On the Second Fundamental Theorem of Asset Pricing Rajeeva L Karandikar

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Communications on Stochastic Analysis

Volume 10 | Number 1 Article 5

3-1-2016 On the second fundamental theorem of asset pricing Rajeeva L Karandikar

B V Rao

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Recommended Citation Karandikar, Rajeeva L and Rao, B V (2016) "On the second fundamental theorem of asset pricing," Communications on Stochastic Analysis: Vol. 10 : No. 1 , Article 5. DOI: 10.31390/cosa.10.1.05 Available at: https://digitalcommons.lsu.edu/cosa/vol10/iss1/5 Communications on Stochastic Analysis Serials Publications Vol. 10, No. 1 (2016) 57-81 www.serialspublications.com

ON THE SECOND FUNDAMENTAL THEOREM OF ASSET PRICING

RAJEEVA L. KARANDIKAR AND B. V. RAO

Abstract. Let X1,...,Xd be sigma-martingales on (Ω, F, P). We show that every bounded martingale (with respect to the underlying ﬁltration) admits an integral representation with respect to X1,...,Xd if and only if there is no equivalent probability measure (other than P) under which X1,...,Xd are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM).

1. Introduction The (first) fundamental theorem of asset pricing says that a market consisting of finitely many stocks satisfies the No Arbitrage (NA) property if and only there exists an Equivalent Martingale Measure (EMM)- i.e. there exists an equivalent probability measure under which the (discounted) stocks are (local) martingales. The No Arbitrage property has to be suitably defined when we are dealing in continuous time, where one rules out approximate arbitrage in the class of admis- sible strategies. For a precise statement in the case when the underlying processes are locally bounded, see Delbaen and Schachermayer [5]. Also see Bhatt and Karandikar [1] for an alternate formulation of a weaker result, where the approximate arbitrage is defined only in terms of simple strategies. For the general case, the result is true when local martingale in the statement above is replaced by sigma-martingale. See Delbaen and Schachermayer [6]. They have an example where the No Arbitrage property holds but there is no equivalent measure under which the underlying process is a local martingale. However, there is an equivalent measure under which the process is a sigma-martingale. The second fundamental theorem of asset pricing says that the market is complete (i.e. every contingent claim can be replicated by trading on the underlying securities) if and only if the EMM is unique. Interestingly, this property was studied by probabilists well before the connection between finance and stochastic calculus was established (by Harrison–Pliska [9]). The completeness of market is same as the question: when is every martingale representable as a stochastic integral with respect to a given set of martingales {M 1,...,M d}. When M 1,...,M d is the d-dimensional Wiener Process, this property was proven by Ito [10]. Jacod

Received 2016-2-19; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 60H05, 62P05; Secondary 60G44, 91B28. Key words and phrases. Equivalent martingale measure, martingale representation, sigma- martingales. 57 58 RAJEEVA L. KARANDIKAR AND B. V. RAO and Yor [13] proved that if M is a P-local martingale, then every martingale N admits a representation as a stochastic integral with respect to M if and only if there is no probability measure Q (other than P) such that Q is equivalent to P and M is a Q-local martingale. The situation in higher dimension is more com- plex. The obvious generalisation to higher dimension is not true as was noted by Jacod–Yor [13]. To remedy the situation, a notion of vector stochastic integral was introduced- where a vector valued predictable process is the integrand and vector valued martingale is the integrator. The resulting integral yields a class larger than the linear space generated by component wise integrals. See [12], [2]. However, one has to prove various properties of the vector stochastic integrals once again. Here we achieve the same objective in another fashion avoiding defining integration again from scratch. In the same breath, we also take into account the general case, when the underlying processes need not be bounded but satisfy the property NFLVR and thus one has an equivalent sigma-martingale measure (ESMM). To the best of our knowledge, the martingale representation property in the framework of sigma-martingales is not available in literature. Indeed, most treatments deal with square integrable martingales where the notion of orthogonality of martingales is available which simplifies the treatment. For a semimartingale X, let L∫(X) denote the class of predictable processes f such that the stochastic integral f dX is defined. A semimartingale Z is said to admit an integral representation with respect to semimartingales (X1,X2,...,Xd) if there exists a semimartingale Y and predictable processes f, gj such that f ∈ L(Y ), gj ∈ L(Xj) ∫ ∑d t j j ∀ ≥ Yt = Y0 + gs dXs , t 0 j=1 0 and ∫ t Zt = Z0 + fs dYs, ∀t ≥ 0. 0 In Theorem 5.3 we will show that for a multidimensional sigma-martingale (X1,X2,...,Xd) all bounded martingales admit a representation with respect to Xj, 1 ≤ j ≤ d if and only if the ESMM is unique. The most critical part of its proof is to show that the class of martingales that admit representation with respect to (X1,X2,...,Xd) is closed under L1 convergence : If M n are martingales such that there exist f n, gn,j, Y n with ∫ ∑d t n n n,j j ∀ ≥ Yt = Y0 + gs dXs , t 0 j=1 0 and ∫ t n n n n ∀ ≥ Mt = M0 + fs dYs , t 0 0 | n − | → and if E[ Mt Mt ] 0, then we need to show that M also admits a representation with respect to (X1,X2,...,Xd). When (X1,X2,...,Xd) is d-dimensional Brownian motion, or when Xj and Xk are orthogonal as martingales, one can deduce that for each j, {gn,j : n ≥ 1} is Cauchy in an appropriate norm and SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 59 thereby complete the proof. This step fails in general, as the Jacod–Yor example shows. So we need to do an orthogonalisation of (X1,X2,...,Xd) to achieve the same. However, (X1,X2,...,Xd) may not be square integrable and thus we need to change the measure to a measure Q (equivalent to the underlying probability measure P) to make it so. Under Q,(X1,X2,...,Xd) need not be martingales. This part is delicately managed by keeping both P, Q in the picture. The rest of the argument is on the lines of Jacod–Yor [13]. When X1,X2,...,Xd represent (prices of) stocks, Y can be thought of as (the price of) a mutual fund or an index fund and the investor is trading on such a fund trying to replicate the security Z. In this framework, Theorem 5.3 gives us the second fundamental theorem of asset pricing : Market is complete if and only if ESMM (equivalent sigma-martingale measure) is unique.

2. Preliminaries and Notation Let us start with some notations. (Ω, F, P) denotes a complete probability space with a filtration (F) = {Ft : t ≥ 0} such that F0 consists of all P-null sets (in F) and ∩t>sFt = Fs ∀s ≥ 0. Thus, we can (and do) assume that all martingales have r.c.l.l. paths. For various notions, definitions and standard results on stochastic integrals, we refer the reader to Meyer [15], Jacod [11] or Protter [16]. Let M denote the class of martingales and Mloc denote the class of local mar- ∈ M L1 tingales. For M loc, let m(M) be the class of predictable processes f such that there exists a sequence of stopping times σ ↑ ∞ with ∫ k σk 1 { 2 } 2 ∞ E[ fs d[M,M]s ] < . ∫ 0 For such an f, N = f dM is defined and is a local martingale. For a semimartingale X, let L(X) denote the class of predictable process f such that X admits a decomposition X = N + A with N ∈ Mloc , A being a process ∈ L1 with finite variation paths with f m(N) and ∫ t |fs|d|A|s < ∞ a.s. ∀t < ∞. (2.1) 0 ∫ ∫ ∫ For f ∈ L(X), the stochastic integral f dX is defined as f dN + f dA. It can be shown that the decomposition X = N + A is not unique, the definition does not depend upon the decomposition. See [11]. For M 1,M 2,...,M d ∈ M let C(M 1,M 2,...,M d) denote the class of martin- ∈ M ∃ j ∈ L1 j ≤ ≤ gales Z such that f m(M ), 1 j d with ∫ ∑d t j j ∀ ≥ Zt = Z0 + fs dMs , t 0. j=1 0 For T < ∞, let ˜ 1 2 d 1 2 d KT (M ,M ,...,M ) = {ZT : Z ∈ C(M ,M ,...,M )}. 60 RAJEEVA L. KARANDIKAR AND B. V. RAO

˜ For the case d = 1, Yor [18] had proved that KT is a closed subspace of 1 ˜ 1 2 d L (Ω, F, P). The problem in case d > 1 is that in general KT (M ,M ,...,M ) need not be closed. Jacod–Yor [13] gave an example where M 1,M 2 are continuous ˜ 1 2 square integrable martingales and KT (M ,M ) is not closed. Jacod [12] defined the vector stochastic integral and Memin [14] proved that with the modified definition of the integral, this space is closed. We will follow a different path. For martingales M 1,M 2,...,M d, let F(M 1,...,M d) be the class of martingales ∈ M ∃ ∈ C 1 d ∈ L1 Z such that Y (M ,...,M ) and f m(Y ) with ∫ t Zt = Z0 + fs dYs, ∀t ≥ 0. 0 Let 1 2 d 1 2 d KT (M ,M ,...,M ) = {ZT : Z ∈ F(M ,M ,...,M )}. The main result of the next section is

1 2 d 1 2 d Theorem 2.1. Let M ,M ,...,M be martingales. Then KT (M ,M ,...,M ) is closed in L1(Ω, F, P). This will be deduced from Theorem 2.2. Let M 1,M 2,...,M d be martingales and Zn ∈ F(M 1,M 2,...,M d) | n − | → ∈ F 1 2 d be such that E[ Zt Zt ] 0 for all t. Then Z (M ,M ,...,M ). When M 1,M 2,...,M d are square integrable martingales, the analogue of The- orem 2.1 for L2 follows from the work of Davis–Varaiya [4]. However, for the EMM characterisation via integral representation, one needs the L1 version, which we deduce using change of measure technique. We will need the Burkholder–Davis–Gundy inequality (see [15]) (for p = 1) which states that there exist universal constants c1, c2 such that for all martingales M and T < ∞,

1 1 2 1 c E[([M,M]T ) 2 ] ≤ E[ sup |Mt|] ≤ c E[([M,M]T ) 2 ]. 0≤t≤T After proving Theorem 2.1, in the next section we will introduce sigma-martingales and give some elementary properties. Then we come to the main theorem on integral representation of martingales. This is followed by the second fundamental theorem of asset pricing.

3. Proof of Theorem 2.1 In this section, we ﬁx martingales M 1,M 2,...,M d. We begin with a few aux- iliary results.

1 d Lemma 3.1. Let Cb(M ,...,M ) be the class of martingales Z such that ∃ bounded predictable processes f j, 1 ≤ j ≤ d with ∫ ∑d t j j ∀ ≥ Zt = Z0 + fs dMs , t 0. j=1 0 SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 61

F 1 d ∈ M ∃ ∈ L1 Let b(M ,...,M ) be the class of martingales Z such that f m(Y ) and 1 d Y ∈ Cb(M ,...,M ) with ∫ t Zt = Z0 + fs dYs, ∀t ≥ 0. 0 1 d 1 d Then Fb(M ,...,M ) = F(M ,...,M ). L1 Proof. Since bounded predictable process belong to m(N) for every martingale 1 d 1 d 1 d N, it follows that Cb(M ,...,M ) ⊆ C(M ,...,M ). Thus Fb(M ,...,M ) ⊆ F(M 1,...,M d). For the other part, let Z ∈ F be given by ∫ t ∈ L1 Zt = Z0 + fs dYs, f m(Y ), 0 where ∫ ∑d t j j Yt = Y0 + gs dMs j=1 0 j ∈ L1 j with g m(M ). Let ∑d gj ξ = 1 + |gj|, hj = s s s s ξ j=1 s and ∫ ∑d t j j Vt = hs dMs . j=1 0 j ∈ C 1 2 d j j Since h are bounded, it follows that V b(M ,M ,...,M ). Using gs = ξshs j ∈ L1 j ∈ L1 and g m(M ), it follows that ξ m(V ) and ∫ t Yt = Y0 + ξs dVs. 0 ∫ ∫ ∈ L1 ∈ L1 □ Since f m(Y ), it follows that fξ m(V ) and f dY = fξdV . Lemma 3.2. Let√Z ∈ M be such that there exists a sequence of stopping times ↑ ∞ ∞ k ∈ F 1 2 d k σk with EP[ [Z,Z]σk ] < and X (M ,M ,...,M ) where Xt =

Zt∧σk . Then Z ∈ F(M 1,M 2,...,M d). ∫ k k k k 1 2 d Proof. Let X = Z0 + f dY for k ≥ 1 with Y ∈ Cb(M ,M ,...,M ) and k ∈ L1 k k,j f m(Y ). Let ϕ be bounded predictable processes such that ∫ ∑d t k k k,j j Yt = Y0 + ϕs dMs . j=1 0 k,1 k,2 k,d j Let ck > 0 be a common bound for ϕ , ϕ , . . . , ϕ . Let us deﬁne η , f by ∑∞ j 1 k,j ηt = ϕt 1{(σk−1,σk]}(t). ck k=1 62 RAJEEVA L. KARANDIKAR AND B. V. RAO ∑∞ k ft = ckft 1{(σ−1,σk]}(t). k=1 ∫ ∑d t j j Yt = ηs dMs . j=1 0 j 1 2 d By deﬁnition, η is bounded by 1 for every j and thus Y ∈ Cb(M ,M ,...,M ). We can note that − k − k Zt∧σ Zt∧σ − = X ∧ X ∧ k k 1 ∫ t σk t σk−1 t k k = fs 1{(σk−1,σk]}(s)dYs ∫0 t

= fs1{(σk−1,σk]}(s)dYs. 0 Thus ∫ t

Zt∧σk = Z0 + 1[0,σk](s)fs dYs 0 and hence ∫ σk 2 [Z,Z]σk = (fs) d[Y,Y ]s. 0 Since by assumption, for all k √ ∞ EP[ [Z,Z]σk ] < ∈ L1 □ it follows that f m(Y ). This proves the required result. n ∈ M | n − | → Lemma 3.3. Let Z be such that E[ Zt Zt ] 0 for all t. Then there j exists a sequence of stopping times σk ↑ ∞ and a subsequence {n } such that for ≥ each k 1, √ ∞ E[ [Z,Z]σk ] < j and writing Y j = Zn , √ j − j − → ↑ ∞ E[ [Y Z,Y Z]σk ] 0 as j . (3.1) 0 | n − | → → ∞ Proof. Let n = 0. For each j, E[ Zj Zj ] 0 as n and hence we can choose nj > n(j−1) such that | nj − | ≤ −j E[ Zj Zj ] 2 . Then using Doob’s maximal inequality we have 2 | nj − | ≥ 1 ≤ j P(sup Zt Zt 2 ) j . t≤j j 2

j As a consequence, writing Y j = Zn , we have ∑∞ | j − | ∞ ∞ ηt = sup Ys Zs < a.s. for all t < . (3.2) ≤ j=1 s t SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 63

Now deﬁne ∑∞ | | | j − | Ut = Zt + Yt Zt . j=1 In view of (3.2), it follows that U is r.c.l.l. adapted process. For any stopping time τ ≤ m, we have ∑∞ | | | j − | E[Uτ ] = E[ Zτ ] + E[ Yτ Zτ ] j=1 ∑∞ ≤ | | | j − | E[ Zm ] + E[ Ym Zm ] j=1 ∑m ∑∞ ≤ | | | j − | −j E[ Zm ] + E[ Ym Zm ] + 2 j=1 j=m+1 < ∞. Here, we have used that Z, Y j − Z being martingales, |Z|, |Y j − Z| are sub- martingales and τ ≤ m. Now deﬁning

σk = inf{t : Ut ≥ k or Ut− ≥ k} ∧ k it follows that σk are bounded stop times increasing to ∞ with ≤ sup Us k + Uσk s≤σk and hence

E[ sup Us] < ∞. (3.3) s≤σk

Thus, for each k ﬁxed E[sup ≤ |Z |] < ∞ and thus by Burkholder–Davis–Gundy s σk s√ ∞ inequality (p = 1 case), we have E[ [Z,Z]σk ] < . In view of (3.2) | j − | lim sup Ys Zs = 0 a.s. j→∞ s≤σk and is dominated by (sup U ) which in turn is integrable as seen in (3.3). Thus s≤σk s by dominated convergence theorem, we have | j − | lim E[ sup Ys Zs ] = 0. j→∞ s≤σk The result (3.1) now follows from the Burkholder–Davis–Gundy inequality (p = 1 case). □

Lemma 3.4. Let V ∈ F(M 1,M 2,...,M d) and τ be a bounded stopping time such that √ E[ [V,V ]τ ] < ∞. 1 2 d Then for ϵ > 0, there exists U ∈ Cb(M ,M ,...,M ) such that √ E[ [V − U,V − U]τ ] ≤ ϵ. 64 RAJEEVA L. KARANDIKAR AND B. V. RAO ∫ t ∈ L1 ∈ C 1 2 d Proof. Let Vt = V0 + 0 f dX where f m(X) and X b(M ,M ,...,M ). Since ∫ t 2 [V,V ]t = |fs| d[X,X]s, 0 the assumption on V gives √∫ τ | |2 ∞ E[ 0 fs d[X,X]s] < . (3.4) k Deﬁning f = fs1{|f |≤k}, let s s ∫ U k = f k dX.

1 2 d k Since X ∈ Cb(M ,M ,...,M ) and f is bounded, it follows that k 1 2 d U ∈ Cb(M ,M ,...,M ). Note that as k → ∞, √ √∫ τ 2 − k − k | | {| | } → E[ [V U , V U ]τ ] = E[ 0 fs 1 fs >k d[X,X]s] 0 in view of (3.4). The result now follows by taking U = U k with k large enough so that √ k k E[ [V − U , V − U ]τ ] < ϵ. □

Lemma 3.5. Suppose Z √∈ M and τ is a bounded stopping time such that Zt = n 1 2 d Zt∧τ for all t ≥ 0 and E[ [Z,Z]τ ] < ∞. Let U ∈ Cb(M ,M ,...,M ) with n U0 = 0 be such that √ n n −n E[ [U − Z,U − Z]τ ] ≤ 4 . ∈ C 1 2 d ∈ L1 Then there exists X b(M ,M ,...,M ) and f m(X) such that ∫ t Zt = Z0 + fs dXs. (3.5) 0 n ∈ C 1 2 d n Proof. Since U b(M ,M ,...,M ) with U0 = 0, get bounded predictable processes {f n,j : n ≥ 1, 1 ≤ j ≤ d} such that ∫ ∑d t n n,j j Ut = fs dMs . (3.6) j=1 0 n n n,j n,j Without loss of generality, we assume that Ut = Ut∧τ and fs = fs 1[0,τ](s). Let ∑∞ √ n n n ζ = 2 [U − Z,U − Z]τ . n=1 Then E[ζ] < ∞ and hence P(ζ < ∞) = 1. Let √ ∑d √ j j η = ζ + [Z,Z]τ + [M , M ]τ j=1 SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 65

Let c = E[exp{−η}] and let Q be the probability measure on (Ω, F) defined by dQ 1 = exp{−η}. dP c 2 Then it follows that α = EQ[η ] < ∞. Noting that ∑d ∑∞ 2 j j 2n n n η ≥ [Z,Z]τ + [M ,M ]τ + 2 [U − Z,U − Z]τ j=1 n=1 j j n we have EQ[[Z,Z]τ ] < ∞, EQ[[M ,M ]τ ] < ∞ for 1 ≤ j ≤ d. Likewise, EQ[[U − n n n j Z,U − Z]τ ] < ∞ and so EQ[[U ,U ]τ ] < ∞. Note that Z,M are no longer martingales under Q, but we do not need that. Let Ωe = [0, ∞) × Ω. Recall that the predictable σ-field P is the smallest σ field on Ωe with respect to which all continuous adapted processes are measurable. We will define signed measures Γ on P as follows: for E ∈ P, 1 ≤ i, j ≤ d let ∫ ij∫ τ i j Γij(E) = 1E(s, ω)d[M ,M ]s(ω)dQ(ω). Ω 0 ∑ d i j Let Λ = j=1 Γjj. From the properties of quadratic variation [M ,M ], it follows that for all E ∈ P, the matrix ((Γij(E))) is non-negative definite. Further, Γij is absolutely continuous with respect to Λ ∀i, j. It follows (see appendix) that we can get predictable processes cij such that dΓ ij = cij dΛ and that C = ((cij)) is a non-negative definite matrix. By construction |cij| ≤ 1 and are predictable. We can diagonalise C (i.e. obtain singular value decomposition) in a measurable way (see appendix) to obtain predictable processes bij, dj such that for all i, k, (writing δik = 1 if i = k and δik = 0 if i ≠ k), ∑d ij kj bs bs = δik, (3.7) j=1 ∑d ji jk bs bs = δik, (3.8) j=1 ∑d ij jl kl i bs cs bs = δikds. (3.9) j,l=1 ij i ≥ ≤ ≤ Since ((cs )) is non-negative definite, it follows that ds 0. For 1 j d, let ∑d ∫ k kl l N = bs dM . l=1 Then N k are P- martingales since bik is are bounded predictable processes. Fur- ther, ∑ ∫ i k ij kl j l [N ,N ] = bs bs d[M ,M ] j,l=1 66 RAJEEVA L. KARANDIKAR AND B. V. RAO and hence for any bounded predictable process h and i ≠ k, we have ∫ ∫ ∫ τ τ ∑d i k ij kl j l EQ[ hs d[N ,N ]] = hs bs bs d[M ,M ]dQ(ω) 0 Ω 0 j,l=1 ∫ ∑d ij kl = h b b dΓjl ¯ Ω j,l=1 ∫ ∑d = h bijbklcjl dΛ ¯ Ω j,l=1 = 0, where the last step follows from (3.9). As a consequence, for bounded predictable processes hi, 1 ≤ i ≤ d, it follows that ∫ ∫ ∑d τ ∑d τ i k i k k 2 k k EQ[ hshs d[N ,N ]s] = EQ[ (hs ) d[N ,N ]s]. (3.10) i,k=1 0 k=1 0

Let us observe that (3.10) holds for any predictable processes {hi : 1 ≤ i ≤ d} i i provided the RHS is finite: we can first note that it holds for h˜ = h {| |≤ } where ∑ 1 h c | | d | i| ↑ ∞ ≥ h = i=1 h and then let c . Note that for n m √ √ √ n m n m n n m m [U − U , U − U ]τ ≤ [U − Z,U − Z]τ + [U − Z,U − Z]τ ≤ 2−mη and hence n m n m −m EQ[[U − U ,U − U ]τ ] ≤ 4 α. (3.11) ∑ n,k d n,j kj Let us define g = j=1 f b . Then note that

∑d ∫ ∑d ∑d ∫ gn,k dN k = f n,jbkj dN k k=1 k=1 j=1 ∑d ∑d ∑d ∫ n,j kj kl l = f b bs dM k=1 j=1 l=1 (3.12) ∑d ∫ = f n,j dM j j=1 = U n, where in the last but one step, we have used (3.8). Using (3.10) for n ≥ m we have ∫ ∑d τ n − m n − m n,k − m,k 2 k k EQ[[U U ,U U ]τ ] = EQ[ (gs gs ) d[N ,N ]s] (3.13) k=1 0 SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 67 and using (3.11), we conclude ∫ ∑d τ 1 Q( (gn,k − gm,k)2 d[N k,N k] ≥ ) ≤ m4E [ [U n − U m,U n − U m] ] s s m4 Q τ k=1 0 ≤ αm44−m. (3.14)

i i n,i Since EQ[[M ,M ]τ ] < ∞ and g are bounded for all n, i, it follows that i i EQ[[N ,N ]τ ] < ∞. Thus defining a measure Θ on P by ∫ [ ∫ ] ∑d τ k k Θ(E) = 1E(s, ω)d[N ,N ]s(ω) dQ(ω) k=1 0 we get (using (3.11) and (3.13)) ∫ (gm+1,k − gm,k)2 dΘ ≤ α4−m and as a consequence, using Cauchy–Schwartz inequality, we get ∫ ∑∞ √ |gm+1,k − gm,k|dΘ ≤ Θ(Ω)¯ α < ∞. m=1 Defining k m,k gs = lim sup gs , m→∞ it follows that gm,k → gk a.s. Θ and as a consequence, taking limit in (3.14) as n → ∞, we get ∫ ∑d τ 1 Q( (gk − gm,k)2 d[N k,N k] ≥ ) ≤ m44−m. (3.15) s s s m4 k=1 0 Since Q and P and equivalent measures, it follows that ∫ ∑d τ 1 P( (gk − gm,k)2 d[N k,N k] ≥ ) → 0 as m → ∞. (3.16) s s s m4 k=1 0 In view of (3.12), we have for m ≤ n ∫ ∑d τ n n n,i n,j i j [U ,U ]τ = gs gs d[N ,N ]s (3.17) i,j=1 0 and ∫ ∑d τ n − m n − m n,i − m,i n,j − m,j i j [U U ,U U ]τ = (gs gs )(gs gs )d[N ,N ]s. (3.18) i,j=1 0 Taking limit in (3.17) as n → ∞, we get (using Fatou’s lemma) √∑ ∫ √ d τ i j i j ≤ EP[ i,j=1 0 gsgs d[N , N ]] EP[ [Z,Z]τ ] (3.19) 68 RAJEEVA L. KARANDIKAR AND B. V. RAO √ √ n n (since (3.1) implies EP[ [U , U ]τ ] → EP[ [Z,Z]τ ]). Let us define bounded predictable processes ϕj and predictable process hn, h and a P-martingale X as follows: ∑d | i | hs = 1 + gs , (3.20) i=1

j j gs ϕs = , (3.21) hs

∫ ∑d t j j Xt = ϕs dNs . (3.22) j=1 0

j j 1 2 d Since ϕ is predictable, |ϕ | ≤ 1 it follows that X ∈ Cb(M ,M ,...,M ) and ∫ ∑d t j k j k [X,X]t = ϕsϕs d[N ,N ]s. (3.23) j,k=1 0

j j Noting that gs = hsϕs by deﬁnition, we conclude using (3.19) that ∫ ∫ t ∑d t 2 j k j k (hs) d[X,X]s = gsgs d[N ,N ]s 0 j,k=1 0 and hence that √∫ √ τ 2 ≤ EP[ 0 (hs) d[X,X]s] EP[ [Z,Z]τ ]. (3.24) ∫ ∈ L1 Since h = h1[0,τ], we conclude that h m(X) and Y = hdX is a martingale with Yt = Yt∧τ for all t. Observe that ∫ ∑d t n n,k j k j [U ,X]t = gs ϕs d[N ,N ]s k,j=1 0 and hence ∫ t n n [U ,Y ]t = hs d[U ,X]s 0 ∫ ∑d t n,k j k j = gs hsϕs d[N ,N ]s k,j=1 0 ∫ ∑d t n,k j k j = gs gs d[N ,N ]s. k,j=1 0 SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 69

As a consequence n n n n n [U − Y,U − Y ]t = [U ,U ]t − 2[U ,Y ]t + [Y,Y ]t ∫ ∫ ∑d t ∑d t n,k n,j k j − n,k j k j = gs gs d[N ,N ]s 2 gs gs d[N ,N ]s k,j=1 0 k,j=1 0 ∫ ∑d t k j k j + gs gs d[N ,N ]s k,j=1 0 ∫ ∑d t n,k − k n,j − j k j = (gs gs )(gs gs)d[N ,N ]s k,j=1 0 and thus using (3.10) we get ∫ ∑d τ n − n − n,k − k 2 k k EQ[[U Y,U Y ]τ ] = EQ[ (gs gs ) d[N ,N ]s. (3.25) k=1 0 Taking lim inf as n → ∞ on the RHS in (3.13) and using (3.11), we conclude ∫ ∑d τ k − m,k 2 k k ≤ −m EQ[ (gs gs ) d[N ,N ]s] α4 k=1 0 and hence (3.25) yields

n n −n EQ[[U − Y,U − Y ]τ ] ≤ α4 . n n Thus [U − Y,U − Y ]τ → 0 in Q-probability and hence in P-probability. By n n assumption, [U − Z,U − Z]τ → 0 in P-probability. Since n n n n [Y − Z,Y − Z]τ ≤ 2([Y − U ,Y − U ]τ + [Z − U ,Z − U ]τ ) for every n, it follows that

[Y − Z,Y − Z]τ = 0 a.s. P. (3.26)

Since Y,Z are P-martingales such that Zt = Zt∧τ and Yt = Yt∧τ , (3.26)∫ implies Yt − Y0 = Zt − Z0 for all t. Recall that by construction, Y0 = 0, Y = hdX and ∈ L1 ∈ C 1 2 d □ h m(X), X b(M ,M ,...,M ). Thus (3.5) holds.

We now come to the proof of Theorem 2.2. Let Zn ∈ F(M 1,M 2,...,M d) be | n− | → ∈ F 1 2 d such that E[ Zt Zt ] 0 for all t. We have to show that Z (M ,M ,...,M ). ↑ ∞ Using Lemma 3.3, get a sequence of stopping times√ σk and a subsequence { j} j nj ≥ ∞ n such that Y = Z satisﬁes for each k 1, E[ [Z,Z]σk ] < and √ j − j − → ↑ ∞ E[ [Y Z,Y Z]σk ] 0 as j . ˜ ˜ ∈ F 1 2 d Fix k and let Zt = Zt∧σk . We will show that Z (M ,M ,...,M ). This will complete the proof in view of Lemma 3.2. By relabelling, we assume that √ n − n − → ↑ ∞ E[ [Z Z,Z Z]σk ] 0 as n . (3.27) 70 RAJEEVA L. KARANDIKAR AND B. V. RAO √ n ∈ F 1 2 d n n ∞ Since Z (M ,M ,...,M ) and E[ [Z , Z ]σk ] < (at least for large n, say ∗ ∗ n 1 2 d n ≥ n ), using Lemma 3.4, for n ≥ n we can get U ∈ Cb(M ,M ,...,M ) such that √ 1 E[ [U n − Zn, U n − Zn] ] ≤ . (3.28) σk n (3.27) and (3.28) give √ n − ˜ n − ˜ → E[ [U Z,U Z]σk ] 0. Thus Z˜ ∈ F(M 1,M 2,...,M d) in view of Lemma 3.5 n n Now we turn to proof of Theorem 2.1. Let ξ ∈ KT be such that ξ → ξ L1 F n n n ∈ F 1 2 d in (Ω, , P). Let ξ = XT where X (M ,M ,...,M ). Let us deﬁne n n n ∈ F 1 2 d n Zt = Xt∧T . Then Z (M ,M ,...,M ) and the assumption on ξ implies n → | F L1 F ∀ Zt Zt = E[ξ t] in (Ω, , P) t. 1 2 d Thus Theorem 2.2 implies Z ∈ F(M ,M ,...,M ) and thus ξ = ZT belongs to KT .

4. Sigma-martingales ∫ Let M be a martingale, f ∈ L(M) and Z = f dM. Then Z is a local mar- ∈ L1 tingale if and only if f m(M). In answer to a question raised by P. A.∫ Meyer, Chou [3] introduced a class Σm of semimartingales consisting of Z = f dM for f ∈ L∫(M). Emery [7] constructed example of f, M such that f ∈ L(M) but Z = f dM is not a local martingale. Such processes occur naturally in mathematical finance and have been called sigma-martingales by Delbaen and Schachermayer[6]. Definition 4.1. A semimartingale X is said to be a sigma-martingale if ∃ϕ ∈ L(X) such that ϕ is (0, ∞) valued and ∫ t Mt = ϕs dXs (4.1) 0 is a martingale. Our first observation is: Lemma 4.2. Every local martingale N is a sigma-martingale. ↑ ∞ Proof. Let ηn be a sequence of stopping times such that Nt∧ηn is a martingale,

σn = inf{t ≥ 0 : |Nt| ≥ n or |Nt−| ≥ n} ∧ n ∧ and τn =√σn ηn. It follows that Nt∧τn is a uniformly integrable martingale and ∞ an = E[ [N,N]τn ] < . Deﬁne ∞ 1 ∑ 1 h = { }(s) + (s). s 1 0 n 1(τn−1,τn] 1 + |N0| 2 (1 + an) n=1 ∫ Then h being bounded belongs to L(N) and M = hdN is a local martingale with √ sup E[ [M,M]t] < ∞. (4.2) t<∞ SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 71

Thus M is a uniformly integrable martingale. Since h is (0, ∞) valued by deﬁnition, it follows that N is a sigma-martingale. □

This leads to Lemma 4.3. A semimartingale X is a sigma-martingale if and only if there exists a uniformly integrable martingale M satisfying (4.2) and a predictable process ψ ∈ L(M) such that ∫ t Xt = ψs dMs. (4.3) 0 Proof. Let X be given by (4.3) with M being a martingale satisfying (4.2) and ψ ∈ L(M), then defining ∫ 1 t gs = 2 ,Nt = gs dXs (1 + (ψs) ) 0 ∫ it follows that N = gψdM. Since gψ is bounded by 1 and M satisfies (4.2), it follows that N is a martingale. Thus X is a sigma-martingale. Conversely, given∫ a sigma-martingale X and a (0, ∞) valued predictable process ∫ϕ such that∫ N = ϕdX is a martingale, get h as in Lemma 4.2 and let M = hdN = hϕdX. Then M is a uniformly integrable martingale that satisfies (4.2) and hϕ is a (0, ∞) valued predictable process. □

From the deﬁnition, it is not obvious that sum of sigma-martingales is also a sigma-martingale, but this is so as the next result shows.

1 2 Theorem 4.4. Let X ,X be sigma-martingales and a1, a2 be real numbers. Then 1 2 Y = a1X + a2X is also a sigma-martingale. Proof. Let ϕ1, ϕ2 be (0, ∞) valued predictable processes such that ∫ t i i i Mt = ϕs dXs, i = 1, 2 0

1 2 i ξs are uniformly integrable martingales. Then, writing ξ = min(ϕ , ϕ ) and ηs = i , ϕs it follows that ∫ ∫ t t i i i i Nt = ηs dMs = ξs dXs 0 0 are uniformly integrable martingales since ηi is bounded by one. Clearly, Y = 1 2 i a1X + a2X is a semimartingale and ξ ∈ L(X ) for i = 1, 2 implies ξ ∈ L(Y ) and ∫ t 1 2 ξs dYs = a1Ns + a2Ns 0 is a uniformly integrable martingale. Since ξ is (0, ∞) valued predictable process, it follows that Y is a sigma-martingale. □

The following result gives conditions under which a sigma-martingale is a local martingale. 72 RAJEEVA L. KARANDIKAR AND B. V. RAO

Lemma 4.5. Let X be a sigma-martingale with X0 = 0. Then X is a local martingale if and only if there exists a sequence of stopping times τn ↑ ∞ such that √ ∞ ∀ E[ [X,X]τn ] < n. (4.4) Proof. Let X be a sigma-martingale and ϕ, ψ, M be such that (4.1), (4.2) holds. 1 Let ψs = ϕ and as noted above, (4.3) holds. Then s ∫ t 2 [X,X]t = (ψs) d[M,M]s. 0 k Defining ψs = ψs1{|ψs|≤k}, it follows that ∫ t k k X = ψs dMs 0 is a uniformly integrable martingale. Noting that for k ≥ 1 ∫ t − k − k 2 [X X ,X X ]t = (ψs) 1{k<|ψs|} d[M,M]s 0 the assumption (4.4) implies that for each n fixed, √ − k − k → → ∞ E[ [X X , X X ]τn ] 0 as k . The Burkholder–Davis–Gundy inequality (p = 1) implies that for each n fixed, | − k| → → ∞ E[ sup Xt Xt ] 0 as k . 0≤t≤τn [n] and as a consequence Xt = Xt∧τn is a martingale for all n and so X is a local martingale. Conversely, if X is a local martingale with X0 = 0, and σn are stop ∞ { times increasing to such that Xt∧σn are martingales then defining ζn = inf t : | | ≥ } ∧ E | | ∞ Xt n and τn = σn ζn, it follows that [ Xτn ] < and since | | ≤ | | sup Xt n + Xτn t≤τn E | | ∞ it follows that [supt≤τn Xt ] < . Thus, (4.4) holds in view of Burkholder– Davis–Gundy inequality (p = 1). □ The previous result gives us: Corollary 4.6. A bounded sigma-martingale X is a martingale. Proof. Since X is bounded, say by K, it follows that jumps of X are bounded by 2K. Thus jumps of the increasing process [X,X] are bounded by 4K2 and thus X satisfies (4.4) for τn = inf{t ≥ 0 : [X,X]t ≥ n}. Thus X is a local martingale and being bounded, it is a martingale. □ Essentially the same argument also gives that if a sigma-martingale X satisfies |Xt| ≤ ξ where ξ is square integrable, then X is a martingale, in fact a square integrable martingale. Here is a variant of the example given by Emery [7] of a sigma-martingale that is not a local martingale. Let τ be a random variable with exponential SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 73 distribution (assumed to be (0, ∞) valued without loss of generality) and ξ with P(ξ = 1) = P(ξ = −1) = 0.5, independent of τ. Let

Mt = ξ1[τ,∞)(t) 1 and F = σ(M : s ≤ t). Easy to see that M is a martingale. Let f = ∞ (t) t ∫ s t t 1(0, ) t and Xt = 0 f dM. Then X is a sigma-martingale and 1 [X,X] = 1 ∞ (t). t τ 2 [τ, ) For any stopping time σ, it can be checked that σ is a constant on σ <√ τ and thus ≥ ∧ ≥ if σ is not identically equal to 0, σ (τ a) for some a > 0. Thus,√ [X,X]σ 1 ∞ τ 1{τ

Es Es 1 d sigma-martingale under Q. Thus P(X) is a convex set. Since (X ,...,X ) = ∩d Es j Es 1 d j=1 (X ) it follows that (X ,...,X ) is convex. Es 1 d E˜s 1 d □ Convexity of P(X ,...,X ) and P(X ,...,X ) follows from this. For sigma-martingales M 1,M 2,...,M d the classes C(M 1,M 2,...,M d) and F(M 1,...,M d) are defined exactly as they were in case M 1,...,M d are martingales. In particular, they remains subsets of the class of martingales M. Lemma 4.9. Let M 1,...,M d be sigma-martingales and let ϕj be (0, ∞) valued predictable processes such that ∫ t j j j Nt = ϕs dMs (4.5) 0 are uniformly integrable martingales. Then C(M 1,M 2,...,M d) = C(N 1,N 2,...,N d), (4.6) F(M 1,M 2,...,M d) = F(N 1,N 2,...,N d). (4.7) ∫ j j −1 j j j ∈ C 1 2 d Proof. Let ψs = (ϕs) . Note that M = ψ dN . If Y (M ,M ,...,M ) is given by ∫ ∑d t j j j ∈ L j Yt = fs dMs , f (M ) (4.8) 0 j=1 ∫ ∫ then defining gj = f jψj, we can see that gj ∈ L(N j) and f j dM j = gj dN j. Thus ∫ ∑d t j j j ∈ L j Yt = gs dNs , g (N ). (4.9) j=1 0 Similarly, if Y ∈ C(N 1,N 2,...,N d) is given by (4.9), then defining f j = ϕjgj, we can see that Y satisfies (4.8). Thus (4.6) is true. Now (4.7) follows from (4.6). □

5. Integral Representation with Respect to Martingales Let M 1,...,M d be sigma-martingales. Deﬁnition 5.1. A sigma-martingale N is said to have an integral representation with respect to M 1,...,M d if N ∈ F(M 1,M 2,...,M d) or in other words, ∃Y ∈ C(M 1,M 2,...,M d) and f ∈ L(Y ) such that ∫ t Nt = N0 + fs dYs ∀t. (5.1) 0 Here is an observation needed later. Lemma 5.2. Let M be a P-martingale. Let Q be a probability measure equivalent dQ | F to P. Let ξ = dP and let Z be the r.c.l.l. martingale given by Zt = EP[ξ t]. Then (i) M is a Q-martingale if and only if MZ is a P-martingale. (ii) M is a Q-local martingale if and only if MZ is a P-local martingale. (iii) If M is a Q-local martingale then [M,Z] is a P-local martingale. (iv) If M is a Q-sigma-martingale then [M,Z] is a P-sigma-martingale. SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 75

Proof. For a stopping time σ, let η be a non-negative Fσ measurable random variable. Then

EQ[η] = EP[ηξ] = EP[ηE[ξ | Fσ] ] = EP[ηZσ].

Thus Ms is Q integrable if and only if MsZs is P-integrable. Further, for any stopping time σ, EQ[Mσ] = EP[MσZσ]. (5.2) Thus (i) follows from the observation that an integrable adapted process N is a martingale if and only if E[Nσ] = E[N0] for all bounded stopping times σ. For (ii), if M is a Q-local martingale, then get stopping times τn ↑ ∞ such that for each n,

Mt∧τn is a martingale. Then we have

EQ[Mσ∧τn ] = EP[Mσ∧τn Zσ∧τn ]. (5.3)

Thus Mt∧τn Zt∧τn is a P-martingale and thus MZ is a P- local martingale. The converse follows similarly. For (iii), note that ∫ ∫ t t MtZt = M0Z0 + Ms− dZs + Zs− dMs + [M,Z]t (5.4) 0 0 and the two stochastic integrals appearing in (5.4) are P local martingales, the ∫result follows from (ii). For (iv), representing the Q sigma-martingale M as M = ψdN, where N is a Q martingale and ψ ∈ L(N), we see ∫ t [M,Z] = ψs d[N,Z]s. 0 By (iii), [N,Z] is a Q sigma-martingale and hence [M,Z] is a Q sigma-martingale. □

The main result on integral representation is:

1 d Theorem 5.3. Let M ,...,M be sigma-martingales on (Ω, F, P). Suppose F0 is trivial. Then the following are equivalent. (i) All bounded martingales admit representation with respect to M 1,...,M d. (ii) All uniformly integrable martingales admit representation with respect to M 1,...,M d. (iii) All sigma-martingales admit representation with respect to M 1,...,M d. (iv) P is an extreme point of the convex set Es(M 1,...,M d). E˜s 1 d { } (v) P(M ,...,M ) = P . Es 1 d { } (vi) P(M ,...,M ) = P . Proof. Since every bounded martingale is uniformly integrable and a uniformly integrable martinagle is a sigma-martingale, we have (iii)⇒ (ii) ⇒ (i). (i) ⇒ (ii) is an easy consequence of Theorem 2.2- given a uniformly integrable martingale Z, let ξ = limt→∞ Zt, so that Zt = E[ξ | Ft]. For n ≥ 1, let us deﬁne martingales Zn as follows: n | F Zt = E[ξ1{|ξ|≤n}] t] 76 RAJEEVA L. KARANDIKAR AND B. V. RAO where take the r.c.l.l. version of the martingale. It is easy to see that Zn are bounded martingales and in view of (i), Zn ∈ F(M 1,M 2,...,M d). Moreover, for n ≥ t | n − | ≤ E[ Zt Zt ] E[ξ1{|ξ|>n}] | n − | → ∈ F 1 2 d and hence for all t, E[ Zt Zt ] 0. Thus Z (M ,M ,...,M ) by Theorem 2.2. This proves (ii). We next prove (ii) ⇒ (iii). Let X be a sigma-martingale. In view of Lemma 4.3, get a uniformly integrable martingale N and a predictable process ψ such that ∫ X = ψdN. ∫ t ∈ C 1 2 d Let Nt = N0 + 0 fs dYs where Y (M ,M ,...,M ). Then we have ∫ t Xt = X0 + fsψdYs 0 and thus X admits an integral representation with respect to M 1,...,M d. s 1 2 d Suppose (v) holds and suppose Q1, Q2 ∈ E (M ,M ,...,M ) and P = αQ1 + (1 − α)Q2. It follows that Q1, Q2 are absolutely continuous with respect to P and ∈ E˜s 1 2 d hence Q1, Q2 P(M ,M ,...,M ). In view of (v), Q1 = Q2 = P and thus P is an extreme point of Es(M 1,M 2,...,M d) and so (iv) is true. Es 1 2 d ⊆ E˜s 1 2 d Since P(M ,M ,...,M ) P(M ,M ,...,M ), it follows that (v) implies (vi). ∈ E˜s 1 2 d On the other hand, suppose (vi) is true and Q P(M ,M ,...,M ). Then 1 ∈ Es 1 2 d Q1 = 2 (Q + P) P(M , M ,...,M ). Then uniqueness in (vi) implies Q1 = P and hence Q = P and thus (v) holds. Till now we have proved (i) ⇐⇒ (ii) ⇐⇒ (iii) and (iv) ⇐ (v) ⇐⇒ (vi). To complete the proof, we will show (iii) ⇒ (v) and (iv) ⇒ (i). ∈ E˜s 1 2 d Suppose that (iii) is true and let Q P(M ,M ,...,M ). Now let ξ be the Radon–Nikodym derivative of Q with respect to P. Let R denote the r.c.l.l. martingale: Rt = E[ξ | Ft]. Since F0 is trivial, N0 = 1. In view of property (iii), we can get Y ∈ C(M 1,M 2,...,M d) and a predictable processes f ∈ L(Y ) such that ∫ t Rt = 1 + fs dYs. (5.5) 0 Note that ∫ t 2 [R,R]t = fs d[Y,Y ]s. (5.6) 0 Since M j is a sigma-martingale under Q for each j, it follows that Y is a Q sigma- martingale. By Lemma 5.2, this gives [Y,R] is a P sigma-martingale and hence ∫ t k Vt = fs1{|fs|≤k} d[Y,R]s (5.7) 0 is a P sigma-martingale. Noting that ∫ t [Y,R]t = fs d[Y,Y ]s 0 SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 77 we see that ∫ t k 2 Vt = fs 1{|fs|≤k} d[Y,Y ]s. (5.8) 0 Thus we can get (0, ∞) valued predictable processes ϕj such that ∫ t k k k Ut = ϕs dVs 0 k k k is a martingale. But U is a non-negative martingale with U0 = 0. As a result U is identically equal to 0 and thus so is V k. It then follows that (see (5.6)) [R,R] = 0 E˜s 1 2 d which yields R is identical to 1 and so Q = P. Thus P(M ,M ,...,M ) is a singleton. Thus (iii) ⇒ (v). To complete the proof, we will now prove that (iv) ⇒ (i). Suppose P is an extreme point of Es(M 1,M 2,...,M d). Since M j is a sigma- martingale under P, we can choose (0, ∞) valued predictable ϕj such that ∫ t j j j Nt = ϕs dMs 0 is a uniformly integrable martingale under P and as seen in Lemma 4.9, we then have F(M 1,M 2,...,M d) = F(N 1,N 2,...,N d). Suppose (i) is not true. We will show that this leads to a contradiction. So suppose S is a bounded martingale that does not admit representation with respect to M 1,M 2,...,M d, i.e. S ̸∈ F(M 1,M 2,...,M d) = F(N 1,N 2,...,N d), then for some T , 1 2 d ST ̸∈ KT (N ,N ,...,N ) 1 2 d 1 We have proven in Theorem 2.1 that KT (N ,N ,...,N ) is closed in L (Ω, F, P). 1 Since KT is not equal to L (Ω, FT , P), by the Hahn–Banach Theorem, there exists ξ ∈ L∞(Ω, F , P), P(ξ ≠ 0) > 0 such that T ∫

ηξdP = 0 ∀η ∈ KT .

Then for all constants c, we have ∫ ∫

η(1 + cξ)dP = ηdP ∀η ∈ KT . (5.9)

Since ξ is bounded, we can choose a c > 0 such that 1 P(c|ξ| < ) = 1. 2 Now, let Q be the measure with density η = (1 + cξ). Then Q is a probability measure. Thus (5.9) yields ∫ ∫

ηdQ = ηdP ∀η ∈ KT . (5.10)

≤ ≤ j ∈ K For any bounded stop time τand 1 j d, Nτ∧T T and hence j j j EQ[Nτ∧T ] = EP[Nτ∧T ] = N0 (5.11) 78 RAJEEVA L. KARANDIKAR AND B. V. RAO

On the other hand, j j EQ[Nτ∨T ] = EP[ηNτ∨T ] j | F = EP[EP[ηNτ∨T T ]] = E [ηE [N j | F ]] P P τ∨T T (5.12) = EP[ηNT ]

= EQ[NT ] j = N0 . j where we have used the facts that η is FT measurable, N is a P martingale and (5.11). Now j j j − j j EQ[Nτ ] = EQ[Nτ∧T ] + EQ[Nτ∨T ] EQ[NT ] = N0 . Thus N j is a Q martingale and since ∫ t j 1 j M = j dNs 0 ϕs it follows that M j is a Q sigma-martingale. Thus Q ∈ Es(M 1,...,M d). Sim- ilarly, if Q˜ is the measure with density η = (1 − cξ), we can prove that Q˜ ∈ Es 1 d 1 ˜ (M ,...,M ). Since P = 2 (Q + Q), this contradicts the assumption that P is an extreme point of Es(M 1,...,M d). Thus (iv) ⇒ (i). This completes the proof. □

6. Completeness of Markets Let the (discounted) prices of d securities be given by X1,...,Xd. We assume that Xj are semimartingales and that they satisfy the property NFLVR so that an ESMM exists. See [6]. Theorem 6.1. (The Second Fundamental Theorem Of Asset Pricing) 1 d F Es 1 d Let X ,...,X be semimartingales on (Ω, , P) such that P(X ,...,X ) is non- empty. Suppose F0 is trivial. Then the following are equivalent:

(a) For all T < ∞, for all FT measurable bounded random variables ξ (bounded by say K), there exist gj ∈ L(Xj) with ∫ ∑d t j j Yt = gs dXs (6.1) j=1 0 ∫ t a constant c and f ∈ L(Y ) such that | fs dYs| ≤ 2K and ∫ 0 T ξ = c + fs dYs. (6.2) 0 Es 1 d (b) The set P(X ,...,X ) is a singleton. Es 1 d { } F Proof. First suppose that P(X ,...,X ) = Q . Given a bounded T measurable random variable ξ, consider the martingale

Mt = EQ[ξ | Ft]. SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 79

Note that M is bounded by K. In view of the equivalence of (i) and (v) in Theorem 5.3, we get that M admits a representation with respect to X1,...,Xd - thus we get gj ∈ L(Xj) and f ∈ L(Y ) where Y is given by by (6.1), with ∫ t Mt = M0 + fs dYs. 0 F ∫Since 0 is trivial, M0 is a constant. Since M is bounded by K, it follows that t 0 fs dYs is bounded by 2K. Thus (b) implies (a). Now suppose (a) is true. Let Q be an ESMM. Let Mt be a martingale. We will show that M ∈ F(X1,...,Xd), i.e. M admits integral representation with respect to X1,...,Xd. In view of Lemmas 3.2 and 4.9, suﬃces to show that for 1 d each T < ∞, N ∈ F(X ,...,X ), where N is deﬁned by Nt = Mt∧T . Let ξ = NT . Then in view of assumption (a), we have ∫ T ξ = c + fs dYs 0 ∫ ∈ L t with Y given by (6.1), a constant c and f (Y ) such that Ut = 0 fs dYs is bounded. Since U is a sigma-martingale that is bounded, it follows that U is a martingale. It follows that ∫ t Nt = c + fs dYs, 0 ≤ t ≤ T. 0 Thus N ∈ F(X1,...,Xd). We have proved that (i) in Theorem 5.3 holds and hence (v) holds, i.e. the ESMM is unique. □

Appendix I: For a non-negative deﬁnite d×d symmetric matrix C, the eigenvalue-eigenvector decomposition gives us a representation C = BT DB, (A.1) where B is a d × d orthogonal matrix (A.2) and D is a d × d diagonal matrix. (A.3) This decomposition is not unique. Note that for each non-negative deﬁnite symmetric matrix C, the set ΓC of pairs (B,D) of d×d matrices satisfying (A.1)-(A.3) is compact. Thus it admits a measurable selection - in other words, for each C, we can pick θ(C) = (B,D) ∈ ΓC in such a way so that θ is a measurable mapping. (See [8] or Corollary 5.2.6 [17]). Of course, (A.1) implies that BCBT is a diagonal matrix.

II: Let D be a σ-ﬁeld on a non-empty set Γ and for 1 ≤ i, j ≤ d, λij be signed D ∈ D measures on (Γ, ) such that for all E ∑, the matrix((λij(E))) is a symmetric d ≤ ≤ non-negative deﬁnite matrix. Let µ(E) = i=1 λii(E). Then for 1 i, j d there 80 RAJEEVA L. KARANDIKAR AND B. V. RAO

ij dλij ∈ exist versions c of the Radon–Nikodym derivate dµ such that for all γ Γ, the matrix ((cij(γ))) is non-negative deﬁnite. To see this, for 1 ≤ i ≤ j ≤ d let f ij be a version of the Radon–Nikodym dλij ji ij derivative dµ and let f = f . For rationals r1, r2, . . . , rd, let ∑ { ij } Ar1,r2,...,rd = γ : rirjf (γ) < 0 . ij

Then µ(Ar1,r2,...,rd ) = 0 and hence µ(A) = 0 where ∪{ } A = Ar1,r2,...,rd : r1, r2, . . . , rd rationals . The required version is now given by

ij ij c (γ) = f (γ)1Ac (γ).

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Rajeeva L. Karandikar: Chennai Mathematical Institute, Chennai 603103, India E-mail address: [email protected] B. V. Rao: Chennai Mathematical Institute, Chennai 603103, India E-mail address: [email protected]