = 0 , , ..., n − Y k +1 = p n/T (cid:0) W ( k +1) T/n − W kT/n (cid:1) , ¯ ξ k +1 = 3 W kT/n + T/ (3 n ) − W kT/n + T/ (2 n ) − W kT/n q Var (cid:0) W kT/n + T/ (3 n ) − W kT/n + T/ (2 n ) − W kT/n (cid:1) , ¯ M k +1 = χ k ( ¯ Y , . . . , ¯ Y k +1 , ¯ M , . . . , ¯ M k , ¯ ξ k +1 ) . Observe that { ¯ Y i } ≤ i ≤ n and { ¯ ξ i } ≤ i ≤ n have the standard normal distribution. Moreover, for any k , ¯ ξ k +1 is independent of { ¯ Y i } ≤ i ≤ k +1 and { ¯ M i } ≤ i ≤ k . From the definition of the functions χ k , k = 0 , , ..., n − 1, we conclude that L (cid:0) { ¯ M i } ≤ i ≤ n , { ¯ Y i } ≤ i ≤ n (cid:1) = L ( { M i } ≤ i ≤ n , { Y i } ≤ i ≤ n ) . (4.14) Next, let ϕ k : R k +1 → R + , k = 0 , , ..., n − 1, be measurable functions such that p E [( M k +1 − M k ) | M , . . . , M k ] = ϕ k ( M , . . . , M k ) , k = 0 , , ..., n − . Introduce the process ν ∈ V by ν t := nT n − X k =0 ϕ k ( ¯ M , . . . , ¯ M k ) { kT/n ≤ t< ( k +1) T/n } , t ∈ [0 , T ] . From (4.14) L ( ν kT/n : k = 0 , . . . , n − 1) = L (cid:16) nT E h ( M k +1 − M k ) | M , . . . , M k i : k = 0 , . . . , n − (cid:17) . Since ν is constant on each of the intervals [ kT /n, ( k + 1) T /n ], k = 0 , . . . , n − 1, we concludethat ˆ E h n n − X k =0 g (cid:16) nσ T ˆ E (cid:2) | M k +1 − M k | | M , . . . , M k (cid:3) (cid:17)i = E P h T Z T g (cid:16) ν t σ (cid:17) dt i . (4.15)Finally, consider the process X ( ν ) . Observe that X ( ν ) kT/n = X + k − X i =0 ¯ Y i +1 ϕ i ( ¯ M , . . . , ¯ M i ) , k = 0 , , ..., n. This together with (4.14) yields that ( X ( ν ) kT/n : k = 0 , . . . , n ) and ( ˆ X k : k = 0 , . . . , n ) have thesame distribution. Therefore, from the fact that f ≥ c , which does not depend on n , such thatˆ E h f n ( M ) i ≤ E P h f n ( X ( ν ) ) i + c ( h ( n ) n ) / + ˆ E h f n ( M ) { max k =0 ,...,n | M k − ˆ X k | > ( h ( n ) n ) / } i ≤ E P h f n ( X ( ν ) ) i + C ( h ( n ) n ) / (4.16)for some constant C which does not depend on n . The last inequality follows from the Cauchy–Schwarz inequality, the linear growth of f , the scaling assumption (2.3) and (4.12)–(4.13).By combining (4.15)–(4.16) we complete the proof of (4.11). (cid:3) We are now ready to prove the upper bound. Proof of the inequality ‘ ≤ ’ in (2.7) . Fix ℓ > 0. By passing to a subsequence (which is stilldenoted by n ) we assume without loss of generality that lim n →∞ π ( n, nℓ ) exists and satisfieslim n →∞ π ( n, nℓ ) > −∞ (otherwise the statement is obvious). Moreover, we assume that n issufficiently large such that ˆ Q ∈ Q n , where, recall that ˆ Q is the unique martingale measure forthe continuous time Bachelier model.Let ( γ, δ ) ∈ A n such that E P h exp (cid:16) − nℓV γ,δT (cid:17)i < ∞ . Using the same arguments as in theproof of Proposition 3.1 we get E ˆ Q h V γ,δT i ≤ 0. Thus, From the Cauchy–Schwarz inequality andthe Jensen inequality for the convex function y → e − nℓy/ we obtain1 ≤ E ˆ Q h exp (cid:16) − nℓV γ,δT / (cid:17)i ≤ (cid:16) E P h exp (cid:16) − nℓV γ,δT (cid:17)i(cid:17) / (cid:18) E P (cid:20)(cid:16) d ˆ Q /d P (cid:17) (cid:21)(cid:19) / , SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 13 and so E P h exp (cid:16) − nℓV γ,δT (cid:17)i is uniformly bounded from below. Thus, from (2.6), (3.1), and (3.3)we get that there exists a sequence of probability measures Q n ∈ Q n , n ∈ N such thatlim n →∞ π ( n, nℓ ) ≤ lim n →∞ E Q n (cid:20) f n ( X ) − nℓ Z T | ψ Q n t | dt (cid:21) . (4.17)Next, fix n sufficiently large and introduce the Q n -martingale M k := X kT/n , k = 0 , , ..., n where,recall that X t = X + σW Q n t + σ Z t ψ Q n s ds + µt, t ∈ [0 , T ] , where W Q n is a Wiener process under Q n . Observe that for any k = 0 , , ..., n − 1, we have E Q n "Z ( k +1) T/nkT/n (cid:16) ψ Q n t + µ/σ (cid:17) dt (cid:12)(cid:12)(cid:12) F kT/n = 0 . This together with Lemma 4.1 and the scaling property of Brownian motion gives E Q n (cid:20)Z T | ψ Q n t | dt (cid:21) = E Q n " n − X k =0 E Q n "Z ( k +1) T/nkT/n | ψ Q n t | dt (cid:12)(cid:12)(cid:12) F kT/n = E Q n " n − X k =0 E Q n "Z ( k +1) T/nkT/n (cid:16) ψ Q n t + µ/σ (cid:17) dt (cid:12)(cid:12)(cid:12) F kT/n − µ T /σ ≥ E Q n " n − X k =0 g (cid:16) nσ T E Q n h | M k +1 − M k | (cid:12)(cid:12)(cid:12) F kT/n i(cid:17) − µ T /σ ≥ E Q n " n − X k =0 g (cid:16) nσ T E Q n h | M k +1 − M k | (cid:12)(cid:12)(cid:12) M , ..., M k i(cid:17) − µ T /σ where the last inequality follows from the Jensen inequality for the convex function g ( · ),Finally, from the assumption that lim n →∞ π ( n, nℓ ) > −∞ and (4.17) it follows that we canapply Lemma 4.2 (i.e. (4.10) holds true for some constant K ). We conclude E Q n (cid:20) f n ( X ) − nℓ Z T | ψ Q n t | dt (cid:21) ≤ C ( h ( n ) n ) / + µ T / (2 ℓσ n ) + sup ν ∈V E P h f n (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt i . The proof is completed by combining (2.3), (4.17) and taking n → ∞ . (cid:3) Lower bound. This section is devoted to the proof of the inequality ‘ ≥ ’ in (2.7). Proof of the inequality ‘ ≥ ’ in (2.7) . Fix ℓ > 0. The proof is done in three steps. In the firststep we construct a sequence of controls on the Brownian probability space which asymptoticallyachieves the supremum on the right-hand side of (2.7) and have a simple structure. In the secondstep we apply the processes θ β , ϑ β from Lemma 4.1 in order to construct a sequence of probabilitymeasures Q n ∈ Q n together with their Girsanov’s kernels ψ Q n . The difficulty in this step stemsfrom the fact that the processes θ β and ϑ β are constructed via the process ¯ W , which in our casetranslates to W Q n . Note that the measure Q n is detemined by the Girsanov’s kernel ψ Q n . Toovercome this technical difficulty, we use integration by parts and introduce a path-dependentSDE. As a by product, our process ψ Q n is measurable with respect to the original filtration F W .Finally, we show convergence of the payoff components. Step 1: For any K > n ∈ N , let V nK ⊂ V be the set of all volatility processes of theform ν t = n − X k =0 φ k (cid:0) W , W T/n , ..., W kT/n (cid:1) t ∈ [ kT/n, ( k +1) T/n ) (4.18)where φ k : R k +1 → [1 /K, K ], k = 0 , , ...., n − 1, are continuous functions.Set ǫ > 0. In this step we argue that there exists K = K ( ǫ ) and N = N ( ǫ ) such that forany n > N sup ν ∈V E P (cid:20) f (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt (cid:21) < ǫ + sup ν ∈V nK E P (cid:20) f n (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt (cid:21) . (4.19)To this end, observe first that, by standard density arguments, we will get the same supremumon the right-hand side of (2.7) if instead of letting ν vary over all of V there, we confine it to beof the form ν t := J − X j =0 φ j (cid:0) W t , . . . , W t j (cid:1) t ∈ [ t j ,t j +1 ) , t ∈ [0 , T ] , where 0 = t < t < · · · < t J = T is a finite deterministic partition of [0 , T ] and each φ j : R j +1 → R + , j = 0 , ..., J − 1, is continuous, bounded and bounded away from zero.Let ν be of the above form. There exists K such that ν ∈ [1 /K, K ] a.s. For any n ∈ N set t nj := min { t ∈ { , T /n, T /n, ..., T } : t ≥ t j } , j = 0 , , ..., J and define ν n ∈ V nK by ν nt := J − X j =0 φ j (cid:16) W t n , . . . , W t nj (cid:17) t ∈ [ t nj ,t nj +1 ) , t ∈ [0 , T ] . Observe that ν n → ν in L ( dt ⊗ P ). This together with the Lipschitz continuity of f and theLipschitz continuity of g on the interval [1 / ( Kσ ) , K/σ ] gives that E P (cid:20) f (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt (cid:21) = lim n →∞ E P (cid:20) f (cid:16) X ( ν n ) (cid:17) − ℓT Z T g (cid:18) ν nt σ (cid:19) dt (cid:21) . Thus, in order to establish (4.19) it remains to show thatlim n →∞ E P h f n (cid:16) X ( ν n ) (cid:17) − f (cid:16) X ( ν n ) (cid:17)i = 0 . Indeed, from the Lipschitz continuity of f , the Burkholder–Davis–Gundy inequality and the factthat ν n ≤ K for all n , it follows that there exists constants ˜ C , ˜ C (independent of n ) such that E P (cid:20)(cid:16) f n (cid:16) X ( ν n ) (cid:17) − f (cid:16) X ( ν n ) (cid:17)(cid:17) (cid:21) ≤ ˜ C E P " max ≤ k ≤ n − sup kT/n ≤ t ≤ ( k +1) T/n (cid:16) X ( ν n ) t − X ( ν n ) kT/n (cid:17) SCALING LIMIT FOR UTILITY INDIFFERENCE PRICES IN THE DISCRETIZED BACHELIER MODEL 15 ≤ ˜ C n − X k =0 E P " sup kT/n ≤ t ≤ ( k +1) T/n (cid:16) X ( ν n ) t − X ( ν n ) kT/n (cid:17) ≤ ˜ C n ˜ C n − = ˜ C ˜ C /n. This completes the proof of (4.19). Step 2: Fix K, A > n ∈ N . Following (4.4)–(4.5) we define the functions β : [1 /K, K ] → R + and θ : [1 /K, K ] × [0 , T /n ] → R by β ( u ) := u/σ | u/σ − | { u = σ } ,θ ( u, t ) := (cid:16) β ( u ) T /n − (cid:16) T /n − t (cid:17)(cid:17) − { u>σ } − (cid:16) β ( u ) T /n + (cid:16) T /n − t (cid:17)(cid:17) − { u<σ } . (4.20)Introduce the map Φ A : [1 /K, K ] × C [0 , T /n ] → C [0 , T /n ], such that for any u ∈ [1 /K, K ] and z ∈ C [0 , T /n ]Φ At ( u, z ) := ( − A ) ∨ (cid:18) θ ( u, t ) z t − θ ( u, z − Z t z s ∂θ ( u, s ) ∂s ds (cid:19) ∧ A, t ∈ [0 , T /n ] . (4.21)Observe that Φ At ( u, z ) depends only on z [0 ,t ] . For a given u ∈ [1 /K, K ] and y ∈ R consider thepath-dependent SDE d ( U t , Y t ) = (0 , dW t ) − (0 , Φ At ( U t , Y ) − µ/σ ) dt, t ∈ [0 , T /n ] , U = u, Y = y. Let us notice that Φ A ( σ , · ) ≡ u = σ there exists ǫ > A : [1 /K ∨ ( u − ǫ ) , ( u + ǫ ) ∧ K ] × C [0 , T /n ] → C [0 , T /n ]is a locally Lipschitz continuous function with respect to the sup-norm. Thus, Theorem 5.2.5from [22] yields a unique strong solution for the above SDE. Moreover, from Theorem 1 in [20]we obtain the existence of a measurable function Ψ A : [1 /K, K ] × R × C [0 , T /n ] → C [0 , T /n ] suchthat for any u > y ∈ R , Y [0 ,T/n ] := Ψ A ( u, y, W [0 ,T/n ] ) is the unique strong solution to theSDE dY t = dW t − (Φ At ( u, Y ) − µ/σ ) dt, t ∈ [0 , T /n ] , Y = y. Next, let ν ∈ V nK be given by (4.18). Define inductively the random variables u A,n,νk = u k and Y A,n,ν [ kT/n, ( k +1) T/n ] = Y A [ kT/n, ( k +1) T/n ] , k = 0 , , ..., n − 1, as follows. Set u := ν , Y A [0 ,T/n ] :=Ψ A ( u , , W [0 ,T/n ] ), and for k = 1 , ...n − u k := φ k (cid:16) Y A , ..., Y AkT/n (cid:17) ,Y A [ kT/n, ( k +1) T/n ] := S k (cid:16) Ψ A (cid:16) u k , Y AkT/n , { W t + kT/n − W kT/n } t ∈ [0 ,T/n ] (cid:17)(cid:17) , where S k : C [0 , T /n ] → C [ kT /n, ( k + 1) T /n ] is the shift operator (bijection) given by ( S k ( z )) t := z t − kT/n .Observe that ( Y At ) t ∈ [0 ,T ] satisfies the equation Y At = W t + µt/σ − n − X k =0 Z t ∧ (( k +1) T/n ) t ∧ ( kT/n ) Φ At (cid:16) u k , S − k (cid:16) Y A [ kT/n, ( k +1) T/n ] (cid:17)(cid:17) dt. (4.22)Since Φ A is a bounded function, then from the Girsanov’s theorem we obtain that there existsa probability measure Q A,n,ν = Q A ∼ P with with finite entropy E Q A [log( d Q A /d P )] < ∞ suchthat W Q A := Y A is a Wiener process under Q A . From (4.21)–(4.22) and the integration by parts formula it follows that ψ Q A t = ( − A ) ∨ Z tkT/n θ (cid:16) φ k ( W Q A , ..., W Q A kT/n ) , s − kT /n (cid:17) dW Q A s ! ∧ A − µ/σ for t ∈ [ kT /n, ( k + 1) T /n ) , k = 0 , , .., n − . (4.23)We end this step with arguing that there exists N = N ( K ) such that for any n > N ( K ) wehave Q A ∈ Q n . First, we establish the martingale property. Indeed, from (4.23) it follows thatfor any k = 0 , , ..., n − 1, and t ∈ [ kT /n, ( k + 1) T /n ), the conditional distribution (under Q A )of ψ Q A t , given F WkT/n , is symmetric around − µ/σ , and so ( X kT/n ) ≤ k ≤ n is Q A -martingale.Finally, we establish the super-martingale property. Clearly, there exists a constant ¯ c > K such that | θ ( u, t ) | ≤ ¯ c for all u ∈ [1 /K, K ] and t ∈ [0 , T /n ]. This togetherwith (2.2) and (4.23) gives that there exists N = N ( K ) such that for any n > N ( K ) E Q A h | X ( k +1) T/n − X kT/n | (cid:12)(cid:12)(cid:12) F WkT/n i ≤ h ( n ) , k = 0 , , ..., n − , as required. Step 3: In this step we fix arbitrary n > N ( K ) and ν ∈ V Kn . Then in view of (4.19), in orderto complete the proof it remains to show that π ( n, nℓ ) ≥ E P (cid:20) f n (cid:16) X ( ν ) (cid:17) − ℓT Z T g (cid:16) ν t σ (cid:17) dt (cid:21) − µ T nℓσ . (4.24)From (2.6) (take ( γ, δ ) ≡ π ( n, nℓ ) ≥ lim inf A →∞ E Q A (cid:20) f n ( X ) − nℓ Z T (cid:12)(cid:12) ψ Q A t (cid:12)(cid:12) dt (cid:21) . (4.25)From (4.20), (4.23), and Lemma 4.1. it follows that for any A > k = 0 , , ..., n − E Q A "Z ( k +1) T/nkT/n (cid:12)(cid:12) ψ Q A t (cid:12)(cid:12) dt (cid:12)(cid:12)(cid:12) F WkT/n ≤ µ Tσ n + g φ k (cid:16) W Q A , ..., W Q A kT/n (cid:17) σ . 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