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Weak Approximation of β Yan Dolinsky Marcel Nutz πΊ-Expectations β β‘ H. Mete Soner March 2, 2011 Abstract We introduce a notion of volatility uncertainty in discrete time and dene the corresponding analogue of Peng's πΊ-expectation. In the continuous-time limit, the resulting sublinear expectation converges weakly to the πΊ-expectation. This can be seen as a Donsker-type result for the πΊ-Brownian motion. Keywords πΊ-expectation, volatility uncertainty, weak limit theorem AMS 2000 Subject Classications 60F05, 60G44, 91B25, 91B30 JEL Classications G13, G32 Acknowledgements Research supported by European Research Council Grant 228053-FiRM, Swiss National Science Foundation Grant PDFM2-120424/1, Swiss Finance Institute and ETH Foundation. 1 Introduction The so-called πΊ-expectation [12, 13, 14] is a nonlinear expectation advanc- ing the notions of backward stochastic dierential equations (BSDEs) [10] π -expectations [11]; see also [2, 16] for a related theory of second order πΊ BSDEs. A πΊ-expectation π 7β β° (π) is a sublinear function which maps random variables π on the canonical space Ξ© = πΆ([0, π ]; β) to the real numbers. The symbol πΊ refers to a given function πΊ : β β β of the form and 1 1 πΊ(πΎ) = (π πΎ + β ππΎ β ) = sup ππΎ, 2 2 πβ[π,π ] 0 β€ π β€ π < β are xed numbers. More generally, the interval [π, π ] D of nonnegative matrices in the multivariate case. The extension to a random set D is studied in [9]. πΊ The construction of β° (π) runs as follows. When π = π (π΅π ), where π΅π is the canonical process at time π and π is a suciently regular function, πΊ then β° (π) is dened to be the initial value π’(0, 0) of the solution of the nonlinear backward heat equation ββπ‘ π’ β πΊ(π’π₯π₯ ) = 0 with terminal condition where is replaced by a set β β β‘ ETH Zurich, Dept. of Mathematics, CH-8092 Zurich, ETH Zurich, Dept. of Mathematics, CH-8092 Zurich, [email protected] [email protected] ETH Zurich, Dept. of Mathematics, CH-8092 Zurich, and Swiss Finance Institute, [email protected] 1 π’(β , π ) = π . The mapping β° πΊ can be extended to random variables of the form π = π (π΅π‘1 , . . . , π΅π‘π ) by a stepwise evaluation of the PDE and then to 1 1 the completion ππΊ of the space of all such random variables. The space ππΊ consists of so-called quasi-continuous functions and contains in particular all bounded continuous functions on functions are included (cf. [3]). Ξ©; however, not all bounded measurable While this setting is not based on a sin- πΊ-Brownian gle probability measure, the so-called motion is given by the πΊ (cf. [14]). It reduces to the standard canonical process π΅ seen under β° πΊ is then the (linear) expectation Brownian motion if π = π = 1 since β° under the Wiener measure. In this note we introduce a discrete-time analogue of the πΊ-expectation and we prove a convergence result which resembles Donsker's theorem for the standard Brownian motion; the main purpose is to provide additional intuition for πΊ-Brownian point is the dual view on motion and volatility uncertainty. πΊ-expectation Our starting via volatility uncertainty [3, 4]: We consider the representation β° πΊ (π) = sup πΈ π [π], (1.1) π βπ« where π« is a set of probabilities on Ξ© such that under any π β π«, the π΅ is a martingale with volatility πβ¨π΅β©/ππ‘ taking values in D = [π, π ], π × ππ‘-a.e. Therefore, D can be understood as the domain of πΊ as the corresponding worst-case (Knightian) volatility uncertainty and β° canonical process expectation. In discrete-time, we translate this to uncertainty about the conditional variance of the increments. Thus we dene a sublinear expectation β°π on the π-step canonical space in the spirit of (1.1), replacing π« by a suit- able set of martingale laws. A natural push-forward then yields a sublinear expectation on the domain D Ξ©, which we show to converge weakly to of uncertainty is scaled by 1/π β°πΊ as π β β, if (cf. Theorem 2.2). The proof relies on (linear) probability theory; in particular, it does not use the central limit theorem for sublinear expectations [14, 15]. The relation to the latter is nontrivial since our discrete-time models do not have independent increments. We remark that quite dierent approximations of the πΊ-expectation (for the scalar case) can be found in discrete models for nancial markets with transaction costs [8] or illiquidity [5]. The detailed setup and the main result are stated in Section 2, whereas the proofs and some ramications are given in Section 3. 2 Main Result π β β and denote by β£ β β£ the Euclidean norm on βπ . π π we denote by π the space of π × π symmetric matrices and by π+ We x the dimension Moreover, its subset of nonnegative denite matrices. We x a nonempty, convex and 2 compact set D β ππ+ ; the elements of D will be the possible values of our volatility processes. Continuous-Time Formulation. Let Ξ© = πΆ([0, π ]; βπ ) be the space of π-dimensional continuous paths π = (ππ‘ )0β€π‘β€π with time horizon π β (0, β), β₯πβ₯β = sup0β€π‘β€π β£ππ‘ β£. We denote by π΅ = (π΅π‘ )0β€π‘β€π the canonical process π΅π‘ (π) = ππ‘ and by β±π‘ := π(π΅π , 0 β€ π β€ π‘) the canonical ltration. A probability measure π on Ξ© is called a martingale law if π΅ is a π -martingale and π΅0 = 0 π -a.s. (All our martingales will start endowed with the uniform norm at the origin.) We set { π«D = π where β¨π΅β© martingale law on } Ξ© : πβ¨π΅β©π‘ /ππ‘ β D, π × ππ‘-a.e. , denotes the matrix-valued process of quadratic covariations. We can then dene the sublinear expectation β°D (π) := sup πΈ π [π] for any random variable π βπ«D π:Ξ©ββ β±π -measurable and πΈ π β£πβ£ < β for all π β π«D . The mapping β°D coincides with the πΊ-expectation (on its domain π1πΊ ) if πΊ : ππ β β is (half ) the support function of D; i.e., πΊ(Ξ) = supπ΄βD trace(Ξπ΄)/2. In- such that π is deed, this follows from [3] with an additional density argument as detailed in Remark 3.6 below. (βπ )π+1 as the canonical space of π-dimensional paths in discrete time π = 0, 1, . . . , π. We π π π π denote by π = (ππ )π=0 the canonical process dened by ππ (π₯) = π₯π for π₯ = (π₯0 , . . . , π₯π ) β (βπ )π+1 . Moreover, β±ππ = π(πππ , π = 0, . . . , π) denes π π the canonical ltration (β±π )π=0 . We also introduce 0 β€ πD β€ π D < β such that [πD , π D ] is the spectrum of D; i.e., Discrete-Time Formulation. Given πD = inf β₯Ξβ1 β₯β1 ΞβD β₯β β₯ π β β, and we consider π D = sup β₯Ξβ₯, ΞβD πD := 0 if D has an element which is not invertible. We note that [πD , π D ] = D if π = 1. π π+1 is called a martingale law if π π Finally, a probability measure π on (β ) π π π π is a π -martingale and π0 = 0 π -a.s. Denoting by Ξππ = ππ β ππβ1 the π increments of π , we can now set { } π martingale law on (βπ )π+1 : for π = 1, . . . , π, π π«D = π ] β D and π2 π β€ β£Ξπ π β£2 β€ π2 π , π -a.s. , πΈ π [Ξπππ (Ξπππ )β² β£β±πβ1 D D π where denotes the operator norm and we set β² Ξπππ is a column vector, so π π+ . We introduce the sublinear expectation where prime ( ) denotes transposition. Note that π π β² that Ξπ (Ξπ ) takes values in π β°D (π) := sup πΈ π [π] π π βπ«D for any random variable 3 π : (βπ )π+1 β β π is β±ππ -measurable and πΈ π β£πβ£ < β for all π β π of β°D as a discrete-time analogue of the πΊ-expectation. such that Remark 2.1. The second condition in the denition of the desire to generate the volatility uncertainty by a π, π«D π π«D and we think is motivated by small set of scenarios; we remark that the main results remain true if, e.g., the lower bound π D omitted and the upper bound πD is replaced by any other condition yielding tightness. Our bounds are chosen so that { π π«D = π martingale law on Continuous-Time Limit. (βπ )π+1 : Ξπ π (Ξπ π )β² β D, π -a.s. if π = 1. To compare our objects from the two formu- lations, we shall extend any discrete path π₯ Λβ Ξ© } by linear interpolation. π₯ β (βπ )π+1 to a continuous path More precisely, we dene the interpolation operator Λ : (βπ )π+1 β Ξ©, π₯ = (π₯0 , . . . , π₯π ) 7β π₯ Λ = (Λ π₯π‘ )0β€π‘β€π , where π₯ Λπ‘ := ([ππ‘/π ] + 1 β ππ‘/π )π₯[ππ‘/π ] + (ππ‘/π β [ππ‘/π ])π₯[ππ‘/π ]+1 [π¦] := max{π β β€ : π β€ π¦} for π¦ β β. In particular, if π π is the π π+1 and π is a random variable on Ξ©, then π(π Λπ ) canonical process on (β ) π π+1 denes a random variable on (β ) . This allows us to dene the following π push-forward of β°D to a continuous-time object, and π π Λπ )) β°ΜD (π) := β°D (π(π for π:Ξ©ββ being suitably integrable. Our main result states that this sublinear expectation with discrete-time volatility uncertainty converges to the πΊ-expectation as the number π of periods tends to innity, if the domain of volatility uncertainty is scaled as D/π := {πβ1 Ξ : Ξ β D}. Theorem 2.2. Let π : Ξ© β β be a continuous function satisfying β£π(π)β£ β€ π (π) β β° (π) as π β β; π(1+β₯πβ₯β )π for some constants π, π > 0. Then β°ΜD/π D that is, Λπ )] β sup πΈ π [π]. sup πΈ π [π(π π π βπ«D/π (2.1) π βπ«D We shall see that all expressions in (2.1) are well dened and nite. Moreover, we will show in Theorem 3.8 that the result also holds true for a strong formulation of volatility uncertainty. Remark 2.3. Theorem 2.2 cannot be extended to the case where π is 1 merely in ππΊ , which is dened as the completion of πΆπ (Ξ©; β) under the π norm β₯πβ₯πΏ1 := sup{πΈ β£πβ£, π β π«D }. This is because β₯ β β₯πΏ1 does not see πΊ πΊ the discrete-time objects, as illustrated by the following example. Assume 4 0 β / D and let π΄ β Ξ© π (π΄) = 0 for any π β π«D , for simplicity that be the set of paths with nite variation. Since we have π := 1 β 1π΄ = 1 in and the right hand side of (2.1) equals one. However, the trajectories of lie in π΄, so that Λπ ) β‘ 0 π(π and the left hand side of (2.1) equals zero. In view of the previous remark, we introduce a smaller space as the completion of β₯πβ₯β := sup πΈ π β£πβ£, πβπ¬ If π π1πΊ Λπ π πΆπ (Ξ©; β) π1β , dened under the norm } { Λπ )β1 : π β π« π , π β β . π¬ := π«D βͺ π β (π D/π is as in Theorem 2.2, then π β π1β (2.2) by Lemma 3.4 below and so the following is a generalization of Theorem 2.2. Corollary 2.4. Let π (π) β β° (π) as π β β. π β π1β . Then β°ΜD/π D Proof. This follows from Theorem 2.2 by approximation, using that sup{πΈ π β£πβ£ : π β π«D } + Λπ )β£ sup{πΈ π β£π(π :π β π , π«D/π π β β} β₯πβ₯β and are equivalent norms. 3 Proofs and Ramications In the next two subsections, we prove separately two inequalities that jointly imply Theorem 2.2 and a slightly stronger result, reported in Theorem 3.8. 3.1 First Inequality In this subsection we prove the rst inequality of (2.1), namely that Λπ )] β€ sup πΈ π [π]. lim sup sup πΈ π [π(π π πββ π βπ«D/π (3.1) π βπ«D The essential step in this proof is a stability result for the volatility (see Lemma 3.3(ii) below); the necessary tightness follows from the compactness of D; i.e., from π D < β. We shall denote πD = {πΞ : Ξ β D} for π β β. Lemma 3.1. Given π β [1, β), there exists a universal constant πΎ > 0 such π, that for all 0 β€ π β€ π β€ π and π β π«D (i) (ii) (iii) πΈ π [supπ=0,...,π β£πππ β£2π ] β€ πΎ(ππ D )π , 2 (π β π)2 , πΈ π β£πππ β πππ β£4 β€ πΎπ D πΈ π [(πππ β πππ )(πππ β πππ )β² β£β±ππ ] β (π β π)D π -a.s. π := π π to ease the notation. (i) Let π β [1, β). By the Burkholder-Davis-Gundy (BDG) there exists a universal constant πΆ = πΆ(π, π) such that [ ] π π 2π πΈ sup β£ππ β£ β€ πΆπΈ π β₯[π]π β₯π . Proof. We set π=0,...,π 5 inequalities In view of π, π β π«D we have β₯[π]π β₯ = β₯ βπ β² π=1 Ξππ (Ξππ ) β₯ (ii) The BDG inequalities yield a universal constant πΆ β€ ππ2 π D π -a.s. such that πΈ π β£ππ β ππ β£4 β€ πΆπΈ π β₯[π]π β [π]π β₯2 . Similarly as in (i), π π β π«D implies that β₯[π]π β [π]π β₯ β€ (π β π)π2 π D π -a.s. (iii) The orthogonality of the martingale increments yields that π πΈ [(ππ β ππ )(ππ β β² ππ ) β£β±ππ ] = π β πΈ π [Ξππ (Ξππ )β² β£β±ππ ]. π=π+1 Since π ] β D π -a.s. πΈ π [Ξππ (Ξππ )β² β£β±πβ1 and since D is convex, ] [ π π πΈ π [Ξππ (Ξππ )β² β£β±ππ ] = πΈ π πΈ π [Ξππ (Ξππ )β² β£β±πβ1 ] β±π again takes values in D. Ξ1 + β β β + Ξπ β πD by convexity. It remains to observe that if Ξ1 , . . . , Ξπ β D, then The following lemma shows in particular that all expressions in Theorem 2.2 are well dened and nite. Lemma 3.2. Let π : Ξ© β β be as in Theorem 2.2. Then β₯πβ₯β < β; that is, Λπ )β£ < β sup sup πΈ π β£π(π π πββ π βπ«D/π sup πΈ π β£πβ£ < β. and (3.2) π βπ«D π . By the assumption on π , there π β β and π β π«D/π constants π, π > 0 such that [ ] ] [ π π Λπ )β£ β€ π + ππΈ π sup β£π Λπ β£π β€ π + ππΈ π πΈ π β£π(π β£ . sup β£π π π‘ Proof. Let 0β€π‘β€π exist π=0,...,π π D/π = π D /π yield that π/2 Λπ )β£ β€ πΎπ πΈ π β£π(π D and the rst claim follows. The second claim similarly π π follows from the estimate that πΈ [sup0β€π‘β€π β£π΅π‘ β£ ] β€ πΆπ for all π β π«D , Hence Lemma 3.1(i) and the observation that which is obtained from the BDG inequalities by using that D is bounded. We can now prove the key result of this subsection. Λπ Lemma 3.3. For each π β β, let {π π = (πππ )π π=0 , π } be a martingale π Λπ on Ξ©. Then with law π π β π«D/π on (βπ )π+1 and let ππ be the law of π (i) the sequence (ππ ) is tight on Ξ©, (ii) any cluster point of (ππ ) is an element of π«D . 6 Proof. (i) Let 0 β€ π β€ π‘ β€ π. As π D/π = π D /π, Lemma 3.1(ii) implies that π Λπ Λ 2 π Λπ 4 πΈ π β£π΅π‘ β π΅π β£4 = πΈ π β£π π‘ β ππ β£ β€ πΆβ£π‘ β π β£ πΆ > 0. for a constant (ii) Let π Hence (ππ ) is tight by the moment criterion. be a cluster point, then π΅ is a π-martingale as a consequence of the uniform integrability implied by Lemma 3.1(i) and it remains to show that πβ¨π΅β©π‘ /ππ‘ β D holds scalar inequalities: given π × ππ‘-a.e. It will be Ξ β ππ , the separating useful to characterize D by hyperplane theorem implies that ΞβD if and only if β β(Ξ) β€ πΆD := sup β(π΄) for all β β (ππ )β , (3.3) π΄βD (ππ )β is the set of all linear functionals β : ππ β β. Let π» : [0, π ] × Ξ© β [0, 1] be a continuous and adapted function and let β β (ππ )β . We x 0 β€ π < π‘ β€ π and denote Ξπ ,π‘ π := ππ‘ β ππ for a process π = (ππ’ )0β€π’β€π . Let π > 0 and let DΜ be any neighborhood of D, then for π where suciently large, Λπ πΈπ ( [ )] π, 0 β€ π’ β€ π β π Λπ )(Ξπ ,π‘ π Λπ )β² π π Λ β (π‘ β π )DΜ πΛ π -a.s. (Ξπ ,π‘ π π’ as a consequence of Lemma 3.1(iii). Since DΜ was arbitrary, it follows by (3.3) that { ( ) }] π[ β lim sup πΈ π π»(π β π, π΅) β (Ξπ ,π‘ π΅)(Ξπ ,π‘ π΅)β² β πΆD (π‘ β π ) πββ Λπ [ = lim sup πΈ π πββ { ( ) }] Λπ ) β (Ξπ ,π‘ π Λπ )(Ξπ ,π‘ π Λπ )β² β πΆ β (π‘ β π ) β€ 0. π»(π β π, π D π(π) = β₯πβ₯2β , we may pass to the limit and conclude that [ ( )] [ ] β πΈ π π»(π β π, π΅) β (Ξπ ,π‘ π΅)(Ξπ ,π‘ π΅)β² β€ πΈ π π»(π β π, π΅) πΆD (π‘ β π ) . (3.4) Using (3.2) with Since π»(π β π, π΅) is β±π -measurable and πΈ π [(Ξπ ,π‘ π΅)(Ξπ ,π‘ π΅)β² β£β±π ] = πΈ π [π΅π‘ π΅π‘β² β π΅π π΅π β² β£β±π ] = πΈ π [β¨π΅β©π‘ β β¨π΅β©π β£β±π ] as π΅ π-martingale, (3.4) is equivalent to ( )] [ ] β π»(π β π, π΅) β β¨π΅β©π‘ β β¨π΅β©π β€ πΈ π π»(π β π, π΅) πΆD (π‘ β π ) . is a square-integrable πΈ [ π π» and dominated convergence as π β 0, we [ ( )] [ ] β πΈ π π»(π , π΅) β β¨π΅β©π‘ β β¨π΅β©π β€ πΈ π π»(π , π΅) πΆD (π‘ β π ) Using the continuity of and then it follows that πΈπ [β« π ] [β« π»(π‘, π΅) β(πβ¨π΅β©π‘ ) β€ πΈ π 0 0 7 π ] β π»(π‘, π΅)πΆD ππ‘ . obtain π» which β holds β(πβ¨π΅β©π‘ /ππ‘) β€ πΆD π × ππ‘-a.e., and since β β (ππ )β was arbitrary, (3.3) shows that πβ¨π΅β©π‘ /ππ‘ β D holds π × ππ‘-a.e. By an approximation argument, this inequality extends to functions are measurable instead of continuous. It follows that We can now deduce the rst inequality of Theorem 2.2 as follows. Proof of (3.1). Let π be as in Theorem 2.2 and let π there exists an π-optimizer π Ξ© under ππ , β π ; i.e., if π«D/π π π By Lemma 3.3, the sequence π«D . For each (ππ ) Λπ )] β π. sup πΈ π [π(π π π βπ«D/π is tight and any cluster point belongs π is continuous and (3.2) implies supπ πΈ ππ β£πβ£ < β, π lim supπ πΈ π [π] β€ supπ βπ«D πΈ π [π]. Therefore, Since yields that πββ Λπ on π then Λπ )] β₯ πΈ π [π] = πΈ π [π(π to π > 0. ππ denotes the law of tightness Λπ )] β€ sup πΈ π [π] + π. lim sup sup πΈ π [π(π π πββ π βπ«D/π Since π>0 π βπ«D was arbitrary, it follows that (3.1) holds. Finally, we also prove the statement preceding Corollary 2.4. Lemma 3.4. Let π : Ξ© β β be as in Theorem 2.2. Then π β π1β . Proof. We show that π π π in the norm β₯ β β₯β as sup{πΈ π [ β ] : π β π¬} π is continuous along the decreasing sequence β£π β π β£, where π¬ is as in (2.2). Indeed, π¬ is tight by (the proof of ) Lemma 3.3. Using that β₯πβ₯β < β by π β β, := (π β§ π) β¨ π converges to or equivalently, that the upper expectation Lemma 3.2, we can then argue as in the proof of [3, Theorem 12] to obtain the claim. 3.2 Second Inequality The main purpose of this subsection is to show the second inequality β₯ of (2.1). Our proof will yield a more precise version of Theorem 2.2. Namely, we will include strong formulations of volatility uncertainty both in discrete and in continuous time; i.e., consider laws generated by integrals with respect to a xed random walk (resp. Brownian motion). In the nancial interpretation, this means that the uncertainty can be generated by models. 8 complete market Strong Formulation in Continuous Time. nian martingales: with { π¬D = π0 β (β« π0 Here we shall consider Brow- denoting the Wiener measure, we dene π (π‘, π΅) ππ΅π‘ )β1 β ) : π β πΆ [0, π ] × Ξ©; D ( } , adapted β β β D = { Ξ : Ξ β D}. (For Ξ β ππ+ , Ξ denotes the unique squareπ root in π+ .) We note that π¬D is a (typically strict) subset of π«D . The elements of π¬D with nondegenerate π have the predictable representation where property; i.e., they correspond to a complete market in the terminology of mathematical nance. We have the following density result; the proof is deferred to the end of the section. Proposition 3.5. The convex hull of π¬π· is a weakly dense subset of π«D . We can now deduce the connection between associated with β°D and the πΊ-expectation D. Remark 3.6. (i) Proposition 3.5 implies that sup πΈ π [π] = sup πΈ π [π], π βπ¬D π βπ«D π β πΆπ (Ξ©; β). πΊ-expectation π β 7 supπ βπ¬βD πΈ π [π] (3.5) In [3, Section 3] it is shown that the as introduced in [12, 13] coincides with the mapping for a certain set satisfying π¬D β π¬βD β π«D . π¬βD In particular, we deduce that the right hand side of (3.5) is indeed equal to the πΊ-expectation, as claimed in Section 2. (ii) A result similar to Proposition 3.5 can also be deduced from [17, Proposition 3.4.], which relies on a PDE-based verication argument of stochastic control. We include a (possibly more enlightening) probabilistic proof at the end of the section. Strong Formulation in Discrete Time. For xed π β β, we consider { } Ξ©π := π = (π1 , . . . , ππ ) : ππ β {1, . . . , π + 1}, π = 1, . . . , π equipped with its power set and let ππ := {(π + 1)β1 , . . . , (π + 1)β1 }π be the product probability associated with the uniform distribution. Moreover, π1 , . . . , ππ be an i.i.d. sequence of βπ -valued random variables on Ξ©π such 2 that β£ππ β£ = π and such that the components of ππ are orthonormal in πΏ (ππ ), βπ for each π = 1, . . . , π. Let ππ = π=1 ππ be the associated random walk. π Then, we consider martingales π which are discrete-time integrals of π of let the form πππ = π β π (π β 1, π)Ξππ , π=1 9 where by π; π¬πD π is measurable and adapted with respect to the ltration generated i.e., π (π, π) depends only on πβ£{0,...,π} . We dene { β = ππ β(π π )β1 ; π : {0, . . . , πβ1}×(βπ )π+1 β D To see that π, π¬πD β π«D we note that measurable, Ξπ π π = π (π β 1, π)ππ } adapted . and the or- ππ yield [ ( )β² ] πΈ ππ Ξπ π π Ξπ π π π(π1 , . . . , ππβ1 ) = π (π β 1, π)2 β D ππ -a.s., thonormality property of while β£ππ β£ = π and π 2 β D imply that ( ) [ ] Ξπ π π Ξπ π π β² = β£π (π β 1, π)ππ β£2 β π2 πD , π2 π D ππ -a.s. Remark 3.7. We recall from [7] that such π1 , . . . , ππ can be constructed as follows. Let π΄ be an orthogonal (π + 1) × (π + 1) matrix whose last row β1/2 , . . . , (π + 1)β1/2 ) and let π£ β βπ be column vectors such is ((π + 1) π that [π£1 , . . . , π£π+1 ] is the matrix obtained from π΄ by deleting the last row. 1/2 π£ Setting ππ (π) := (π + 1) ππ for π = (π1 , . . . , ππ ) and π = 1, . . . , π, the above requirements are satised. We can now formulate a result which includes Theorem 2.2. Theorem 3.8. Let lim π : Ξ© β β be as in Theorem 2.2. Then Λπ )] = lim sup πΈ π [π(π πββ π βπ¬π Λπ )] sup πΈ π [π(π πββ π βπ« π D/π D/π π = sup πΈ [π] π βπ¬D = sup πΈ π [π]. (3.6) π βπ«D Proof. Since π π¬πD/π β π«D/π for each π β₯ 1, the inequality (3.1) yields that Λπ )] β€ sup πΈ π [π]. lim sup sup πΈ π [π(π πββ π βπ¬π D/π π βπ«D As the equality in (3.6) follows from Proposition 3.5, it remains to show that lim inf Λπ )] β₯ sup πΈ π [π]. sup πΈ π [π(π πββ π βπ¬π D/π To this end, let π β π¬D ; i.e., π π βπ¬D is the law of a martingale of the form β« π= π (π‘, π ) πππ‘ , 10 where π function. We shall construct tend to For β ) ( π β πΆ [0, π ] × Ξ©; D is an adapted (π) whose laws are in π¬π martingales π D/π and is a Brownian motion and π. π β₯ 1, let (π) ππ = βπ π=1 ππ be the random walk on (Ξ©π , ππ ) as intro- duced before Remark 3.7. Let [ππ‘/π ] (π) ππ‘ := π β1/2 β ππ , 0β€π‘β€π π=1 be the piecewise constant càdlàg version of the scaled random walk and let (π) Λ (π) := πβ1/2 πΛ π be its continuous counterpart obtained by linear interpo- lation. It follows from the central limit theorem that ( ) Λ (π) β (π, π ) π (π) , π π·([0, π ]; β2π ), on the space of càdlàg paths equipped with the Skorohod topology. Moreover, since ( π is continuous, we also have that ( )) Λ (π) β (π, π (π‘, π )) π (π) , π [ππ‘/π ]π /π, π on 2 π·([0, π ]; βπ+π ). Thus, if we introduce the discrete-time integral (π) ππ := π ) β )( (π) ( Λ Λ (π) Λ (π) π β π π (π β 1)π /π, π ππ /π (πβ1)π /π , π=1 it follows from the stability of stochastic integrals (see [6, Theorem 4.3 and Denition 4.1]) that ( (π) π[ππ‘/π ] ) 0β€π‘β€π βπ Moreover, since the increments of π (π) on π·([0, π ]; βπ ). uniformly tend to 0 as π β β, it also follows that Λ(π) β π π As π 2 /π takes values in D/π, the law of on π (π) Ξ©. is contained in π¬πD/π and the proof is complete. It remains to give the proof of Proposition 3.5, which we will obtain by a randomization technique. Since similar arguments, at least for the scalar case, can be found elsewhere (e.g., [8, Section 5]), we shall be brief. Proof of Proposition 3.5. We may assume without loss of generality that there exists an invertible element (3.7) D is a convexβ© subset of ππ+ , we observe that (3.7) is πΎ = {0} for πΎ := ΞβD ker Ξ. If π = dim πΎ > 0, a change Indeed, using that equivalent to Ξβ β D. 11 of coordinates bring us to the situation where π coordinates of βπ . πΎ corresponds to the last We can then reduce all considerations to βπβπ and thereby recover the situation of (3.7). 1. Regularization. We rst observe that the set { π β π«D : πβ¨π΅β©π‘ /ππ‘ β₯ π1π π × ππ‘-a.e. is weakly dense in π«D . (Here a martingale whose law is in for some 1π denotes the unit matrix.) π«D . Recall (3.7) and let π } π>0 (3.8) Indeed, let π be be an independent continuous Gaussian martingale with πβ¨π β©π‘ /ππ‘ = Ξβ . ππ + (1 β π)π D is convex. and is contained in the set (3.8), since tends to the law of π For π β 1, the law of 2. Discretization. Next, we reduce to martingales with piecewise constant volatility. Let π be a martingale whose law belongs to (3.8). We have β« π= where π (π) ππ‘ πππ‘ ππ‘ := for β β« πβ¨π β©/ππ‘ and π := ππ‘β1 πππ‘ , π is a Brownian motion by Lévy's theorem. For π β₯ 1, we introduce β« (π) = ππ‘ πππ‘ , where π (π) is an ππ+ -valued piecewise constant process satisfying ( (π) )2 ππ‘ [( = Ξ D π π β« ππ /π )2 ] ππ ππ , ( ] π‘ β ππ /π, (π + 1)π /π (πβ1)π /π π β D is the Euclidean projection. π = 1, . . . , π β 1, where Ξ D : πβ (π) [0, π /π] one can take, e.g., π := Ξβ . We then have β« π β© βͺ ππ‘ β π (π) 2 ππ‘ β 0 πΈ π β π (π) π = πΈ π‘ for On 0 and in particular π (π) converges weakly to π. 3. Randomization. Consider a martingale of the form where π π π = β« ππ‘ πππ‘ , is a Brownian motion on some given ltered probability space and β is an adapted π= πβ1 β D-valued process which is piecewise constant; i.e., 1[π‘π ,π‘π+1 ) π(π) for some 0 = π‘0 < π‘1 < β β β < π‘π = π π=0 π β₯ 1. Consider also a second probability space carrying a BrowΛ and a sequence π 1 , . . . , π π of βπ×π -valued random variables π π such that the components {πππ : 1 β€ π, π β€ π; 1 β€ π β€ π} are i.i.d. uniformly Λ. distributed on (0, 1) and independent of π and some nian motion Using the existence of regular conditional probability distributions, we β 2 2 Ξπ : πΆ([0, π‘π ]; βπ ) × (0, 1)π × β β β × (0, 1)π β D Λ β£[0,π‘ ] , π 1 , . . . , π π ) satisfy that the random variables π Λ (π) := Ξπ (π π { } { } Λ ,π π Λ (0), . . . , π Λ (π β 1) = π, π(0), . . . , π(π β 1) in law. (3.9) can construct functions such 12 We can then consider the volatility corresponding to a xed realization of π 1, . . . , π π. 2 π’ = (π’1 , . . . , π’π ) β (0, 1)ππ , let ( ) Λ β£[0,π‘ ] , π’1 , . . . , π’π π Λ (π; π’) := Ξπ π π β« π’ β Λπ’ = π Λ π‘ , where π π Λπ‘ ππ Λ π’ := πβ1 Λ (π; π’). π=0 1[π‘π ,π‘π+1 ) π Indeed, for and consider ( πΉ β πΆπ Ξ©; β), For any the equality (3.9) and Fubini's theorem yield that [ ( (π 1 ,...,π π ) )] Λ πΈ[πΉ (π )] = πΈ πΉ π = β€ β« (0,1)ππ2 Λ π’ )] ππ’ πΈ[πΉ (π sup Λ π’ )]. πΈ[πΉ (π π’β(0,1)ππ2 π is contained in the weak Λ π’ : π’ β (0, 1)ππ2 }. We note { π β« Λ π’ is of the form π Λ π’ = π(π‘, π Λ ) ππ Λ π‘ with a measurable, adapted, that π β D-valued function π , for each xed π’. 4. Smoothing. As π¬π· is dened through continuous functions, it remains β to approximate π by a continuous function π . Let π : [0, π ] × Ξ© β D be a measurable adapted function and πΏ > 0. 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