LOCAL MARTINGALES IN DISCRETE TIME VILMOS PROKAJANDJOHANNESRUF 7 Abstract. Foranydiscrete-timeP–localmartingaleS thereexistsaprobabilitymeasureQ∼ 1 PsuchthatS isaQ–martingale. Anewproofforthisresultisprovided. Thisproofalsoyields 0 2 that, for any ε>0, themeasure Q can be chosen so that dQ/dP≤1+ε. n a J 5 1. Introduction and related literature 1 Let (Ω,F,P) denote a probability space equipped with a discrete-time filtration (F ) , R] where Ft ⊂ F. Moreover, let S = (St)t∈N0 denote a d-dimensional P–local martingale,twth∈eNr0e P d ∈N. ThenthereexistsaprobabilitymeasureQ,equivalenttoP,suchthatS isaQ–martingale. . This follows from more general results that relate appropriate no-arbitrage conditions to the h t existence of an equivalent martingale measure; see Dalang et al. (1990) and Schachermayer a (1992) for the finite-horizon case and Schachermayer (1994) for the infinite-horizon case. These m results are sometimes baptized fundamental theorems of asset pricing. [ More recently, Kabanov (2008) and Prokaj and Ra´sonyi (2010) have provided a direct proof 1 for the existence of such a measure Q; see also Section 2 in Kabanov and Safarian (2009). v The proof in Kabanov (2008) relies on deep functional analytic results, e.g., the Krein-Sˇmulian 5 2 theorem. The proof in Prokaj and Ra´sonyi (2010) avoids functional analysis but requires non- 0 trivial measurable selection techniques. 4 As this note demonstrates, in one dimension, an important but special case, the Radon- 0 . Nikodym derivative Z∞ = dQ/dP can be explicitly constructed. Moreover, in higher dimensions, 1 themeasurable selection results can besimplified. This is donehereby appropriately modifying 0 7 an ingenious idea of Rogers (1994). 1 More precisely, the following theorem will be proved in Section 3. : v Xi Theorem 1. For all ε > 0, there exists a uniformly integrable P–martingale Z = (Zt)t∈N0, bounded from above by 1+ε, with Z = lim Z > 0, such that ZS is a P–martingale and ∞ t↑∞ t r such that E [Z |S |p]< ∞ for all t,p ∈ N . a P t t 0 The fact that the bound on Z can be chosen arbitrarily close to 1 seems to be a novel observation. Considering a standard random walk S directly yields that there is no hope for a stronger version of Theorem 1 which would assert that ZS is not only a P–martingale but also a P–uniformly integrable martingale. A similar version of the following corollary is formulated in Prokaj and Ra´sonyi (2010); it would also bea direct consequence of Kabanov and Stricker (2001). To state it, let us introduce the total variation norm k·k for two equivalent probability measures Q ,Q as 1 2 kQ1−Q2k= EQ1[|dQ2/dQ1−1|]. Corollary 2. For all ε > 0, there exists a probability measure Q, equivalent to P, such that S is a Q–martingale, kP−Qk< ε, and E [|S |p]< ∞ for all t,p ∈ N . Q t 0 Date: January 17, 2017. 2010 Mathematics Subject Classification. Primary: 60G42; 60G48. We thank Yuri Kabanov for many helpful comments. J.R. is grateful for the generous support provided by the Oxford-Man Instituteof QuantitativeFinance at theUniversity of Oxford. 1 2 VILMOSPROKAJANDJOHANNESRUF To reformulate Corollary 2 in more abstract terms, let us introduce the spaces M = {Q ∼ P : S is a Q–local martingale}; loc Mp = {Q ∼ P : S is a Q–martingale with E [|S |p]< ∞ for all t ∈ N }, p > 0. Q t 0 Then Corollary 2 states that the space Mp is dense in M with respect to the total p>0 loc variation norm k·k. T Proof of Corollary 2. Consider the P–uniformly integrable martingale Z of Theorem 1, with ε replaced by ε/2. Then the probability measure Q, given by dQ/dP= Z∞, satisfies the conditions of the assertion. Indeed, we only need to observe that E [|Z −1|] = 2E [(Z −1)1 ]≤ ε, P ∞ P ∞ {Z>1} where we used that E [Z −1] = 0 and the assertion follows. (cid:3) P ∞ 2. Generalized conditional expectation and local martingales For sake of completeness, we review the relevant facts related to local martingales in discrete time. To start, note that for a sigma algebra G ⊂ F and a nonnegative random variable Y, not necessarily integrable, we can define the so called generalized conditional expectation E [Y |G] = limE [Y ∧k|G]. P P k↑∞ Next, for a general random variable W with E [|W||G] < ∞, but not necessarily integrable, P we can define the generalized conditional expectation E [W |G]= E [W+|G]−E [W−|G]. P P P For a stopping time τ and a stochastic process X we write Xτ to denote the process obtained from stopping X at time τ. Definition 3. A stochastic process S = (S ) is t t∈N0 • a P–martingale if E [|S |]< ∞ and E [S |F ] = S for all t ∈ N ; P t P t+1 t t 0 • a P–local martingale if there exists a sequence (τ ) of stopping times such that n n∈N lim τ = ∞ and Sτn1 is a P–martingale; n↑∞ n {τn>0} • a P–generalized martingale if E [|S ||F ]< ∞ and E [S |F ] = S for all t ∈ N . P t+1 t P t+1 t t 0 Proposition 4. Any P–local martingale is a P–generalized martingale. This proposition dates back to Theorem II.42 in Meyer (1972); see also Theorem VII.1 in Shiryaev (1996). Its reverse direction would also be true but will not be used below. A direct corollary of the proposition is that a P–local martingale S with E [|S |] < ∞ for all t ∈ N is P t 0 indeed a P–martingale. For sake of completeness, we will provide a proof of the proposition here. Proof of Proposition 4. Let S denote a P–local martingale. Fix t ∈ N and a localization 0 sequence (τ ) . For each n∈ N, we have, on the event {τ > t}, n n∈N n E [|S ||F ] = limE [|S |∧k|F ]= lim E [|Sτn |∧k|F ] = E [|Sτn ||F ] < ∞. P t+1 t P t+1 t P t+1 t P t+1 t k↑∞ k↑∞ Since lim τ = ∞, we get E [|S ||F ]< ∞. n↑∞ n P t+1 t The next step we only argue for the case d = 1, for sake of notation, but the general case follows in the same manner. As above, again for fixed n ∈ N, on the event {τ > t}, we get n E [S |F ] = lim E [S+ ∧k|F ]−E [S− ∧k|F ] P t+1 t P t+1 t P t+1 t k↑∞ Ä = limä EP[(Stτ+n1∧k)∨(−k)|Ft]= St. k↑∞ Thanks again to lim τ = ∞, the assertion follows. (cid:3) n↑∞ n LOCAL MARTINGALES IN DISCRETE TIME 3 Example 5. Assume that (Ω,F,P) supports two independent random variables U and θ such that U is uniformly distributed on [0,1], and P[θ = −1] = 1/2 = P[θ = 1]. Moreover, let us assume that F = {∅,Ω}, F = σ(U), and F = σ(U,θ) for all t ∈ N\{1}. Then the stochastic 0 1 t process S = (St)t∈N0, given by St = θ/U1t≥2 is easily seen to be a P–generalized martingale and a P–local martingale with localization sequence (τ ) given by n n∈N τ = 1×1 +∞×1 . n {1/U>n} {1/U≤n} However, we have EP[|S2|] = EP[1/U] = ∞; hence S is not a P–martingale. Now, consider the process Z = (Z ) , given by Z = 1 +2U1 . A simple computation t t∈N0 t t=0 t≥1 shows that Z is a strictly positive P–uniformly integrable martingale. Moreover, since Z S = t t 2θ1 , we have E [Z |S |] ≤ 2 for all t ∈ N and ZS is a P–martingale. If we require the t≥2 P t t 0 Radon-Nikodym to be bounded by a constant 1+ε ∈ (1,2], we could consider Z = (Z ) t t∈N0 with Zt = 1t=0 +(U∧ε)/(ε−ε2/2)1t≥1. This illustrates the validity of Theorem 1 in the context of this example. “ “ To see a difficulty in provingTheorem 1, let us consider a local martingale S′ = (S′) with “ t t∈N0 two jumps instead of one; for example, let us define θ S′ = 1 −1 1 + 1 . t {U>1/2} {U<1/2} t≥1 U t≥2 Again, it is simple to see that tÄhis specification maäkes S′ indeed a P–local and P–generalized martingale. However, now we have EP[Z1S1′] = 1/2 6= 0; hence ZS′ is not a P–martingale. Similarly, neither is ZS′. Nevertheless, as Theorem 1 states, there exists a uniformly integrable P–martingale Z′ such that Z′S′ is a P–martingale. More details on th“e previous example are provided in Ruf (2017). 3. Proof of Theorem 1 We start with three preparatory lemmata. Lemma 6. Let Q denote some probability measure on (Ω,F), let G,H be sigma algebras with G ⊂ H ⊂ F, let W denote a H –measurable d-dimensional random vector with E [|W||G] < ∞ and E [W |G]= 0. (3.1) Q Q Suppose that (α ) is a bounded family of H –measurable random variables with lim α = k k∈N k↑∞ k 1. Then for any ε> 0 there exists a family (V ) of random variables such that k k∈N (i) V is H –measurable and takes values in (1−ε,1) for each k ∈N; k (ii) limk↑∞1{EQ[VkαkW|G]=0} = 1. We shall provide two proofs of this lemma, the first one applies only to the case d = 1, but avoids the technicalities necessary for the general case. Proof of Lemma 6 in the one-dimensional case. With the convention 0/0 := 1, define, for each k ∈ N, the random variable E [α W+|G] Q k C = k E [α W−|G] Q k and note that E [W+|G] 1 lim|C −1| = Q −1 = E [W+|G]−E [W−|G] = 0. k↑∞ k (cid:12)EQ[W−|G] (cid:12) EQ[W−|G] Q Q (cid:12) (cid:12) (cid:12) (cid:12) Next, set (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) V = (1−ε)∨ 1 (1∧C−1)+1 (1∧C ) , k {W≥0} k {W<0} k and note that on the event {1−ε Ä≤ Ck ≤ 1/(1−ε)} ∈ G we indeed haväe EQ[VkαkW |G] = 0, which concludes the proof. (cid:3) 4 VILMOSPROKAJANDJOHANNESRUF Proof of Lemma 6 in the general case. Theproof is similar to theproof of theDalang–Morton– Willinger theorem based on utility maximisation, see Rogers (1994) and Delbaen and Schacher- mayer (2006, Section 6.6) for detailed exposition. But instead of using the exponential utility, we choose a strictly convex function (the negative of the utility) which is smooth and whose derivative takes values in (1−ε,1). Indeed, in what follows we fix the convex function ε π f(a)= a 1+ arctan(a)− , a ∈ R. π 2 Then f is smooth and direct comÅputatiÅon shows, thatãfã is convex with derivative f′ taking values in the interval (1−ε,1). We formulated the statement with generalized conditional expectations. However, changing the probability appropriately with a G–measurable density we can assume, without loss of generality, that W ∈ L1(Q). Indeed, the probability measure Q′, given by dQ′ e−EQ[|W||G] = , dQ EQ[e−EQ[|W||G]] satisfies that W ∈ L1(Q′). Moreover, the (generalized) conditional expectations with respect to G arethe same underQ and Q′. So in whatfollows we assumethat |W|is an integrable random variable. For W there is a maximal G–measurable orthogonal projection R of Rd such that RW = 0 almost surely. For a proof, see Proposition 2.4 in Rogers (1994) or Section 6.2 in Delbaen and Schachermayer (2006). The orthocomplement of the range of R is called the predictable range of W. Let B denote the d–dimensional Euclidean unit ball and set α = 1. For each k ∈ N∪{∞}, ∞ consider the random function (or field) h over B, defined by the formula k h (u,·) = h (u) = E [f(α W ·u)|G]+|Ru|2 for all u ∈B. k k Q k Since f is continuous, for each k ∈ N∪{∞}, h has a version that is continuous in u for each k ω ∈ Ω; see Lemma 9 below. Then for each compact subset C of B and each k ∈ N∪{∞} there is a G–measurable random vector UC taking values in C such that h (UC) = min h (u). k k k u∈C k This is a kind of measurable selection; for sake of completeness we give an elementary proof below in Lemma 11. Next, for each k ∈ N, let U be a G–measurable minimiser of h in the unit ball B and define k k V = f′(α W ·U ). k k k With this definition, (i) follows directly. For (ii) we prove below that E [V α W |G]+2RU = 0, on {|U | < 1}, k ∈ N; (3.2) Q k k k k lim U = 0, almost surely. (3.3) k k↑∞ Then, on the event {|U |< 1}, (3.2) and the G–measurability of R yield k |E [V α W |G]|2 = −E [V α W |G]·Ru= −E [V α RW |G]·u= 0, Q k k Q k k Q k k giving us (ii). Thus, in order to complete the proof it suffices to argue (3.2)–(3.3). For (3.2), note that h k is continuously differentiable almost surely for each k ∈ N, see Lemma 10 below; morever, its derivative attheminimumpointU , whichequals theleft-hand sideof (3.2), mustbezero when k U is inside the ball B. k For (3.3) observe that h has a unique minimiser over B which is the zero vector. To see ∞ this, observe that h (u) = E [f(W ·(I −R)u)|G]+|Ru|2, ∞ Q LOCAL MARTINGALES IN DISCRETE TIME 5 where I denotes the d-dimensional identity matrix. So to see that the zero vector is the unique minimiser it is enough to show that inf h (u) > 0 = h (0) almost surely for any δ ∈(0,1]. |u|≥δ ∞ ∞ Let U be a G–measurable minimiser of h over {u : |u| ∈ [δ,1]}. Then ∞ E [f(W ·(I −R)U)|G]> 0, on {(I −R)U 6= 0}; Q |RU|2 ≥ δ2 > 0, on {(I −R)U = 0}. The first part follows from the strict convexity of f in conjunction with Jensen’s inequality, taking into account that E [W |G] = 0 and that W ·(I −R)U has non-trivial conditional law Q on {(I −R)U 6= 0} by the maximality of R. Whence inf h (u) > 0 = h (0), as required. |u|≥δ ∞ ∞ Finally, as lim α = 1 and f is Lipschitz continuous we have k↑∞ k lim sup|h (u)−h (u)| = lim sup |h (u)−h (u)| = 0 almost surely. k ∞ k ∞ k↑∞u∈B k↑∞u∈B∩Qd Hence, any G–measurable sequence (U ) of minimisers of h converges to zero, the unique k k∈N k minimiser of h , almost surely. This shows (3.3) and completes the proof. (cid:3) ∞ Lemma 7. Let Q denote some probability measure on (Ω,F), let G,H be sigma algebras with G ⊂ H ⊂ F, let Y denote a one-dimensional random variable with Y ≥ 0 and E [Y |H ] < ∞, Q and let W denote a H –measurable d-dimensional random vector such that (3.1) holds. Then, for any ε > 0, there exists a random variable z such that (i) z is H –measurable and takes values in (0,1+ε); (ii) Q[z < 1−ε] < ε; (iii) E [z|G]= 1; Q (iv) E [zW |G] = 0; Q (v) E [zY |G]< ∞. Q Proof. For each k ∈ N, define the (0,1]–valued, H –measurable random variable 1 αk = 1{EQ[Y|H]≤k}+ E [Y |H ]1{EQ[Y|H]>k} Q and note that lim α = 1. Lemma 6 now yields the existence of a family (V ) of k↑∞ k k k∈N H –measurable random variables such that Vk ∈ (1/(1+ε/2),1) and limk↑∞1{EQ[VkαkW|G]=0} = 1. Note that this yields a G–measurable random variable K, taking values in N, such that EQ[VKαKW |G] =0, EQ[VKαK |G] >1/(1+ε), and Q[EQ[Y |H ] > K]< ε. Setting now V α K K z = E [V α |G] Q K K yields a random variable with the claimed properties. (cid:3) Lemma 8. Fix n ∈ N , let Q denote some probability measure on (Ω,F) such that S is a Q– 0 local martingale, and let Y denote a nonnegative random variable with E [Y |F ]< ∞. Then, Q n for each ε > 0, there exists a probability measure Q′, equivalent to Q, with density Z(n) = dQ′/dQ such that (i) Z(n) ∈ (0,1+ε); (ii) Q[Z(n) < 1−ε] < ε; (iii) S is a Q′–local martingale; (iv) EQ′[Y]< ∞. Proof. Inthisproof,we usetheconvention F = {∅,Ω} and∆S = 0. Setε > 0besufficiently −1 0 small such that (n+1)ε ≤ ε, (1+ε)n+1 ≤ 1+ε, (1−ε)n+1 ≥ 1−eε. With εreplaced by εandwithG = F , H = F , andW = ∆S , henceE [|W||G] < ∞and n−1 n n Q E [W |G] = 0 by Proposeition 4, let z deenote the correspondingerandom variable of Lemma 7. Q n If n = 0, we define Qe′ by dQ′/dQ = z, and the lemma is proven. 6 VILMOSPROKAJANDJOHANNESRUF Ifn> 0, weproceediteratively. Considert ∈ {0,··· ,n−1} andassumethatwehaverandom variables z ,··· ,z such that, in particular, E [Y n z | F ] < ∞. We now obtain a t+1 n Q i=t+1 i t random variable z by again applying Lemma 7, with ε replaced by ε and with G = F , t t−1 H = F , W = ∆S , and Y replaced by Y n z . Q t t i=t+1 i With the family (z0,··· ,zn) now given, lQet us define Z(n) = ni=0zi aend Q′ by dQ′/dQ = Z(n). To argue that S is a Q′–local martingale, we may consider any sequence of stopping times that Q localizes S. Since all other assertions follow directly from the construction of Z(n) and the choice of ε, the lemma is proven. (cid:3) Proof of Theorem 1. We inductively construct a sequence (Q(n)) of probability measures, e n∈N0 equivalenttoP,andasequence(ε(n)) ofpositiverealsusingLemma8. Tostart, setQ(−1) = n∈N0 P. Now, fixn ∈ N forthemomentandsupposethatwehave Q(n−1) and(ε(m)) suchthat 0 0≤m<n n−1(1+ε(m))< 1+ε. Choose ε(n) to be sufficiently small such that n (1+ε(m)) < 1+ε, m=0 m=0 and for any A ∈ F with Q(n−1)[A] ≤ ε(n) we have P[A] < 2−n. Then apply Lemma 8 with ε Q Q replaced by ε(n), and with Q = Q(n−1) and Y = e|Sn| to obtain a probability measure Q(n) with density Z(n), that is dQ(n) =Z(n)dQ(n−1) = ( n Z(m))dP. m=0 Due to the fact Q P |1−Z(n)| > ε(n) ≤ 2−n as Q(n−1) |1−Z(n)| > ε(n) ≤ ε(n), the Borel-Cantelli lemma yields |1 − Z(n)| < ∞; hence the infinite product Z = î ó n∈N0 î ó ∞ ∞ Z(n) converges and is positive P–almost surely. It is clear that Z ≤ 1+ε. n=0 P ∞ We define the probability measure Q by dQ/dP = Z∞ and denote the corresponding density Q process by Z = E [Z |F ], for each t ∈ N . As Z(m) < 1+ε we have Q ≤ (1+ε)Q(t) t P ∞ t 0 m>t and as a result Q E Z e|St| = E e|St| ≤(1+ε)E e|St| < ∞ P t Q Q(t) by the choice of Q(t); hence E [Z |S |p] < ∞ for all t,p ∈ N . î P tó t î ó 0î ó It remains to argue that ZS is a P–martingale or, equivalently, that S is a Q–martingale. Sincewealreadyhave establishedE [|S |]< ∞forallt ∈ N , itsufficestofixt ∈ Nandtoprove Q t 0 that E [S |F ] = S . To this end, recall that S is a Q(n)–local martingale for each n ∈ N Q t t−1 t−1 0 by Lemma 8(iii) and note that dominated convergence, Bayes formula, and Proposition 4 yield n E [S |F ]Z = E [S Z |F ]= lim E S Z(m) F Q t t−1 t−1 P t ∞ t−1 P t t−1 n↑∞ " (cid:12) # mY=0 (cid:12) dQ(n) (cid:12)(cid:12) n = lim E [S |F ] = S l(cid:12)im E Z(m) F n↑∞ Q(n) t t−1 dP (cid:12)(cid:12)Ft−1 t−1n↑∞ P"mY=0 (cid:12)(cid:12) t−1# (cid:12) (cid:12) = St−1Zt−1. (cid:12) (cid:12) (cid:12) (cid:12) This completes the proof. (cid:3) Appendix A In this appendix, we provide some measurability results necessary for the proof of Lemma 6. Lemma 9. Let G be a sigma algebra with G ⊂ F and let ξ be a random element in C(K), where (K,m) isacompact metric space. Suppose thatE [sup |ξ(u)|] < ∞andletη(u) = E [ξ(u)|G] P u∈K P for all u ∈K. Then (η(u)) has a continuous modification. u∈K Proof. Let D be a countable dense subset of K. We show that there is Ω′ ∈ G with full probability such that (η(u)) is uniformly continuous over D on Ω′. Then we can define u∈D lim η(u ) on Ω′, n η˜(u) = uunn→∈Du  0 otherwise.   LOCAL MARTINGALES IN DISCRETE TIME 7 It is a routine exercise to check that η˜ is well defined and a continuous modification of η. One way to get Ω′ is the following. Let µ be the modulus of continuity of ξ, that is, µ(δ) = sup |ξ(u)−ξ(u′)|, δ > 0. u,u′∈K,m(u,u′)≤δ Obviously µ(δ) → 0 everywhere as δ ↓ 0. Dominated convergence, in conjunction with the bound µ ≤ 2sup |ξ(u)|, yields µ˜(δ) = E[µ(δ)|G]→ 0 as δ ↓0 almost surely. Now define u∈K 1 1 Ω′ = lim µ˜ = 0 ∩ |η(u)−η(u′)| ≤ µ˜ . n↑∞ n n ß Å ã ™ Ñn\∈N u,u′∈D,m\(u,u′)≤1/nß Å ã™é Clearly Ω′ has full probability and the claim is proved. (cid:3) In the setting of Lemma 9 when K ⊂ Rd and ξ is a random element in C1(K) then under mild conditions η(u) = E[ξ(u)|G] has a version taking values in C1(K). This is the content of the next lemma. Recall that a function f defined on K belongs to C1(K) if f is continuous and there is continuous Rd–valued function on K which agrees with the gradient f′ of f in the interior of K. Lemma 10. Let G be a sigma algebra with G ⊂ F and let ξ be a random element in C1(K), where K ⊂ Rd is a compact subset set. Suppose that E sup|ξ(u)| +E sup|ξ′(u)| < ∞ P P u∈K u∈K ñ ô ñ ô and let η(u) = E [ξ(u)|G] for all u∈ K. Then (η(u)) has a version taking values in C1(K) P u∈K and the continuous version of (E[ξ′(u)|G]) gives the gradient of η almost surely. u∈K Proof. By Lemma 9 both η(u) = E[ξ(u)|G] and η′(u) = E[ξ′(u)|G] have continuous versions. We prove that, apart from a null set, η′ is indeed the gradient of η. To this end, let D be a countable dense subset of the interior of K and denote by I(a,b) a directed segment going from a to b, for each a,b ∈ K. Then, by assumption, for a,b ∈ D, with I(a,b) ⊂ intK we get η(b)−η(a) = E[ξ(a)−ξ(b)|G] =E ξ′(u)du G = η′(u)du, almost surely. ñZI(a,b) (cid:12)(cid:12) ô ZI(a,b) Hence, there exists an event Ω′ ∈ G with P[Ω′] = 1 su(cid:12)(cid:12)ch that (cid:12) η(b,ω)−η(a,ω) = η′(u,ω)du, for all a,b ∈D, with I(a,b) ⊂ intK and ω ∈ Ω′. ZI(a,b) By continuity this idenity extends to all a,b ∈ intK with I(a,b) ⊂ intK on Ω′. Using again the continuity of η′(.,ω) yields that η′ is indeed the gradient of η on Ω′. (cid:3) Lemma 11. Let (K,m) be a compact metric space and η a random element in C(K). Then there is a measurable minimiser of η, that is a random element U in K, such that η(U) = min η(u). u∈K Proof. To shorten the notation, for each x ∈ K and δ ≥ 0, let B(x,δ) = {u ∈K : m(u,x) ≤δ}, η(x,n) = min{η(u) : u∈ B(x,2−n)}. For each n ∈N let D be a finite 2−n-net in K; that is, K ⊂ B(x,2−n). For each n ∈ N n x∈Dn fix an order of the finite set D . We shall use the fact that for any closed set F the minimum n S over F, that is, min η(u), is a random variable. u∈F We define a sequence (U ) of random elements in K by recursion, such that n n∈N • η(U ,n)= min η(u), and n u∈K • m(U ,U )≤ 2−n +2−(n+1). n n+1 8 VILMOSPROKAJANDJOHANNESRUF Then (U ) has a limit U which is a measurable minimiser of η over K. n n∈N For n = 1 let U be the first element in 1 {v ∈ D : η(v,1) = minη(u)}. 1 u∈K Since this set is not empty, U is well defined. If U ,...,U are defined for some n ∈ N set U 1 1 n n+1 to be the first element in v ∈ D : η(v,n+1) = minη(u), m(v,U )≤ 2−n +2−(n+1) n+1 n u∈K This set is not eßmpty as ™ B(U ,2−n)⊂ B(v,2−(n+1)), n v∈D[n+1 m(v,Un)≤2−n+2−(n+1) so U is well defined. We conclude that the sequence with the above properties exists and its n+1 limit is a measurable minimiser. (cid:3) References Dalang, R. C., A. Morton, and W. Willinger (1990). Equivalent martingale measures and no- arbitrage in stochastic securities market models. Stochastics Stochastics Rep. 29(2), 185–201. Delbaen, F. and W. Schachermayer (2006). The Mathematics of Arbitrage. Springer. Kabanov, Y. (2008). In discrete time a local martingale is a martingale under an equivalent probability measure. Finance Stoch. 12(3), 293–297. Kabanov, Y. and M. Safarian (2009). Markets with Transaction Costs. Springer-Verlag, Berlin. Kabanov,Y.andC.Stricker(2001). Onequivalentmartingalemeasureswithboundeddensities. In S´eminaire de Probabilit´es, XXXV, Volume 1755 of Lecture Notes in Math., pp. 139–148. Springer, Berlin. Meyer, P.-A. (1972). Martingales and Stochastic Integrals. I. Lecture Notes in Mathematics. Vol. 284. Springer-Verlag, Berlin-New York. Prokaj, V. and M. Ra´sonyi (2010). Local and true martingales in discrete time. Teor. Veroyatn. Primen. 55(2), 398–405. Rogers, L.C. G.(1994). Equivalent martingale measures and no-arbitrage. Stochastics Stochas- tics Rep. 51(1-2), 41–49. Ruf, J. (2017). Local martingales and their uniform integrability. A mini course. Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Math. Econom. 11(4), 249–257. Schachermayer, W. (1994). Martingale measures for discrete-time processes with infinite hori- zon. Math. Finance 4(1), 25–55. Shiryaev, A. N. (1996). Probability (Second ed.), Volume 95 of Graduate Texts in Mathematics. Springer-Verlag, New York. Translated from the first (1980) Russian edition by R. P. Boas. Vilmos Prokaj, Department of Probability Theory and Statistics, Eo¨tvo¨s Lora´nd University, Budapest E-mail address: [email protected] JohannesRuf,DepartmentofMathematics,London SchoolofEconomicsandPoliticalScience E-mail address: [email protected]