aa r X i v : . [ q -f i n . M F ] J a n Instantaneous Arbitrage and the CAPM
Lars Tyge NielsenDepartment of MathematicsColumbia UniversityJanuary 2019 ∗ Abstract
This paper studies the concept of instantaneous arbitrage in con-tinuous time and its relation to the instantaneous CAPM. Absenceof instantaneous arbitrage is equivalent to the existence of a tradingstrategy which satisfies the CAPM beta pricing relation in place of themarket. Thus the difference between the arbitrage argument and theCAPM argument in Black and Scholes [3, 1973] is this: the arbitrageargument assumes that there exists some portfolio satisfying the capmequation, whereas the CAPM argument assumes, in addition, that thisportfolio is the market portfolio.
This paper studies the concept of instantaneous arbitrage in continuous timeand its relation to the instantaneous CAPM.An instantaneous arbitrage trading strategy is an instantaneously risklesstrading strategy whose instantaneous expected excess return is always non-negative and sometimes positive.We define a market to be instantaneously arbitrage-free if it is not possibleto construct a zero-value instantaneous arbitrage trading strategy. Thisdefinition is independent of any choice of a potentially non-unique interest ∗ Previous Version May 2006
We consider a securities market where the uncertainty is represented by acomplete probability space (Ω , F , P ) with a filtration F = {F t } t ∈T and a K -dimensional process W , which is a Wiener process relative to F .A cumulative dividend process is a measurable adapted process D with D (0) = 0.Suppose a security has cumulative dividend process D and price process S .Define the cumulative gains process G of this security as the sum of the priceprocess and the cumulative dividend process: G = S + D Assume that G is an Itˆo process. It follows that G will be continuous,adapted, and measurable. Since D is adapted and measurable, so is S .Since D (0) = 0, G (0) = S (0).An ( N + 1)-dimensional securities market model (based on F and W ) willbe a pair ( ¯ S, ¯ D ) of measurable and adapted processes ¯ S and ¯ D of dimension N + 1, interpreted as a vector of price processes and a vector of cumulativedividends processes, such that ¯ D (0) = 0 and such that ¯ G = ¯ S + ¯ D is an Itˆoprocess with respect to F and W . The process ¯ G = ¯ S + ¯ G is the cumulativegains processes corresponding to ( ¯ S, ¯ D ).Write ¯ G ( t ) = ¯ G (0) + Z t ¯ µ ds + Z t ¯ σ dW where ¯ µ is an N + 1 dimensional vector process in L and ¯ σ is an ( N + 1) × K dimensional matrix valued process in L .Here, L is the set of adapted, measurable, and pathwise integrable pro-cesses, and L is the set of adapted, measurable, and pathwise square inte-grable processes. 4 trading strategy is an adapted, measurable ( N +1)-dimensional row-vector-valued process ¯∆.The value process of a trading strategy ¯∆ in securities model ( ¯ S, ¯ D ) is theprocess ¯∆ ¯ S .The set of trading strategies ¯∆ such that ¯∆¯ µ ∈ L and ¯∆¯ σ ∈ L , will bedenoted L ( ¯ G ).In general, if X is an n -dimensional Itˆo process, X ( t ) = X (0) + Z t a ds + Z t b dW then L ( X ) is the set of adapted, measurable, ( n × K )-dimensional processes γ such that γa ∈ L and γb ∈ L .If ¯∆ is a trading strategy in L ( ¯ G ), then the cumulative gains process of¯∆, measured relative to the securities market model ( ¯ S, ¯ D ), is the process G ( ¯∆; ¯ G ) defined by G ( ¯∆; ¯ G )( t ) = ¯∆(0) ¯ G (0) + Z t ¯∆ d ¯ G for all t ∈ T .A trading strategy ¯∆ in L ( ¯ G ) is self-financing with respect to ( ¯ S, ¯ D ) if¯∆ ¯ S = G ( ¯∆; ¯ G )or ¯∆( t ) ¯ S ( t ) = ¯∆(0) ¯ S (0) + Z t ¯∆ d ¯ G Generally, if ¯∆ is a trading strategy in L ( ¯ G ) which may not be self-financing,then the cumulative dividend process of ¯∆ with respect to ( ¯ S, ¯ D ) is theprocess D ( ¯∆; ¯ S, ¯ D ) defined by¯∆ ¯ S + D ( ¯∆; ¯ S, ¯ D ) = G ( ¯∆; ¯ G )The process D ( ¯∆; ¯ S, ¯ D ) is adapted and measurable and has initial value D ( ¯∆; ¯ S, ¯ D )(0) = 0.A money market account for ( ¯ S, ¯ D ) is a self-financing trading strategy ¯ b (or a security that pays no dividends) whose value process is positive and5nstantaneously riskless (has zero dispersion). We denote its value processby M : M = ¯ b ¯ S .If M is the value process of a money market account, then it must have theform M ( t ) = M (0) exp (cid:26)Z t r ds (cid:27) for some r ∈ L (the interest rate process) and some M (0) > D be a cumulative dividend process. Then Dr ∈ L if and only if D ∈ L (1 /M ) = L ( M ). If so, then the cumulative dividend process in unitsof the money market account is D /M ( t ) = D ( t ) /M ( t ) + Z t D rM ds
If (
S, D ) is a security model with D ∈ L ( M ), and if the associated gainsprocess G = S + D has differential dG = µ dt + σ dW then the cumulative gains process in units of the money market account is G /M ( t ) = G ( t ) /M ( t ) + Z t D rM ds and dG /M ( t ) = µ − rSM dt + σM dW To begin with, we define instantaneous arbitrage in a way that does notassume the existence of a money market account or an instantaneous interestrate process. On this basis we show that absence of instantaneous arbitrageimplies uniqueness of the instantaneous interest rate process and of the valueprocess of the money market account (up to a positive scaling factor). Wethen re-formulate instantaneous arbitrage in various equivalent ways whichdo involve the instantaneous interest rate process.6f ¯∆ is a trading strategy (not necessarily self-financing), then the processes¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ and √ ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ will be called the instantaneous dollar returnvariance , and the instantaneous dollar return standard deviation of ¯∆, re-spectively.A trading strategy ¯∆ is instantaneously riskless if ¯∆¯ σ = 0 almost every-where.A zero-value instantaneous arbitrage trading strategy is an instantaneouslyriskless trading strategy ¯∆ such that ¯∆ ¯ S = 0 almost everywhere, ¯∆¯ µ ≥ µ > S, ¯ D ) is instantaneously arbitrage-free if there exists no zero-value instantaneous arbitrage trading strategy. Proposition 1
Suppose the securities market model ( ¯ S, ¯ D ) is instantaneouslyarbitrage-free. If ¯ b and ¯ b are money market accounts with interest rate pro-cesses r and r and value processes M and M , then r and r are almosteverywhere identical, and M /M (0) and M /M (0) are indistinguishable. The proofs of Propostion 1 and other results in this and the following sectionare in Appendix A.Now let ¯ b be a money market account with value process M and interestrate process r .If ¯∆ is a trading strategy (not necessarily self-financing), then the processes¯∆(¯ µ − r ¯ S ) will be called the instantaneous excess expected dollar return of¯∆.Assume that ¯ D ∈ L (1 /M ).An instantaneous arbitrage trading strategy is an instantaneously risklesstrading strategy ¯∆ ∈ L ( ¯ G /M ) such that ¯∆(¯ µ − r ¯ S ) ≥ µ − r ¯ S ) > d ¯ G /M = 1 M (¯ µ − r ¯ S ) dt + 1 M ¯ σ dW
7t follows that a zero-value trading strategy is in L ( ¯ G ) if and only if it isin L ( ¯ G /M ). Therefore, the definition of an instantaneous arbitrage tradingstrategy is consistent with the earlier definition of a zero-value instantaneousarbitrage trading strategy. A zero-value instantaneous arbitrage tradingstrategy is nothing other than an instantaneous arbitrage trading strategy¯∆ such that ¯∆ ¯ S = 0 almost everywhere.The following proposition shows that a number of possible definitions ofthe concept of an instantaneously arbitrage-free securities market model areequivalent. Proposition 2
The following statements are equivalent:1. ( ¯ S, ¯ D ) is instantaneously arbitrage-free (there exists no zero-value in-stantaneous arbitrage trading strategy)2. There exists no instantaneous arbitrage trading strategy3. There exists no self-financing instantaneous arbitrage trading strategy4. Every instantaneously riskless trading strategy has zero instantaneousexpected excess return almost everywhere A vector of instantaneous prices of risk is an adapted measurable K -dimensionalrow-vector-valued process λ such that¯ µ − r ¯ S = ¯ σλ ⊤ almost everywhere .Proposition 3 states the main characteristics of an instantaneously arbitrage-free securities market model. Proposition 3
The following statements are equivalent:1. ( ¯ S, ¯ D ) is instantaneously arbitrage free Some authors, including Nielsen [7, 1999], require in addition that λ ∈ L . . There exists a vector of instantaneous prices of risk3. There exists a trading strategy ¯ ψ such that ¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯ ψ ⊤ almost everywhere If ¯ ψ is a trading strategy such as the one in (3) of Proposition 3, thenaccording to Proposition 4 below, the process λ ∗ = ¯ ψ ¯ σ will be the minimalvector of prices of risk, in the sense that for any other vector λ of prices ofrisk, λ ∗ λ ∗⊤ ≤ λλ ⊤ almost everywhere. Proposition 4
Suppose λ is a vector of instantaneous prices of risk and ¯ ψ is a trading strategy such that ¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯ ψ ⊤ almost everywhere. Set λ ∗ = ¯ ψ ¯ σ . Then λ ∗ λ ∗⊤ ≤ λλ ⊤ almost everywhere. If ¯∆ is a trading strategy, let b ¯∆ be the vector of fundamental betas of theindividual securities with respect to ¯∆: b ¯∆ = (cid:26) σ ¯ σ ⊤ ¯∆ ⊤ ¯ σ ¯ σ ⊤ ¯∆ ⊤ if ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ = 00 otherwiseSay that a trading strategy ¯∆ satisfies the CAPM equation if¯ µ − r ¯ S = b ¯∆ ¯∆(¯ µ − r ¯ S )almost everywhere. Theorem 1 ( ¯ S, ¯ D ) is instantaneously arbitrage free if and only if there ex-ists a trading strategy ¯∆ which satisfies the CAPM equation. Theorem 1 implies that the difference between the arbitrage argument andthe CAPM argument in Black and Scholes [3, 1973] is this: the arbitrageargument assumes that there exists some portfolio satisfying the capm equa-tion, whereas the CAPM argument assumes, in addition, that this portfoliois the market portfolio. 9
Appendix A: Proofs
Most proofs are in this appendix. The proof of Proposition 3 depends onsome concepts of “measurable linear algebra,” which are developed in Ap-pendix B, culminating in Proposition 7.
Proof of Proposition 1 :Suppose it is not true that r and r are almost everywhere identical. As-sume without loss of generality that r > r on a set of positive measure.Define a trading strategy ¯ b by¯ b = 1 r ≥ r (cid:18) M ¯ b − M ¯ b (cid:19) + 1 r >r (cid:18) M ¯ b − M ¯ b (cid:19) Then¯ b ¯ S = 1 r ≥ r (cid:18) M ¯ b ¯ S − M ¯ b ¯ S (cid:19) + 1 r >r (cid:18) M ¯ b ¯ S − M ¯ b ¯ S (cid:19) = 1 r ≥ r (cid:18) M M − M M (cid:19) + 1 r >r (cid:18) M M − M M (cid:19) = 0¯ b ¯ µ = 1 r ≥ r (cid:18) M ¯ b ¯ µ − M ¯ b ¯ µ (cid:19) + 1 r >r (cid:18) M ¯ b ¯ µ − M ¯ b ¯ µ (cid:19) = 1 r ≥ r (cid:18) M r M − M r M (cid:19) + 1 r >r (cid:18) M r M − M r M (cid:19) = 1 r ≥ r ( r − r ) + 1 r >r ( r − r )which is non-negative almost everywhere and positive on a set of positivemeasure, and¯ b ¯ σ = 1 r ≥ r (cid:18) M ¯ b ¯ σ − M ¯ b ¯ σ (cid:19) + 1 r >r (cid:18) M ¯ b ¯ σ − M ¯ b ¯ σ (cid:19) = 0almost everywhere. Hence, ¯∆ is a zero-value instantaneous arbitrage tradingstrategy, a contradiction.It follows that r and r are almost everywhere identical. This implies that M /M (0) and M /M (0) are indistinguishable. (cid:3) emma 1 Let ¯ b be a money market account with value process M . Assumethat ¯ D ∈ L (1 /M ) . Let ¯∆ be a trading strategy. The trading strategy ¯Θ = ¯∆ + D (cid:16) ¯∆; ¯ S/M, ¯ D /M (cid:17) ¯ b is self-financing with ¯Θ ¯ S/M = G (cid:16) ¯∆; ¯ G /M (cid:17) If ¯∆ is an instantaneous arbitrage trading strategy, then so is ¯Θ . Proof:
It is easily seen that D (cid:16) ¯Θ; ¯ S/M, ¯ D /M (cid:17) = D (cid:16) ¯∆; ¯ S/M, ¯ D /M (cid:17) − D (cid:16) ¯∆; ¯ S/M, ¯ D /M (cid:17) = 0which implies that ¯Θ is self-financing. The process D (cid:16) ¯∆; ¯ S/M, ¯ D /M (cid:17) ¯ b has zero cumulative gains process with respect to ( ¯ S/M, ¯ D /M ). Hence,¯Θ ¯ S/M = G (cid:16) ¯Θ; ¯ G /M (cid:17) = G (cid:16) ¯∆; ¯ G /M (cid:17) Observe that ¯Θ(¯ µ − r ¯ S ) = ¯∆(¯ µ − r ¯ S )and ¯Θ¯ σ = ¯∆¯ σ almost everywhere. It follows that if ¯∆ is an instantaneousarbitrage trading strategy, then so is ¯Θ. (cid:3) Proof of Proposition 2 :It is useful to observe that Statement (4) is equivalent to the following:(5) There exists no trading strategy ¯∆ such that on a set of positive mea-sure, ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ = 0 and ¯∆(¯ µ − r ¯ S ) > A ⊂ Ω × T be the set of ( ω, t ) such that¯∆( ω, t )(¯ µ ( ω, t ) − r ( ω, t ) ¯ S ( ω, t )) > A is a measurable and adapted process, and A is measurable with positive measure.Define the process ¯Θ by ¯Θ = ¯∆ on A and ¯Θ = 0 outside of A . Then ¯Θ ismeasurable and adapted, and, hence, it is a trading strategy. It is an in-stantaneously riskless trading strategy with positive instantaneous expectedexcess return on a set of positive measure, contradicting (4).(2) equivalent to (3): Follows from Lemma 1.(5) implies (2): Obvious.(2) implies (1):If ( ¯ S, ¯ D ) is not instantaneously arbitrage free, then there exists a zero-value instantaneous arbitrage trading strategy, which is, in particular, aninstantaneous arbitrage trading strategy.(1) implies (5):Suppose there exists a trading strategy ¯∆ such that on a set of positivemeasure, ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ = 0 and ¯∆(¯ µ − r ¯ S ) > A ⊂ Ω × T be the set of ( ω, t ) such that¯∆( ω, t )¯ σ ( ω, t )¯ σ ( ω, t ) ⊤ ¯∆( ω, t ) ⊤ = 0and ¯∆( ω, t )(¯ µ ( ω, t ) − r ( ω, t ) ¯ S ( ω, t )) > A is a measurable and adapted process, and A is measurable with positive measure.Define a process ¯Θ by ¯Θ = ¯∆ − ¯∆ ¯ SM ¯ b on A and ¯Θ = 0 outside of A . Then ¯Θ is measurable and adapted, and,hence, it is a trading strategy.On A , ¯Θ¯ σ ¯ σ ⊤ ¯Θ ⊤ = 0, ¯Θ ¯ S = ¯∆ ¯ S − ¯∆ ¯ SM ¯ b ¯ S = 0and ¯Θ¯ µ = ¯∆¯ µ − ¯∆ ¯ SM ¯ b ¯ µ = ¯∆(¯ µ − r ¯ S ) > A , ¯Θ = 0, and, hence, ¯Θ¯ σ ¯ σ ⊤ ¯Θ ⊤ = 0, ¯Θ ¯ S = 0, and ¯Θ¯ µ = 0.This implies that ¯Θ is a zero-value instantaneous arbitrage trading strategy.Hence, ( ¯ S, ¯ D ) is not instantaneously arbitrage free. (cid:3) Proof of Proposition 3 :(3) implies (2):Set λ = ¯ σ ¯ ψ .(2) implies (1):If an instantaneous arbitrage trading strategy ¯∆ exists, then¯∆(¯ µ − r ¯ S ) = ¯ δ ¯ σλ ⊤ = 0almost everywhere, a contradiction.(1) implies (3):From (5) of the proof of Propostion 2, we know that there exists no tradingstrategy ¯∆ such that on a set of positive measure, ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ = 0 and ¯∆(¯ µ − r ¯ S ) = 1From Proposition 7 in Appendix B, it then follows that there exists anadapted, measurable process (a trading strategy) ¯ ψ such that¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯ ψ ⊤ almost everywhere. (cid:3) Proof of Proposition 4 :Observe that ( λ − ¯ ψ ¯ σ )¯ σ ⊤ = (¯ µ − r ¯ S ) ⊤ − λ ∗ ¯ σ ⊤ = 0almost everywhere. Hence, λλ ⊤ = [( λ − ¯ ψ ¯ σ ) + ¯ ψ ¯ σ ][( λ − ¯ ψ ¯ σ ) + ¯ ψ ¯ σ ] ⊤ = ( λ − ¯ ψ ¯ σ )( λ − ¯ ψ ¯ σ ) ⊤ + ¯ ψ ¯ σ ¯ σ ⊤ ¯ ψ ⊤ ≥ ¯ ψ ¯ σ ¯ σ ⊤ ¯ ψ ⊤ = λ ∗ λ ∗⊤ (cid:3) Proof of Theorem 1 :Suppose the trading strategy ¯∆ exists.Pick a measurable set
N ⊂ Ω × T with zero measure such that¯ µ − r ¯ S = b ¯∆ ¯∆(¯ µ − r ¯ S )outside of N .Suppose ¯ γ is a trading strategy such that on a set C of positive measure,¯ γ ¯ σ ¯ σ ⊤ ¯ γ ⊤ = 0 and ¯ γ (¯ µ − r ¯ S ) >
0. Then ¯ µ − r ¯ S = 0 and, hence, ¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ > C \ N . But then¯ γ (¯ µ − r ¯ S ) = ¯ γb ¯∆ ¯∆(¯ µ − r ¯ S ) = ¯ γ ¯ σ ¯ σ ⊤ ¯∆ ⊤ σ ¯ σ ⊤ ¯∆ ⊤ ¯∆(¯ µ − r ¯ S ) = 0on C \ N , a contradiction.Conversely, if ( ¯ S, ¯ D ) is instantaneously arbitrage free, then it follows fromProposition 3 that there exists a trading strategy ¯∆ such that¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯∆ ⊤ almost everywhere.Pick a measurable set N ⊂ Ω × T with zero measure such that¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯∆ ⊤ outside of N .Set A = { ( ω, t ) ∈ Ω × T : ¯∆( ω, t )¯ σ ( ω, t )¯ σ ( ω, t ) ⊤ ¯∆( ω, t ) ⊤ > } and B = { ( ω, t ) ∈ Ω × T : ¯∆( ω, t )¯ σ ( ω, t )¯ σ ( ω, t ) ⊤ ¯∆( ω, t ) ⊤ = 0 } Then the indicator functions 1 A and 1 B are measurable and adapted pro-cesses, and in particular, A and B are measurable.On B \ N , ¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯∆ ⊤ = 0 = b ¯∆ ¯∆(¯ µ − r ¯ S )14n A \ N ,¯ µ − r ¯ S = ¯ σ ¯ σ ⊤ ¯∆ ⊤ = ¯∆(¯ µ − r ¯ S )¯∆¯ σ ¯ σ ⊤ ¯∆ ⊤ ¯ σ ¯ σ ⊤ ¯∆ ⊤ = b ¯∆ ¯∆(¯ µ − r ¯ S ) (cid:3) B Appendix B: Measurable Linear Algebra
A linear equation may have zero, one, or infinitely many solutions. Thisappendix shows that where at least one solution exists, a particular solutionmay be chosen as a measurable function of the parameters of the equation.In the case where the parameters are stochastic processes, we give a dualcharacterization of the existence of a solution process.Let A M,K ⊂ R M × R M × K be the set of pairs ( y, V ) of an M -dimensional column vector y and an( M × K )-dimensional matrix V such that y is in the span of the columns of V , or equivalently, such that there exists a K -dimensional column vector x with y = V x . Proposition 5
The set A M,K is measurable, and there exists a measurablemapping φ M,K : A M,K → R K such that y = V φ ( y, V ) for all ( y, V ) ∈ A M,K . Proposition 5 follows directly from Proposition 6 below by transposition.Let B M,K ⊂ R K × R M × K be the set of pairs ( y, V ) of a K -dimensional row vector y and an ( M × K )-dimensional matrix V such that y is in the span of the rows of V , orequivalently, such that there exists an M -dimensional row vector x with y = xV . 15 roposition 6 The set B M,K is measurable, and there exists a measurablemapping ψ M,K : B M,K → R M such that y = ψ ( y, V ) V for all ( y, V ) ∈ B M,K . The proof of Proposition 6 will be given below after a series of lemmas.
Proposition 7
Let Y and Σ be adapted, measurable processes with valuesin R M and R M × K , respectively. There exists an adapted, measurable process X with values in R K such that Y = Σ X almost everywhere if and only ifthere does not exist an adapted measurable process Z with values in R M suchthat ZY = 1 and Z Σ = 0 on a set of positive measure.
Proof:
By Proposition 5, A M,K is measurable, and the mapping φ M,K : A M,K → R K is measurable and has the property that y = V φ
M,K ( y, V ) forall ( y, V ) ∈ A M,K .If ( Y, Σ) ∈ A M,K almost everywhere, then define X = φ ( Y, Σ). Then X ismeasurable and adapted and Y = Σ X almost everywhere. Hence, a processlike X exists.Suppose a process like X exists. Then almost everywhere, ( Y, Σ) ∈ A M,K .If a process like Z exists, then on a set of positive measure, ( Y, Σ)
6∈ A
M,K ,a contradiction. Hence, no process like Z exists.Finally, if no process like Z exists, then ( Y, Σ) ∈ A M,K almost everywhere.To prove this, assume to the contrary that ( Y, Σ)
6∈ A
M,K on a set of positivemeasure.Let B be the set where ( Y, Σ)
6∈ A
M,K . Then the process 1 B is measurableand adapted.By Proposition 6, B M,K +1 ⊂ R K +1 × R ( K +1) × M is measurable, and the mapping ψ M,K +1 : B M,K +1 → R M has the property that(1 ,
0) = ψ M,K +1 ((1 , , ( y, V ))( y, V )16or all ( y, V ) ∈ R M × R M × K such that ((1 , , ( y, V )) ∈ B M,K +1 . By elemen-tary linear algebra, these are exactly those ( y, V ) such that ( y, V )
6∈ A
M,K .Define a process Z by Z = 0 outside of B and Z = ψ M,K +1 ((1 , , ( Y, Σ))on B . Since 1 B is measurable and adapted, so is Z . On B ,(1 ,
0) = Z ( Y, Σ)which is equivalent to ZY = 1 and Z Σ = 0. (cid:3)
We proceed to the proof of Proposition 6.Let 0 ≤ r ≤ min { M, K } .Lemma 2 says that we can select, in a measurable manner, a set of r inde-pendent linear combinations of the rows of an ( M × K )-dimensional matriceswith rank r .Let ˜ D r ⊂ R M × K denote the set of ( M × K )-dimensional matrix with rank r . Then ˜ D r is a measurable subset of R M × K . Lemma 2
There exists a measurable mapping H : ˜ D r → R r × M such that for each V ∈ ˜ D r , the rows of H ( V ) V span the same linear subspaceof R K as the rows of V . Proof:
Let j be the first of the numbers { , . . . , M } such that the j th row V j − of V is non-zero. Let j be the first of the numbers { j +1 , . . . , M } suchthat V j − and V j − are linearly independent. Whenever j , . . . , j n have beenchosen, and n < r , let j n +1 be the first of the numbers { j n + 1 , . . . , M } suchthat V j − and V j n +1 − are independent. Set H ( V ) i = e j i for each i = 1 , . . . , r .Then H is a measurable mapping, and for each i = 1 , . . . , r , H ( V ) i V = e j i V = V j i − Hence, the r rows of H ( V ) V are a linearly independent subset of the rowsof V . (cid:3) r × K )-dimensional matrix with rank r in a measurable manner.Let D r ⊂ R r × K be the set of matrices with rank r . Lemma 3
There exists a measurable mapping G : D r → R r × r such that for each V ∈ D r , the rows of G ( V ) V are orthonormal and spanthe same linear subspace of R K as the rows of V . Proof:
We shall use the Gramm-Schmidt orthonormalization procedure toconstruct G and the mapping θ : V G ( V ) V : D r → R r × K simultaneously. The mapping θ will be measurable and will have the prop-erty that for each V ∈ D r , the rows of θ ( V ) are orthonormal and have thesame span as the rows of V .For notational simplicity, in this proof, write V i = V i − for the i th row of V ,and write θ i = θ ( V ) i − for the i th row of θ = θ ( V ), i = 1 , . . . , r .Set x = V and θ = 1 k x k x Then θ is a measurable function of V . Set G = 1 k x k (1 , , . . . , G is a measurable function of V , and G V = 1 k x k V = θ Next, project V on θ , let x be the residual, and let θ be the normalizedresidual. Specifically, V = t , θ + x x is orthogonal to θ . Now, V θ ⊤ = t , θ θ ⊤ + x θ ⊤ = t , θ θ ⊤ so that t , = V θ ⊤ θ θ ⊤ which is a measurable function of V . Furthermore, x = V − t , θ = 0since V and θ are independent. Set θ = 1 k x k x Then is θ is a measurable function of V . Set G = 1 k x k [(0 , , , . . . , − t , G ]Then G is a measurable function of V , and G V = 1 k x k [ V − t , θ ] = 1 k x k x = θ Once θ , . . . , θ n have been chosen, where n < r , construct θ n +1 inductivelyas follows. Project V ( n +1) on θ , . . . , θ n , let x n +1 be the residual, and let θ n +1 be the normalized residual. Specifically, V n +1 = n X j =1 t n +1 ,j θj + x n +1 where x n +1 is orthogonal to θ j for j = 1 , . . . , n . Now, for k = 1 , . . . , n , V n +1 θ k = n X j =1 t n +1 ,j θjθ k + x n +1 θ k = t n +1 ,k θ k θ ⊤ k so that t n +1 ,k = V n +1 θ ⊤ k θ k θ ⊤ k which is a measurable function of V . Furthermore, x n +1 = V n +1 − n X j =1 t n +1 ,j θ j = 019ince V n +1 is not spanned by θ , . . . , θ n . Set θ n +1 = 1 k x n +1 k x n +1 Then is θ n +1 is a measurable function of V . Set G n +1 = 1 k x n +1 k e n +1 − n X j − t n +1 ,j G j Then G n +1 is a measurable function of V , and G n +1 V = 1 k x n +1 k V n +1 − n X j − t n +1 ,j θ j = 1 k x n +1 k x n +1 = θ n +1 This completes the construction of G and θ . (cid:3) Proof of Proposition 6 :For each r with 0 ≤ r ≤ min { M, K } , let B r = { ( y, V ) ∈ B M,K : rank( V ) = r } Then B r is measurable, and B M,K is measurable since B M,K = min { M,K } [ r =0 B r To complete the proof, it suffices to show that for each r with 0 ≤ r ≤ min { M, K } , there exists a measurable mapping ψ r : B r → R M such that y = ψ r ( y, V ) V for all ( y, V ) ∈ B r .Let H : ˜ D r → R r × M be the mapping from Lemma 2, and let G : D r → R r × r
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