Classifying Financial Markets up to Isomorphism
aa r X i v : . [ q -f i n . M F ] J u l Classifying Financial Markets up to Isomorphism
John Armstrong
Abstract
Two markets should be considered isomorphic if they are financiallyindistinguishable. We define a notion of isomorphism for financial marketsin both discrete and continuous time. We then seek to identify the distinctisomorphism classes, that is to classify markets.We classify complete one-period markets. We define an invariant ofcontinuous time complete markets which we call the absolute market priceof risk. This invariant plays a role analogous to the curvature in Rieman-nian geometry. We classify markets when the absolute market price ofrisk is deterministic.We show that, in general, markets with non-trivial automorphismgroups admit mutual fund theorems. We prove a number of such the-orems.
Introduction
Two financial markets should be considered equivalent if there is a bijective cor-respondence between the investment strategies in each market which preservesboth the costs and the payoff distributions of these strategies. This intuitionallows us to define a formal notion of isomorphism for financial markets. In thelanguage of category theory [7], we shall define the category of financial markets.We shall then demonstrate that one can prove financially interesting classi-fication theorems. We classify Gaussian markets and complete one-period mar-kets. We also prove a partial classification theorem for complete continuous-timemarkets with a fixed risk-free rate which we will now describe.The minimum number of assets required to replicate an arbitrary contingentclaim gives one basic invariant of such markets, the dimension. The next usefulinvariant we identify is the length of the market-price-of-risk vector, which wecall the absolute market price of risk . While it is easy to define other invariants,this has the advantage of being a local invariant , by which we mean that it canbe calculated from the coefficients of an SDE defining the asset price dynamicsby simple algebra and differentiation. The absolute market price of risk gives abasic invariant of markets up to isomorphism. In this sense it is analogous to theRiemannian curvature, which gives a basic invariant of Riemannian manifoldsup to isometry.We classify continuous-time complete markets whose absolute market priceof risk is deterministic. Markets with constant absolute market price of risk are1etermined up to isomorphism by just their dimension, the risk-free rate andthe absolute market price of risk and are isomorphic to Black–Scholes–Mertonmarkets.Our classification theorems have a number of interesting financial applica-tions.Firstly, one can often use a classification theorem to illuminate a mathemat-ical proof using without-loss-of-generality arguments. We will see that one canspecify an n -dimensional Black–Scholes–Merton market up to isomorphism us-ing only the parameters of dimension, risk-free rate and absolute market price ofrisk. This allows one to prove financial results for these markets by consideringonly markets with particularly simple forms. Our classification of one-periodcomplete markets admits similar applications.Secondly, we establish a connection between the automorphisms of a mar-ket and mutual-fund theorems. We prove that investment strategies solvinginvariant convex optimization problems in a market can be assumed to be in-variant under automorphisms. For markets with large symmetry groups such asBlack–Scholes–Merton markets, this imposes strong limitations on the form ofoptimal investment strategies, giving a significant generalization of the classicalmutual-fund theorems.Thirdly, we will see that a surprisingly large number of markets are iso-morphic to a Black–Scholes–Merton market, and so financial results proved forsuch markets can be applied more widely than one might expect. In particu-lar given any diffusion model, one can obtain a related Black–Scholes–Mertonmarket by making an appropriate choice of drift. This is significant since thedrift is difficult to estimate from statistical evidence and its functional form isusually chosen for parsimony. This result implies that one can find, for exam-ple, stochastic-volatility models which are isomorphic to Black–Scholes–Mertonmarkets.The effect of transformations on a market has been considered by many au-thors. If one considers the asset prices as stochastic trajectories in R n one canask how the dynamics change under diffeomorphisms of R n . Stochastic differen-tial equations (SDEs) on manifolds have been studied extensively and this haslead to a variety of geometric formulations [3, 5, 8, 9, 13, 18, 17]. The diffusionterm of a non-degenerate SDE on a manifold can be interpreted as defining aRiemannian metric and this yields a connection between Riemannian curvatureand SDEs. The geometric theory of SDEs on manifolds has been successfullyapplied to finance in, for example, [16]. However, the maps induced by diffeo-morphisms of R n are hard to interpret financially since financially importantproperties, such as whether a process is a martingale, are not preserved bydiffeomorphisms.In this paper, the transformations we consider are those that do preserve fi-nancially important properties. They are given by maps between the underlyingprobability spaces defining the markets rather than on the space R n . The ob-jects in our categories are given by filtered probability spaces equipped with costfunctionals. This probabilistic definition of a market is influenced by the work2f Pennanen [25, 26]. The morphisms we define are built upon the theory ofprobability-space homomorphisms developed by Rokhlin in [27] (who extendedthe work of von Neumann in [29]).Rokhlin’s classifications of standard probability spaces and their homomor-phisms are key ingredients in our classification of one-period complete markets.While Rokhlin’s results are all we need for this paper, we note that the cate-gory theory of probability has been developed by other authors, for example thetheory of stochastic processes is explored from a categorical viewpoint in [12].Category theory was applied to financial markets in [2], which classifiesMarkowitz markets and relates the classical mutual-fund theorems to the sym-metries of the market. The formulation of this earlier paper is purely algebraic.The probabilistic formulation we will develop is more fundamental and moregeneral.The structure of the paper is as follows.In Section 1 we define the category of discrete-time markets. We prove ageneral mutual-fund theorem for markets with automorphisms. We prove theequivalence (or more precisely the duality) between the formulation of categoriesgiven in this paper and the algebraic approach of [2]. We illustrate our mutual-fund theorem with the example of Gaussian markets.In Section 2 we classify complete one-period markets. We first give a sim-plified classification by assuming that one can additionally invest in a “casino”,which is a complete market where the P and Q measures coincide. This form ofthe classification is sufficient for most applications. We give a full classificationfor markets without using the casino. Our general mutual-fund theorem onlyapplies to convex optimization problems, but for complete one-period marketswe are also able to prove a mutual-fund theorem for problems where all agentshave monotonic preferences. This generalization is useful for solving problemsinvolving investors with S-shaped utility, as motivated by the theory of Kahne-man and Tversky [21].In Section 3 we extend out category to multi-period and continuous-timemarkets. We classify complete continuous-time markets with constant absolutemarket price of risk.Since the primary novelty of this paper is our definitions of financial cat-egories, the resulting classification results and their financial applications, theproofs of our results have been placed in an online appendix. If the reader isunfamiliar with category theory, a short review of the basic terminology we re-quire can be found in a second appendix. The appendices may be found on thejournal website or in the arXiv version of this manuscript [1]. In this section we give a coordinate-free definition of a one-period financialmarket and relate this to the elementary, coordinate-based approach of defininga market in n -assets using a probability distribution on R n . We will illustrate3ith the example of the Markowitz model. We will use this to demonstrate therelationship between invariant investment strategies and mutual-fund theorems.We begin by recalling a number of definitions due to Rokhlin [27] for mor-phisms between probability spaces. Definition 1.1.
Let (Ω , F , P ) and (Ω , F , P ) be two probability spaces.A map φ : Ω → Ω is called a homomorphism if φ is measurable and if P ( φ − U ) = P ( U ) for all U ∈ F . A homomorphism φ is called an isomor-phism if it is bijective and its inverse is a homomorphism. We call φ a mod 0isomorphism if there are subspaces Ω ′ ⊆ Ω and Ω ′ ⊆ Ω both of full measuresuch that φ restricted to Ω is an isomorphism to Ω .From the point of view of probability theory, two probability spaces should beconsidered as equivalent if they are mod 0 isomorphic. We define the cate-gory Prob to have objects given by probability spaces and morphisms given byalmost-sure equivalence classes of homomorphisms. Rohklin’s definition of amod 0 isomorphism does not coincide exactly with the set of isomorphisms inProb. The next lemma explains how the two notions are related. Lemma 1.2.
A measurable function is a mod 0 isomorphism if and only if itsalmost-sure equivalence class is a
Prob isomorphism.
An important functor is the contravariant functor L which maps the cate-gory Prob to the category Vec of vector spaces. L acts on the objects of Probby mapping a probability space to its vector space of almost-sure equivalenceclasses of measurable functions. Given a Prob morphism f : Ω → Ω and X ∈ L (Ω ) we define a linear transformation L ( f ) : L (Ω ) → L (Ω ) by L ( f )( X ) = X ◦ f . Definition 1.3. A one-period financial market ((Ω , F , P ) , c ) consists of: aprobability space (Ω , F , P ); a function c : L (Ω; R ) → R ∪ {±∞} . We call c − ( R ∪ {−∞} ) the domain of c , denoted dom c .We interpret a real random variable X on Ω as an investment strategy withpayoff X ( ω ) in scenario ω ∈ Ω. c ( X ) denotes the up front cost of strategy X and is equal to ∞ if one cannot pursue a strategy. A strategy with c ( X ) = −∞ results in liabilities so bad that the market is willing to pay arbitrarily largeincentives to encourage someone to take these liabilities on. A typical investmentstrategy is the purchase of an asset or of a portfolio of assets which are thensold at a final time T . In this case c ( X ) would be the cost of purchasing theasset. However, one can also model a commitment to pursue a continuous-timetrading strategy as yielding a single payoff at the final time T and our definitionof a market is flexible enough to include such strategies.This definition is deliberately minimal. To obtain interesting markets onewould typically want to impose additional conditions, such as that the mar-ket should be arbitrage free. This condition can be expressed as: for randomvariables X , if X ≥ X = 0 then c ( X ) > efinition 1.4. A morphism of markets M = ((Ω , F , P ) , c ) and M =((Ω , F , P ) , c ) is a Prob morphism φ : Ω → Ω satisfying c ( X ) ≥ c ( X ◦ φ )for all X ∈ L (Ω ; R ).Financially, a market morphism ψ : M → M represents an inclusion ofthe market M in M : given an investment strategy represented by the randomvariable X in M , we have the investment strategy X ◦ ψ in M which hasidentical payoff distribution but which has lower up-front cost. So if one canafford to pursue the strategy X , one can also afford to pursue X ◦ ψ . Thecontravariance between ψ and the financial notion of inclusion stems from thecontravariance of the functor L .Our primary interest in this paper is in market isomorphisms. We maydescribe them as follows. Lemma 1.5.
An isomorphism of markets ((Ω , F , P ) , c ) and ((Ω , F , P ) , c ) is the almost-sure equivalence class of a mod 0 isomorphism φ : Ω → Ω satis-fying c ( X ) = c ( X ◦ φ ) for all X ∈ L (Ω ; R ) . In finance, optimal investment problems are often convex optimization prob-lems (see for example [25]). A convex optimization problem is a problem requir-ing finding the set of minimizers of a convex objective function over a convex do-main. For example, a risk-averse agent will have a concave utility function, andso the objective in expected utility maximization problems can be expressed asthe minimization of their convex expected disutility function. Cost constraintsare typically linear, and hence define a convex domain. Additional constraintsimposed by a risk manager will further restrict the domain, but if one usesexpected-shortfall constraints, or any other coherent, or simply convex, riskmeasure (see [10]), this too will yield a convex domain.The solution set of a convex optimization problem is itself a convex set. Wealso expect that if the solution set is financially meaningful, it will be invariantunder the automorphism group of the market. Our next result will show thatone may then find an element of the solution set which is itself invariant.To state our result, let us define the necessary terminology. A measurablegroup G has a left-invariant probability measure, G if for all measurable sets A ⊆ G and elements h ∈ G we have G ( A ) = G ( hA ). A representation of sucha group on a Banach space V is a group homorphism ρ : G → Aut V , whereAut V is the group of linear isometries of V . We think of ρ as defining an actionof G on V on the left, given by gv = ρ ( g ) v . Theorem 1.6.
Let G be a measurable group with a left-invariant probabilitymeasure G . Let ρ : G → Aut V be a representation. Suppose that for all v in V the map g → ρ ( g ) v is measurable.If S is a non-empty G -invariant convex subset of V , then S contains a G -invariant element.If G is a finite group, we only need require that V is a vector space and G acts by linear automorphisms. G to be a subgroup of the auto-morphism group of the market which admits a left-invariant density and ρ tobe the standard action of G on L (Ω; R ). This allows us to simplify invariantconvex optimization problems by restricting attention to invariant investmentstrategies.We will see a number of applications of this general result throughout thispaper. In this section we will use this result to prove the classical two-mutual-fund theorem of [24]. A similar argument was used in [2] to prove the classicaltwo-mutual-fund theorem but the notion of isomorphism was different. Beforeproving the two-mutual-fund theorem we will show how the notion of isomor-phism in [2] relates to our new definition. We will do this by defining a generalnotion of a “finite-dimensional linear market” and giving a classification resultfor such markets and their isomorphisms. Definition 1.7.
A one-period financial market M = ((Ω , F , P ) , c ) is separated if there is a subset ˚Ω ⊂ Ω of full measure such that for any distinct ω , ω ∈ ˚Ωthere exists X ∈ dom c with X ( ω ) = X ( ω ).A one-period financial market is linear if dom c is a linear subspace of L (Ω; R ) and c is linear on dom c . The dimension of a linear market is thedimension of dom c .On a linear market, we may define a map π from Ω to (dom c ) ∗ , the algebraicdual space of dom c , by π ( ω )( X ) = X ( ω ) (1)for X ∈ dom C and ω ∈ Ω. One checks that π ( ω )( αX + X ) = ( αX + X )( ω ) = αX ( ω ) + X ( ω ) = απ ( X ) + π ( X ), so π ( ω ) ∈ (dom c ) ∗ as claimed. The map π induces a sigma algebra and measure on (dom c ) ∗ . We write d M for thismeasure, which we call the distribution of the market. If M is separated, then π is a mod 0 isomorphism.Financially, a market is linear if all traded assets can be bought and soldin unlimited quantities at a fixed price per unit. A market is separated if theprobability space contains no information other than that captured by assetprices.A finite-dimensional real vector space has a natural topology defined by therequirement that linear isomorphisms to R n are homeomorphisms. We wouldlike to require that the measure d M is in some sense compatible with this topol-ogy. To be precise we recall the following definition. Definition 1.8. (see [19]) A regular probability measure is a probability mea-sure arising as the Lebesgue extension of a Borel probability measure on atopological space.We would like to be able to ensure that d M is a regular probability measure.To do this we require an additional condition on the probability space (Ω , F , P ).6 efinition 1.9. (see [27] and [19]) A probability space (Ω , F , P ) is standard ifit is isomorphic mod 0 to either: the Lebesgue measure on [0 , R n andthe Wiener measure on C [0 , ∞ ). Finite and countable products of standardspaces are standard. A non-null measurable subset of a standard probabilityspace becomes a standard probability space when endowed with the conditionalmeasure. For proofs of these assertions see [27] or [19]. Lemma 1.10. If M is a finite-dimensional linear market based on a standardprobability space, then d M ∈ P ((dom c ) ∗ ) where P ( S ) denotes the set of regularprobability measures on S . Definition 1.11.
A regular probability measure on a finite-dimensional vectorspace, V , is said to be non-degenerate if for any X, Y ∈ V ∗ , X = Y almosteverywhere implies X = Y .Degenerate probability measures arise when the measure is concentrated ona vector subspace. Definition 1.12.
VecM is defined to be the category with objects consistingof triples (
V, d, c ) with V a finite-dimensional vector space, d ∈ P ( V ) with d non-degenerate and c ∈ V . VecM is equipped with a notion of morphism givenby linear transformations T : ( V , d , c ) → ( V , d , c ) satisfying:(i) for any Borel measurable set A ⊆ V d ( A ) = d ( T − ( A )); (2)(ii) the vectors c and c are related by c = T ( c ) . (3) Definition 1.13.
DualM is defined to be the category with objects consistingof triples (
V, d ∗ , c ∗ ) with V a finite-dimensional vector space, d ∗ ∈ P ( V ∗ ) with d ∗ non-degenerate and c ∗ ∈ V ∗ . Morphisms T : ( V , d ∗ , c ∗ ) → ( V , d ∗ , c ∗ ) inDualM are given by a linear transformation T : V → V whose whose vectorspace dual T ∗ is a VecM morphism T ∗ : ( V ∗ , d ∗ , c ∗ ) → ( V ∗ , d ∗ , c ∗ ). Definition 1.14.
FinM is defined to be the category with objects given byseparated finite-dimensional linear markets whose probability space is standard,and morphisms given by market morphisms.7or any element M of FinM defineVec( M ) = ((dom c ) ∗ , d M , c ) . In the opposite direction, for any element ((
V, d, c )) of VecM we defineFin((
V, d, c )) = (( V, F , d ) , c )where F is the sigma algebra associated with d and the map c : L ( V ; R ) → R satisfies c ( X ) = ( X ( c ) if X is equal to a linear map almost everywhere, ∞ otherwise . Theorem 1.15 (Equivalence of vector space and probabilistic categories ofmarket) . Vec( M ) lies in VecM and the map
Vec : FinM → VecM defines abijection on isomorphism classes.
Fin((
V, d, c )) lies in FinM . We may extend
Vec and
Fin to functors by defining their action on morphisms such that
Vec and
Fin define an equivalence of categories. Similarly the map
Dual : ob(FinM) → ob(DualM) given by Dual( M ) = (dom c, d M , c ) may be extended to a give aduality of the categories FinM and
DualM . To interpret this result financially, we suppose that we have a market of n assets. The space of portfolios in these assets is an n -dimensional vector space V . The cost of a portfolio defines a linear functional c ∗ on this vector space.The eventual payoff of a portfolio gives rise to a random linear functional actingon the space of portfolios. The distribution of this payoff functional is given by d ∗ . Together this data defines an element ( V, d ∗ , c ∗ ) ∈ ob(DualM). Thinking ofthe space of portfolios as a vector space with no preferred basis represents thefinancial idea that a portfolio of assets can be viewed as an asset in its own right.The category DualM is therefore the appropriate category to use if one believesthat the distinction between an asset traded on the market and a portfolio ofassets is not financially significant.The significance of Theorem 1.15 is that it shows the notion of equivalence ofmarkets obtained by treating all portfolios as equally valid investment strategiesis the same as the notion of equivalence given in Definition 1.4. This relatesthe definitions of [2] to the definitions in this paper. The advantage of our newDefinition 1.4 is that it can be applied to infinite markets, as we shall see whenwe discuss complete markets later, and to non-linear markets.The proof of Theorem 1.15 shows that morphisms in VecM are surjective lin-ear transformations. It follows that the morphisms of DualM are injective. Thisbacks up the claim we made earlier that market morphisms are a contravariantrepresentation of market inclusion.We now apply this general theory to the case of assets following a multivari-ate normal distribution, as considered by Markowitz [22].Let g µ be the multivariate normal distribution with mean µ ∈ R n and co-variance matrix given by the identity id n . We say that a market is Gaussian if it8s isomorphic to a market on R n with density g µ . Trivially any Gaussian marketis isomorphic to a market of the form Fin( R n , g µ , c ) for some µ, c ∈ R n . Let { e i } be the standard basis for R n . Since isometries of R n preserve the Gaussianmeasure, we may apply a rotation so that µ lies in the span of e and c lies inthe span of e and e . This shows that any Gaussian market can be written inthe form Fin( R n , g α e , β e + γ e ) , α, β, γ ∈ R . (4)We now have the following classification theorem. Theorem 1.16 (Classification of Markowitz markets) . Let M ∈ FinM be amarket and suppose that { X i } is a basis for dom c given by assets following amultivariate normal distribution. Then M is Gaussian, and hence is isomorphicto a market of the form (4) . This theorem is essentially a restatement of the main classification result of [2]in the language of one-period markets.
Corollary 1.17.
All invariant investment strategies X ∈ dom c in a Gaussianmarket lie in a two-dimensional vector subspace of dom c . Corollary 1.18. (Two-mutual-fund theorem [24]) Suppose we have n assets ofa given cost whose payoffs follow a multivariate normal distribution. We wishto find the portfolio of assets with minimum variance but with a given expectedpayoff C and cost C . There are two portfolios X and X independent of C and C such that we can solve these mean–variance optimization problems forany C and C simply by considering linear combinations of X and X . The portfolios X and X are the two “mutual funds” that give this theoremits name.We remark that Corollary 1.17 is a much stronger result than the classicaltwo-mutual-fund theorem. The paper [2] gives numerous concrete examples offinancially interesting results arising from invariance arguments other than justthe two-mutual-fund-theorem.We also remark that the concrete isomorphism found in Theorem 1.16 makesit extremely easy to solve the classical mean-variance optimization problemdirectly, thereby recovering the full set of results found in [24]. This approachis pursued in [2]. Definition 2.1.
A one-period market M = ((Ω , F , P ) , c ) is complete if thereexists a measure Q on Ω equivalent to P , and C > c ( X ) = ( C ( E Q ( X + ) − E Q ( X − )) one of E Q ( X ± ) is finite ∞ otherwise . (5)9n this formula X + and X − denote the positive and negative parts of the randomvariable X . We note that c (1) = C , so we interpret ( C −
1) as a deterministicinterest rate.
Example 2.2.
Let I be the market given by taking the P and Q measure to bothbe equal to the Lebesgue measure on [0 ,
1) and with cost of the constant functionwith value 1, equal to 1. In this market prices are given by expectations, so wecall I a casino . (Our casino is of course an idealized one, in which the profitsand losses of a typical client form a martingale rather than a supermartingale.)Given a complete market M we may define a new complete market M × I by taking the product measures for both the P and the Q measures and takingthe constant C to be that given by the market M .From a financial point of view the market M × I represents the marketobtained by considering investment strategies where one first invests in themarket M and then places a bet at the casino.In applications it is not unreasonable to assume that there is a casino avail-able should a trader wish to use it. So classifying complete markets of the form M × I should be just as useful in practice as a full classification. The theorembelow gives a classification for markets of this form. Theorem 2.3 (Classification of complete markets up to a casino) . Let M be acomplete market on a standard probability space. Then M × I is isomorphic to ˜ M × I , where ˜ M is the market with probability space given by ˜Ω = [0 , equippedwith the Lebesgue measure and with pricing function ˜ c ( X ) = C Z F − d Q d P X ( x )d x. Here F − d Q d P is the inverse distribution function of d Q d P on M . The first step toward proving this is to observe that we may recover Q from c since for any measurable set A ⊂ Ω we have Q ( A ) = E Q (1 A ) = c (1 A ) c (1) . It follows that two one-period complete markets ((Ω i , F i , P i ) , c i ) ( i = 1 ,
2) areisomorphic if and only if (a) there is a mod 0 isomorphism for the P i measureswhich is also a mod 0 isomorphism for the Q i measures; and (b) the cost of theconstant function with value 1 is equal in both markets.There may be more than just 2 measures on the market which are of financialinterest. A trader with views about the market represented by a measure P maybe constrained by a risk manager or regulator with different views about themarket. These can be represented by alternative measures. Let us state aclassification result similar to Theorem 2.3 that applies to this situation. Theorem 2.4 (Classification of complete markets with multiple views) . Let I denote the interval [0 , with the Lebesgue measure. We suppose that P , P , . . . , P n re equivalent probability measures on (Ω , F ) . We assume P is standard. Thenthere is a unique Lebesgue measure P ′ on Ω ′ = (0 , ∞ ) n such that P × I and P ′ × I are mod 0 isomorphic via an isomorphism which also acts as a mod 0isomorphism between the measures P i × I and P ′ i × I where P ′ i is the Lebesguemeasure given by P ′ i ( A ) = Z (0 , ∞ ) n ω i A ( ω ) d µ. In this formula, A is a measurable set, A is the indicator function A and ω i is the i -th coordinate function on R n . Note that we must have E P ′ ( ω i ) = 1 forthese P ′ i to be probability measures. We note the following financial implication (using the notation of Theorem2.4).
Corollary 2.5 (Convex mutual-fund theorem for complete markets) . Let A bea non-empty convex subset of the space of P -integrable random variables on Ω .Suppose that A is also invariant under mod 0 isomorphisms that preserve allthe P i . Then A contains an element which can be written as a function of theRadon–Nikodym derivatives d P i d P . For example, A might arise as the optimal investment strategies in a convexoptimization problem with a cost constraint and risk-management constraintsimposed by a number of regulators and risk managers given in terms of the P i .A special case of the result above is the problem of expected-utility optimi-sation in a complete market subject to a single cost constraint for a concave,increasing utility function. In this case it is well-known that the optimal invest-ment has a payoff function given as a function of the Radon–Nikodym derivative(see [11]).Let us now give the definitions needed to state a full classification for com-plete one-period markets. Write S for the set of mod 0 isomorphism classes ofstandard probability spaces. We call S the moduli space of standard probabilityspaces.Given m ∈ S , we define m to be the measure of the continuous componentof m (or zero if it has no continuous component) and we define m i for i > i -th largest atom in our probability space (or 0 if thereless than i atoms). Thus we have identified a correspondence between S andsets of numbers m i ( i ∈ N ) which satisfy m i ∈ [0 , ∀ i ∈ N + , m i ≥ m i +1 ; and m = 1 − ∞ X i =1 m i . (6)We give S the topology induced by thinking of it as a subset of R ∞ in this way.Thus we may talk about measurable maps to S , or S -valued random variables.The theory of disintegration of measure tells us that for a complete market M based on a standard probability space, there is a µ M -almost-surely uniquemeasurable function m M : (0 , ∞ ) → S m M ( x ) given by the mod 0 isomorphism class of the P conditional measureconditioned on the value of d Q d P = x and where µ M denotes the measure on (0 , ∞ )induced by d Q d P . Definition 2.6.
Let Measures( n ) be the set consisting of pairs ( µ, m ) where:(i) µ is a regular probability measure on (0 , ∞ ) n satisfying E µ ( ω i ) = 1 forthe i th coordinate function ω i on R n ;(ii) m is an S valued µ random variable. Theorem 2.7 (Generalised classification of complete markets) . Standard prob-ability spaces (Ω , F , P ) equipped with n -additional equivalent measures P , . . . , P n are classified up to joint P -, . . . , P n - mod 0 isomorphism by elements ( µ q , m q ) ∈ Measures( n ) . Here µ q is the measure on (0 , ∞ ) n induced by the R n vector valued function q with i -th component given by the Radon–Nikodymderivative d P i d P . The proof uses Rokhlin’s theory of the decomposition of measure.
We show in this section that Theorem 2.4 allows us to identify a mutual-fundtheorem that applies to optimization in complete markets when we assume thatthe problem is “monotonic” rather than convex.We have in mind applications to behavioural economics based on the ob-servations of Kahneman and Tversky in [21]. For examples of applications ofKahneman and Tversky’s ideas to mathematical finance and risk management,see, for example, [20], the review [30], and [4] which contains numerous furtherreferences.It has been observed in this literature (see for example [15]) that the solutionto optimal investment problems in complete markets involving S-shaped utilityfunctions can be obtained by considering monotonic functions of the Radon–Nikodym derivative d Q d P . The aim of this section is to show how these resultsarise from general monotonicity properties, automorphism invariance and ourclassification theorems. We take the opportunity to show how these results canbe generalized to situations where there are more than two measures P and Q ,for example, to the case where risk managers and traders have different beliefsabout the future evolution of the market.Given two random variables X , Y on a probability space (Ω , F , P ) we write d P ( X ) (cid:22) d P ( Y )if F X ( k ) := P ( X ≤ k ) ≥ P ( Y ≤ k ) =: F Y ( k ) for all k . The notation d P ( X ) isintended to suggest “the P -distribution of X ”. Given a third random variable Z we write d P ( X | Z ) (cid:22) d P ( Y | Z )if P ( X ≤ k | Z ) ≥ P ( Y ≤ k | Z ) almost surely for all k .12e suppose that market participants such as traders and risk managersimpose some form of relation (cid:22) ′ on random variables to express their preferencesbetween different investment opportunities. One might reasonably expect that X (cid:22) Y = ⇒ X (cid:22) ′ Y. (7)If this condition holds, we will say that (cid:22) ′ is increasing . We say that (cid:22) ′ is decreasing if the reversed relation is increasing. We say that a relation onrandom variables is monotonic if it is either increasing or decreasing. We saythat the sign of a monotonic relation is 1 if it increasing or − Definition 2.8 (Rearrangement) . Let m be a Lebesgue probability measure on(0 , ∞ ). Let F m denote the cumulative distribution function of m . Write x, y forthe coordinate functions on (0 , ∞ ) × [0 , U m : (0 , ∞ ) × [0 , → [0 , U m ( ω ) = (1 − y ( ω )) lim x ′ → x ( ω ) − F m ( x ′ ) + y ( ω ) lim x ′ → x ( ω )+ F m ( x ′ ) .U m is well-defined since F m is c`adl`ag. We write P m for the product measure on(0 , ∞ ) × [0 , E P m ( x ( ω )) = 1 (8)then x is the Radon–Nikodym derivative of an equivalent measure we call Q m .Given X ∈ L P m ((0 , ∞ ) × [0 , R ) we define the increasing and decreasing rear-rangements of X by R + m ( X ) = F − X ( U m ) , R − m ( X ) = − F − − X ( U m )respectively, where F − X is the P m inverse distribution function of X .Our next theorem shows that the notion of rearrangement can be general-ized to situations when there are more than two probability measures underconsideration. Theorem 2.9 (Monotone mutual-fund theorem for complete markets) . Let (Ω , F , P ) be a standard probability space equipped with n equivalent measures P i ( ≤ i ≤ n ). Let I = [0 , . Let (cid:22) i ( ≤ i ≤ n ) be monotonic relations onthe set of probability distributions on R . Write sign i for the sign of (cid:22) i . Thereexists a mapping R : L (Ω × I ) → L (Ω × I ) , which we call rearrangement , withthe following properties.(i) Rearrangment does not change P distributions: d P ( X ) = d P ( R ( X )) . (ii) Rearrangement increases or decreases P i distributions according to the signof (cid:22) i : d P i ((sign i ) X ) (cid:22) d P i ((sign i ) R ( X )) , ≤ i ≤ n. (iii) Let q denote the vector of n Radon–Nikodym derivatives d P i d P . Define (cid:22) on R n by x (cid:22) y if (sign i ) x i ≤ (sign i ) y i for all components i , and hencedefine ≺ on R n . Then R ( X ) satisfies R ( X )( ω ) ≤ R ( X )( ω ′ ) if q ( ω ) ≺ q ( ω ′ ) . P i , we canrestrict our attention to strategies that lie in the image of R . We interpret thisas a mutual-fund theorem since it says that, for a general class of optimizationproblems, we can safely restrict attention to a subset of the random variablesavailable in the market.The assumption that there is a casino can be dropped in many cases since,as one might intuitively expect, one often doesn’t take any real advantage ofthe casino. This is formalized in the next corollary. Corollary 2.10.
Let (Ω , F , P i ) ( ≤ i ≤ n ) be as in the previous Theorem 2.9We can find a map ˜ R : L (Ω) → L (Ω) which shares properties (i), (ii) and(iii) described in Theorem 2.9 so long as either: (a) P is atomless and n = 1 ; or (b) for some j , the distribution of d P j d P condi-tioned on the value of all the other Radon–Nikodym derivatives is almost surelycontinuous. In case (b) ˜ R can be assumed to depend only on the value of q . Note that the theory of conditional distributions detailed in [19] ensures thatthe conditional distribution exists in case (b).
Let us extend our definitions of markets to the multi-period setting.
Definition 3.1.
A multi-period market consists of the following.(i) A filtered probability space (Ω , F t , P ) where t ∈ T ⊆ [0 , T ] for some indexset T containing both 0 and T . We write F = F T . We require F = {∅ , Ω } . (ii) For each X ∈ L (Ω; R ), an F t -adapted process c t ( X ) defined for t in T \ T .Random variables X ∈ L (Ω , F T ; R ) are interpreted as contracts which havepayoff X at time T . The cost of this contract at time t is c t ( X ).We note that this is deliberately bare-bones definition of a market. In prac-tice would want to impose additional conditions on the c t . For example, onewould normally wish to forbid arbitrage and to impose “the usual conditions”on the filtered probability space. Definition 3.2. A filtration isomorphism of filtered spaces (Ω , F , F t , P ) where t ∈ T for some index set T is a mod 0 isomorphism for F which is also a mod 0isomorphism for each F p . An isomorphism of multi-period markets is a filtrationisomorphism that preserves the cost functions.Given a one-period market ((Ω , F , P ) , c ) we can trivially define a filtration F = {∅ , Ω } , F = F indexed by { , } and we may define c = c . Hence we candefine a multi-period market in a canonical fashion from a one-period market.The notion of isomorphism is preserved. In this sense, our definition of multi-period markets and their isomorphisms is a generalization of the correspondingnotions for one-period market. 14 efinition 3.3 (Exchange market) . Let (Ω , F t , P ) be n -dimensional Wienerspace, that is the probability space generated by the n -dimensional Brownianmotion W t . Let X t be an n -dimensional stochastic processes defined by astochastic differential equation of the formd X t = µ ( X t , t ) d t + σ ( X t , t ) d W t . (9)Here µ is an R n -vector valued function and σ is an invertible-matrix valuedfunction. We assume the coefficients µ and σ are sufficiently well-behaved forthe solution of the equation to be well-defined on [0 , T ]. The components, X it ,of the vector X t are intended to model the prices of n -assets.The exchange market for (9) with risk-free rate r over a time period [0 , T ] isgiven by defining c t : L (Ω; R ) → R for t ∈ [0 , T ) by c t ( X ) = ( α e − r ( T − t ) + P ni =1 α i X it if X = α + P ni =1 α i X iT , (10a) ∞ otherwise. (10b)This is well-defined so long as we assume that X iT are linearly independentrandom variables. This will be the case in all situations of interest.The market defined above is called an exchange market because it modelsthe basic assets that can be purchased directly on an exchange, but does nottake into account the possibility of replicating payoffs via hedging. The nextdefinition does take this into account. Definition 3.4 (Superhedging market) . The superhedging market for (9) withrisk-free rate r over a time period [0 , T ] is given by defining c t ( X ) to be theinfimum of the cost at time t of self-financing trading strategies that superhedge X . See [14] for a definition of a self-financing trading strategy. A self-financingtrading strategy superhedges X ∈ L (Ω; F T ) if the final payoff of the strategyis always greater than or equal to X .Thus the superhedging market represents the effective market of derivativesthat a trader can achieve given the exchange market. The cost function c t forsuch a market is the superhedging price. Of particular interest are completemarkets where any contingent claim may be both superhedged and subhedged.One expects that the price in an arbitrage-free market can be expressed as arisk-neutral probability. These remarks motivate the next definition. Definition 3.5.
A continuous-time market (Ω , F t , P ) , c t ) on [0 , T ] is called a continuous-time complete market with risk-free rate r if there exists a measure Q equivalent to P with c t ( X ) = e − r ( T − t ) E Q ( X | F t ) (11)for Q -integrable random variables X and equal to ∞ otherwise. We follow ourusual conventions on expectations to allow −∞ when the positive part of anexpectation is finite and the negative part is infinite.15sing our new terminology, the theory of Harrison and Pliska [14] shows howthe superhedging market associated with the SDE (9) gives rise to a continuous-time complete market, subject to sufficient regularity assumptions on the coef-ficients. Definition 3.6.
The continuous-time complete market with risk-free rate andcost function given by the superhedging market is called the complete marketassociated with the SDE (9) (subject to the required regularity assumptions for q t to be a well-defined P -martingale).We differ slightly in our presentation from Harrison and Pliska [14] in thatthey discuss replication and we consider superhedging. This is why we arewilling to ascribe a cost of −∞ to some X ∈ L (Ω , F T ), whereas if one insists onreplication, X must be absolutely integrable. The definition of the superhedgingmarket associated to a given market can be applied equally well to incompletemarkets where there is a more meaningful difference between replication andsuperhedging. This is why we prefer to think in terms of superhedging, and inthis we are influenced by the presentation of [26]. Definition 3.7.
The absolute market price of risk in the complete market as-sociated with the SDE (9) is the element of L (Ω × [0 , T ] , P ) defined byAMPR t = | σ − ( r X t − µ ) | . Theorem 3.8.
Let F be the contravariant functor mapping a continuous-timemarket, M with underlying probability space Ω to the vector space L (Ω × [0 , T ] , P × λ ) where [0 , T ] is equipped with the Lebesgue measure λ , and where F acts on morphisms φ : Ω → Ω by F ( φ )( X ) = X ◦ ( φ × id) for X ∈ L (Ω × [0 , T ] , P × λ ) . We recall that elements of L (Ω × [0 , T ] , P × λ ) are definedto be almost-sure equivalence classes. Write AMPR( M ) ∈ L (Ω × [0 , T ] , P × λ ) then for any market isomorphism φ AMPR( φ ( M )) = F ( φ − )AMPR( M ) . We summarize this by saying that the absolute market price of risk is an invariantly-defined element for F . Theorem 3.8 can be viewed as an analogue of Gauss’s Theorema Egregiumfor the category of continuous-time complete markets. Of course, we are onlyclaiming that this is an analogy. We have not established any relationshipbetween markets and Gaussian curvature. If one is interested in direct relation-ships between curvature and finance, one can consider the theory of SDEs onmanifolds, the Riemannian metric defined by a non-degenerate volatility termand the corresponding curvature tensor (see, for example, [16]). Note that theRiemannian metric arising in this way is independent of the choice of drift term,and so one may have non-zero curvature even when P = Q .The proof of Theorem 3.8 suggests we extend the definition of AMPR t toall complete markets as follows. 16 efinition 3.9. In a continuous-time complete market we define AMPR t ∈ L ≥ (Ω × R ) (if it exists) to be the solution of Z t Q t d[ Q, Q ] t := Z t AMPR t d t, (12)where Q t := d Q d P (cid:12)(cid:12)(cid:12) F t . (13)An additional invariant we need to consider is the dimension of our market.This is given by the number of independent Brownian motions n . Our nextresult shows that this is an invariant of the market; indeed it is an invariant ofthe filtered probability space (Ω , F t , P ). Definition 3.10.
The n -dimensional Wiener space on [0 , T ] is the filtered prob-ability space generated by n independent standard Brownian motions on [0 , T ].A filtered probability space is called a Wiener space if it is filtration isomorphicto an n -dimensional Wiener space. Theorem 3.11.
The dimension of a Wiener space is invariant under filtrationisomorphisms.
We are now ready to state a classification theorem for complete markets withdeterministic absolute market price of risk. We recall that { e i } is the standardbasis for R i and id n is the identity matrix. Theorem 3.12 (The test case) . Let M be a continuous-time complete marketwith risk-free rate r , time period T based on a Wiener space of dimension n andwith AMPR given by
AMPR t = A ( t ) ≥ for a bounded measurable function of time A ( t ) . Suppose that the process q t is continuous. In these circumstances M is isomorphic to the complete marketassociated with the SDE (9) with µ = r X t + A ( t ) e , and σ = id n and X = 0 . We call markets of this form canonical Bachelier markets .The key step in the proof of this theorem is to invariantly define a Brownianmotion corresponding to the first component of W t . To do this, one shows that − Z t A ( s ) d(log Q ) s is a Brownian motion using Levy’s characterisation of Brownian motion.We have called Theorem 3.12 “the test case” as it is an analogous result tothe theorem in differential geometry that a Riemannian manifold with vanishingcurvature is flat. This latter result is called “the test case” in [28].17 xample 3.13. The n -dimensional Black–Scholes–Merton market is isomor-phic to a Bachelier market, since market price of risk in the Black–Scholes–Merton market is a deterministic constant vector. Example 3.14.
Given a positive real number A , an invertible matrix σ anda vector X , the set of vectors µ satisfying | σ − ( r X − µ ) | = A is non-empty;indeed, it as an ellipsoid. Hence given a complete continuous-time market mod-elled by an SDE, we may modify the drift to obtain a market isomorphic to aBlack–Scholes–Merton market with market price of risk A .It is difficult to estimate the drift of a volatile asset. As a result, the func-tional form of the drift is usually chosen for parsimony; one then uses long-termdata to calibrate this functional form. If one is following this approach, in theabsence of statistical evidence to the contrary it might be be reasonable to choosethe functional form of the drift to ensure that the resulting model has a constantmarket price of risk, and hence is isomorphic to a Black–Scholes–Merton model.Our result shows that the many financial results that have been proved forthe Black–Scholes–Merton model can be applied to a far wider range of marketsthan one might at first sight expect. Even markets which seem superficially verydifferent from the Black–Scholes–Merton market, such as stochastic-volatilitymodels, may still provide isomorphic investment opportunities.These observations suggests that one should separate optimal investmentproblems into two components. One has the strategic problem of optimal in-vestment for a particular isomorphism class of market. Additionally one hasthe tactical problem of finding a concrete realisation (or approximate realisa-tion) of the strategy, which can be interpreted as the task of finding a concretemorphism. This division of investment problems into strategic and tacticalproblems is already widely used in practice (see [6]).Although we have restricted ourselves to considering markets with determin-istic absolute market price of risk, this approach can be generalized. Rather thanattempt to model asset price dynamics directly, one may choose a market modelby attempting to model invariantly-defined quantities. For example, if one hasa view on the dynamics of the absolute market price of risk, one may developa market model to reflect this. We expect this approach to yield a systematicmethod for developing low-dimensional (and hence numerically tractable) mar-ket models which still capture the essential features of the market. We willexplore this in future research.As an application of our classification theorem, we may now prove a mutual-fund theorem. Theorem 3.15 (Continuous-time one-mutual-fund theorem) . Let M be a com-plete continuous-time market with continuous q t and with deterministic, boundedabsolute market price of risk. Let X it for ( ≤ i ≤ n ) be a collection of squareintegrable stochastic processes representing n basic assets, then there exist n pre-dictable real valued processes α it such that any invariant, non-empty, convex setof martingales contains an element which can be replicated by a continuous-time rading strategy using only the risk-free asset and the portfolio consisting of α it units of asset X it .In complete markets arising from SDEs of the form (9) which also have adeterministic absolute market price of risk, we may take the portfolio α withcomponents α i to be given by the vector ( σσ ⊤ ) − ( r X t − µ ) . We note that a convex set of martingales can be interpreted as a convex setof self-financing trading strategies or as a convex set of derivative securities.We call this result a one-mutual-fund theorem because it shows that a fundmanager can create a single mutual fund that can be used to implement thesetrading strategies. A key difference between our result and the classical one-mutual-fund theorem is that an investor needs to trade in our mutual fund incontinuous time.This result explains the general form of the solution to the portfolio op-timization problem studied by Merton in [23]. However, it goes considerablybeyond this.As an example, consider the problem of managing the investment and pen-sion payments for a collective pension. Suppose that the fund is heterogenous,so each individual may have a distinct mortality distribution, initial wealth andrisk appetite. Assume that fund may invest in a Black–Scholes–Merton mar-ket, and that the individuals preferences and mortality are independent of thismarket. Assume that the investors preferences are convex. Our theorem nowshows that one need only consider investments in the risk-free asset and the mu-tual fund we have identified when deciding how to manage the pension. We cansay this without actually formulating an optimal investment problem describinghow such a heterogeneous fund should be managed.In summary, our classification theorems have identified interesting isomor-phisms between markets that are not obviously related. We have found largefamilies of automorphisms for the classical markets of Markowitz and Black–Scholes–Merton. We have shown that considering these automorphisms allowsone to prove very general mutual-fund theorems.
I would like to thank both the anonymous referees and Andrei Ionescu for theiruseful comments and corrections, and to thank Markus Riedle and Nick Bing-ham for their valuable advice.
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Appendix A Proofs
A.1 Proofs for Section 1
Proof of Lemma 1.2.
Let f : Ω → Ω be a Prob isomorphism. We can thenfind a homomorphism g : Ω → Ω such that g ◦ f = id almost surely and21 ◦ g = id almost surely. Define Ω ′ to be the set of points where f g ( x ) = x andΩ ′ to be the set of points where gf ( y ) = y . Ω ′ and Ω ′ will be of full measure.If x , x ∈ Ω and f ( x ) = f ( x ), then gf ( x ) = gf ( x ), hence x = x . Thus f is injective on Ω ′ . If y ∈ Ω ′ then f g ( y ) = y , so gf g ( y ) = g ( y ) and hence g ( y ) ∈ Ω ′ with y = f g ( y ). Thus f maps Ω ′ onto Ω ′ . Hence f is a mod 0isomorphism.The converse follows trivially from the definitions. Proof of Lemma 1.5.
Let φ : Ω → Ω be a Prob morphism with two-sidedinverse φ − . Suppose φ is, moreover, a market isomorphism. Using the fact that φ and φ − are both market morphisms, we have that for any X ∈ L (Ω ; R ) wehave c ( X ) = c ( X ◦ φ ◦ φ − ) ≤ c ( X ◦ φ ) ≤ c ( X ) . Hence we must have equality throughout. Hence c ( X ) = c ( X ◦ φ ).The result now follows from Lemma 1.2. Proof of Theorem 1.6.
Given h ∈ G , define φ h : G → G by left multiplication,so φ h ( g ) = hg . Let A be a measurable set and let 1 A denote the indicatorfunction of A then 1 A ◦ φ h = 1 h − A . We deduce that E ( X ◦ φ h ) = E ( X ) (14)if X is an indicator function of a set, and hence this holds for all integrablerandom variables X .By assumption S is non-empty, so we may choose an element s ′ ∈ S . Wedefine a random variable X : G → V by X ( g ) = ρ ( g ) s ′ . (15)Because G acts by isometries on V , k X ( g ) k = k ρ ( g ) s ′ k = k s ′ k for all g . Henceby the dominated convergence theorem we may define an element s by s := E G ( X ) . (16)By the convexity of S , s ∈ S . Given h ∈ G , we now compute that s = E G ( X ) = E G ( X ◦ φ h ) = E G ( ρ ( hg ) s ′ ) = E G ( ρ ( h ) ρ ( g ) s ′ ) = ρ ( h ) E G ( ρ ( g ) s ′ ) = ρ ( h ) s, using (14), (15), that ρ is a homomorphism, linearity of expectation, and finally(15) and (16). So s is invariant under G .If G is finite, the expectation is a finite sum, so we do not need the dominatedconvergence theorem. Proof of Lemma 1.10.
We recall that a perfect probability measure is a completeprobability measure, µ on a set S such that for every measurable map f : S → R the image measure is a regular measure on R . Lemma 2.4.3. of [19] proves thatall standard probability spaces are perfect. Let S be a perfect probability space22nd let V be a finite-dimensional real vector space, then Exercise 3.1(iii) of [19]shows that any measurable map f : S → V induces a regular measure on V .Thus it suffices to show that π defined by (1) is measurable.Choose a basis { X i } for dom c . Define a map X : Ω → R n by requiring thatthe i -th component of X ( ω ) is given by X ( ω ) i = X i ( ω ). This map is measurablesince each X i is measurable. Define a map X ∗∗ : (dom c ) ∗ → R n by requiringthat the i -th component of X ∗∗ ( f ) is given by X ∗∗ ( f ) i = f ( X i ). ( X ∗∗ ) − is alinear isomorphism and so is measurable by the definition of the topology ondom c . Since π = ( X ∗∗ ) − ◦ X , π is measurable. Proof of Theorem 1.15.
We first show that Vec( M ) lies in VecM.We have already seen in Lemma 1.10 that d M is regular.We must also show that d M is non-degenerate. Given X ∈ dom c we maydefine a linear functional X ∗∗ ∈ (dom c ) ∗∗ by X ∗∗ ( f ) = f ( X ). Double dualityis an isomorphism, so given distinct ˜ X, ˜ Y ∈ (dom c ) ∗∗ we may find distinct X, Y ∈ (dom c ) with X ∗∗ = ˜ X and Y ∗∗ = ˜ Y . For any Z ∈ (dom c ), Z ∗∗ ◦ π = Z . M is separated, so π is a mod 0 isomorphism. Since X and Y are not equal, itthen follows that ˜ X = X ∗∗ and ˜ Y = Y ∗∗ are not equal almost everywhere. So d M is non-degenerate, as claimed.This completes the proof that Vec( M ) lies in VecM.We now define an additional map, also denoted Vec, which sends morphismsof FinM to morphisms of VecM. Given a market morphism T between twosuch markets M i = ((Ω i , F i , P i ) , c i ) ∈ FinM ( i = 1 ,
2) we define T ∗ : dom c → dom c by T ∗ ( f ) = f ◦ T. We define Vec( T ) = T ∗∗ : (dom c ) ∗ → (dom c ) ∗ to be the ordinary vector space dual of T ∗ . We wish to show that Vec( T ) is amorphism in VecM. Since T is a market morphism we compute that for any v ∈ (dom c ) ∗ (Vec( T ) c )( v ) = T ∗∗ ( c )( v ) = c ( T ∗ v ) = c ( v ◦ T ) ≤ c ( v ) . Applying the same calculation to − v and using linearity, we also have (Vec( T ) c )( v ) ≤− c ( v ) . Hence (Vec( T ) c )( v ) = c ( v ) . (17)We note that T ∗∗ ( v ) = w ⇐⇒ ∀ f ∈ V ∗ , T ∗ f ( v ) = f ( w ) ⇐⇒ ∀ f ∈ V ∗ , f T ( v ) = f ( w ) ⇐⇒ T ( v ) = w. It follows that given a set A ⊆ V ( T ∗∗ ) − ( A ) = T − ( A ) . So if A is Borel measurable we have d (Vec( T ) − A ) = d (( T ∗∗ ) − ( A )) = d ( T − ( A )) = d ( A ) . (18)Together (17) and (18) show that Vec( T ) is a morphism in VecM, as claimed.23e must show that Fin(( V, d, c )) is an element of FinM. We first note thatthe probability space underlying Fin((
V, d, c )) is standard, since a regular distri-bution on a real vector space always defines a standard probability distribution.Since all elements of VecM have non-degenerate distributions, dom c ⊂ L ( V ; R )is equal to V ∗ (rather than a non-trivial quotient space of V ∗ by equivalencealmost everywhere). The dual space of a finite-dimensional vector space sepa-rates the points of the vector space, so Fin(( V, d, c )) is separated. It is now clearthat Fin((
V, d, c )) lies in FinM.We define a mapping on morphisms, also called Fin, by Fin( T ) = T for anymorphism T of VecM. We must show that a VecM morphism is automaticallya market morphism. Equation (2) shows that Fin( T ) is a Prob morphism.Next observe that a VecM morphism is automatically surjective. Supposefor contradiction that T is not surjective, then we can find a non-zero linearfunctional X which annihilates Im( T ). Since Im( T ) is of full measure, X isalmost-surely zero, and hence d is degenerate, yielding the desired contradic-tion.Now let T : V → V be a morphism in VecM and X ∈ L ( V ; R ). Firstsuppose X is linear, then equation (3) shows that c ( X ◦ T ) = c ( X ). Nextsuppose X is not linear, so we may find v, w ∈ V and α ∈ R with X ( αv + w ) = αX ( v ) + X ( w ). Since T is surjective we may find v ′ , w ′ ∈ V with T v ′ = v and T w ′ = w . Then XT ( αv ′ + w ′ ) = αXT ( v ′ ) + XT ( w ′ ). So XT is also non-linear, and hence c ( X ◦ T ) = ∞ = c ( X ). Thus c ( X ◦ T ) = c ( X ) for all X ∈ L ( V ; R ). So Fin( T ) is a market morphism as claimed.Since dom c = V ∗ , we have (dom c ) ∗ = V ∗∗ . Hence the composition Vec ◦ Finis given by double duality of vector spaces. In particular Vec ◦ Fin(
V, d, c ) isnaturally isomorphic to (
V, d, c ).We note that Fin ◦ Vec( M ) is naturally isomorphic to M with the isomor-phism given by π defined in (1).We have now shown that Vec and Fin define an equivalence of the categoriesFinM and VecM. It is trivial to check that vector-space duality defines a dualityof the categories VecM and DualM. The statement that Vec and Dual definebijections follows by elementary category theory [7]. Proof of Theorem 1.16.
Let Cov : dom c × dom c → R be given by the covari-ance. This is a non-degenerate symmetric bilinear form and hence defines aninner product on dom c . All real inner-product spaces of dimension n are iso-morphic to the standard Euclidean space R n , hence we can find a second basis { Y i } for dom c with covariance matrix id n . The distribution of these assets willstill be a multivariate normal distribution, but now with covariance matrix id n .This shows that the market is Gaussian. Proof of Corollary 1.17.
It suffices to prove the result for markets of the form244). Let φ : R n → R n be the linear transformation given by the matrix φ ij = i = j and i, j ≤ , − i = j and i, j > , .φ defines an automorphism of any market of the form (4). Any invariant in-vestment strategy must be invariant under φ ∗ . φ ∗ has the same matrix repre-sentation as φ when written with respect to the standard dual basis { e ∗ i } for( R n ) ∗ . If X is an invariant investment strategy, its components ( X ) i writtenwith respect to this basis satisfy X i = 0 for i > A.2 Proofs for Section 2
We review the features of the theory of disintegration of measures we will need.
Definition A.1.
Let { S α } be a countable collection of subsets of a set S . Wewrite ζ ( { S α } ) for the collection of sets of the form ∞ \ i =1 S ′ α , ( S ′ α = S α or S ′ α = S \ S α ) . These sets are disjoint and cover S so they define a decomposition of S calledthe decomposition generated by { S α } . A decomposition of a measurable set S generated by a countable collection of measurable sets is called a measurabledecomposition . Here we are using the terminology of [27] p5 and p26. Thesedecompositions are called separable decompositions in [19]. We say that twomeasurable decompositions ζ and ζ ′ of probability spaces Ω and Ω ′ are mod 0isomorphic if there is a mod 0 isomorphism of Ω mapping the elements of ζ tothe elements of ζ ′ .Given a decomposition ζ of a probability space Ω we may define a projectionmap, π ζ : Ω → ζ by sending a point ω to the element of ζ containing ω . Thisprojection map induces a measure µ ζ on ζ . Rokhlin refers to the resultingmeasurable space as the quotient space Ω /ζ (see p4 of [27]). Definition A.2.
Let ζ be a decomposition of a standard probability space Ω.Let µ C be a set of measures defined indexed by C ∈ ζ . We say that µ C is canonical with respect to ζ if the following hold.(i) µ C is a standard probability space for µ ζ -almost-all C ∈ ζ .(ii) If A is a measurable subset of Ω then:(a) the set A ∩ C is µ C measurable for µ ζ -almost-all C ;(b) µ C ( A ∩ C ) defines a µ ζ -measurable function acting on C ∈ ζ ;(c) the measure A can be recovered by integrating over ζ , i.e. µ ( A ) = Z ζ µ C ( A ∩ C ) d µ ζ . Theorem A.3.
Let Ω be a standard probability space. There exists a set ofmeasures µ C canonical with respect to ζ if and only ζ is a measurable decompo-sition ([27] p26). Moreover, µ C is defined essentially uniquely: if µ C and µ C ′ are both canonical for ζ then µ C is mod 0 isomorphic to µ C ′ for µ ζ -almost-all C ([27] p25). Theorem A.4.
Let Ω be a standard probability space and ζ a measurable de-composition. Let m ζ : ζ → S be given by mapping the measure µ C to the elementof S corresponding to its isomorphism class. Then m ζ is µ ζ measurable. Twodecompositions ζ and ζ ′ are mod 0 isomorphic if and only µ ζ and µ ζ ′ are mod0 isomorphic via a map sending m ζ to m ′ ζ ([27] p40). Finally, Theorem 3.3.1 of [19] tells us that if X is a real random variable, andif we define ζ to be the set of sets of the form X − ( x ) then ζ is a measurabledecomposition. When we apply Theorem [27] to the level sets of a randomvariable ζ , the measure µ X − ( x ) on the level set X − ( x ) for x ∈ R is called the conditional probability measure , conditioned on X = x (see [19] Section 3.5).Note that in this case the projection map sending the level set X − ( x ) to x defines a mod 0 isomorphism between ζ with measure µ ζ and the probabilitymeasure on R induced by X . Proof of Theorem 2.7.
First note that ( µ q , m q ) ∈ Measures( n ) is manifestly aninvariant of Ω.Given a pair M = ( µ, m ) ∈ Measures( n ), let us see how to define Ω( M ) with( µ q , m q ) = M .Let a be the probability space [0 , i >
0, let a i be a probability spaceconsisting of a single atom. We take as probability spaceΩ( M ) = (0 , ∞ ) n × ( ⊔ ∞ i =0 a i ) . This has a measure we denote by ( µ × λ ) induced by taking the standard con-struction of product measures and measures on disjoint unions and then ob-taining the Lebesgue extension. Using our concrete realisation of S , given in(6), we define the components m i of the function m for i ∈ N ∪ {∞} . Let π : Ω M → (0 , ∞ ) n denote the projection onto the (0 , ∞ ) n component. Wethen obtain measurable functions m i ◦ π defined on Ω. Given a Lebesgue mea-26urable subset A of Ω, we define a measure P ( A ) by P ( A ) := Z Ω ∞ X i =0 ( m i ◦ π ) · A ∩ ((0 , ∞ ) n × a i ) d( µ × λ )= Z (0 , ∞ ) n ∞ X i =0 ( m i ◦ π ) Z a i A ∩ ((0 , ∞ ) n × a i ) d( µ × λ | a i )= Z (0 , ∞ ) n ∞ X i =0 m i P a i ( A ∩ π − ( ω ) ∩ a i ) d µ = Z (0 , ∞ ) n P m ( A ∩ π − ( ω )) d µ. (19)Let ζ be the decomposition of Ω( M ) given by the pre-images π − ( ω ) for ω ∈ (0 , ∞ ) n . For ω ∈ (0 , ∞ ) n , let µ π − ( ω ) be the measure m ( ω ). We observe that m ( ω ) is canonical with respect to ζ . We explicitly check the requirements givenin Definition A.2. Property (i) follows since π − ( ω ) is always standard. Similarlyproperty (ii) (a) follows since A ∩ π − ( ω ) is always measurable. Property (ii)(b) follows from Fubini’s theorem, as used in the derivation of equation (19)above. Property (ii) (c) is given by (19) itself.For 1 ≤ i ≤ n , we define measures P i,M by P i,M ( A ) = ω i E µ ( π · A ) (20)where ω i is the i th coordinate function on (0 , ∞ ) n as before. This is an equiv-alent probability measure to P since ω i is positive and has P expectation of1. We see that Ω( M ) equipped with these measures satisfies ( µ q , m q ) = M .Suppose Ω is a probability space with n additional equivalent measures P i .Let M = ( µ q , m q ). By Theorem A.4 we can find a mod 0 isomorphism, φ , fromΩ to Ω M equipped with measure P which also sends q to π for each i . Itfollows from (20) that φ must be a P i -isomorphism too. Proof of Theorem 2.4.
Let S be a standard probability space and ζ a decom-position of S . Let T be another standard probability space. We write ζ ⋆ T for the decomposition of S × T given by taking the product of elements of ζ with T . Given a set of measures µ C on ζ we write µ C × µ T for the productmeasures. It is clear that if µ C is canonical with respect to ζ then µ C is canon-ical with respect to ζ ⋆ T . Thus the conditional measures of d P i d P on Ω × I areall given by products with the standard measure on I . Hence ( m Ω × I ) = 1 µ Ω × I -almost-everywhere.On the other hand, taking the product of a Ω with I does not affect thedistribution µ Ω . So if we take Ω ′ to be the space defined in the statement ofTheorem 2.4, we will have that the invariants of Ω ′ × I are equal to the invariantsof Ω × I . The result now follows from Theorem 2.7. Proof of Theorem 2.3.
Pick a Lebesgue measure P on (0 , ∞ ) with E P ( ω ) = 1.The coordinate function ω on (0 , ∞ ) is just the identity. Define a measure Q by requiring that the Radon–Nikodym derivative is d Q d P = ω = id.27et F be the distribution function of this measure and F − : [0 , → (0 , ∞ )its inverse distribution function. We equip the interval [0 ,
1] with the Lebesguemeasure P ′ and a measure Q ′ given by requiring that the Radon–Nikodymderivative d Q ′ d P ′ = F − .If we can find a simultaneous mod 0 isomorphism between the measures( P , Q ) on M × I and ( P ′ , Q ′ ) on ˜ M × I we see that Theorem 2.3 follows fromTheorem 2.4. We take P = P and P = Q when applying Theorem 2.4.We will now find the required isomorphism. In what follows, if X is a setwith measure µ we will write X µ to emphasize the measure on X .Let 0 ≤ p ≤ p ≤ p and p are the two ends of a connected component of Im F then F is continuous between p and p and so F defines a mod 0 isomorphismbetween [ F − ( p ) , F − ( p )) P and [ p , p ) P ′ . So ( F − [ p , p )) P × I is mod 0isomorphic to ( p , p ) P ′ × I via F × id. This isomorphism maps the randomvariable ω = id to F − X . Hence it is also a mod 0 isomorphism for the measures Q and Q ′ .Suppose that p and p are the two ends of a connected component of [0 , \ Im F . ( F − [ p , p )) P is mod 0 isomorphic to the atom { F − ( p ) } P with mass( p − p ). So ( F − [ p , p )) P × I is mod 0 isomorphic to [ p , p ) P ′ which in turnis mod 0 isomorphic to [ p , p ) P ′ × I . The Q -measure on the atom { F − ( p ) } is equal to F − ( p ), which is equal to F − ( p ) for all p ≤ p ≤ p . Hence( F − [ p , p )) Q × I is simultaneously mod 0 isomorphic to [ p , p ) Q ′ × I .We may therefore cover [0 , × I with a countable set of disjoint intervalsof the form [ p , p ) × I which are simultaneously P / Q mod 0 isomorphic to( F − [ p , p )) × I .We may therefore combine these mod 0 isomorphisms on intervals to obtainthe desired mod 0 isomorphism for the P and Q measures. Proof of Corollary 2.5.
We have the obvious inclusion ι : L P (Ω) → L P (Ω × I ).Any element of L P (Ω × I ) which can be written as a function of the Radon–Nikodym derivatives d P i d P must lie in the image of ι . Hence it suffices to provethat ιA contains an element which can be written as a function of these Radon–Nikodym derivatives.By Theorem 2.4 we may assume without loss of generality that the marketΩ × I is given by Ω ′ × I = (0 , ∞ ) n × I and P ′ i as described in Theorem 2.4. Inthis case the Radon–Nikodym derivatives are given by the coordinate functions ω i . Let G = S ∼ = R / Z with measure given by the quotient measure. Since eachelement of R / Z has a unique representative on [0 , G is strictly isomorphicto [0 ,
1) as a probability space. Hence we may define an action of G on anyproduct space X × I by using the action on the right-hand side of the product.We can apply Theorem 1.6 with this choice of G and taking as ιA as the convexset. The result now follows.We collect together the key properties of rearrangement in a single lemma.28 emma A.5. Let m be a Lebesgue measure on (0 , ∞ ) satisfying condition (8) .Then U m is a uniformly-distributed random variable. Let X be a random vari-able in X ∈ L P m ((0 , ∞ ) × [0 , R ) .The P m distribution is left fixed by rearrangement of X . The Q m distribu-tions are increased or decreased according to whether one applies the increasingor decreasing rearrangement. Symbolically: d P m ( X ) = d P m ( R ± m ( X )) , (21) d Q m ( X ) (cid:22) d Q m ( R + m ( X )) , (22) d Q m ( X ) (cid:23) d Q m ( R − m ( X )) . (23) In addition: d Q m d P m ( ω ) < d Q m d P m ( ω ′ ) = ⇒ R ± m ( ± X ( ω )) ≤ R ± m ( ± X ( ω ′ )) , (24) d P m ( X ) (cid:22) d P m ( Y ) = ⇒ d Q m ( R + m ( X )) (cid:22) d Q m ( R + m ( Y )) , (25) F d Q m d P m is continuous at x = ⇒ R ± m ( X )( x, y ) = R ± ( X ) m ( x, y ) ∀ y , y . (26) Proof.
Pick z ∈ (0 , F m is an increasing function, we can find x ∈ (0 , ∞ ) with lim x ′ → x − F m ( x ) ≤ z ≤ lim x ′ → x + F m ( x ). Hence we can find y with U m ( x , y ) = z . Since F m is increasing, we deduce that P m ( U m ( ω ) ≤ z ) = P m ( x ( ω ) < x or ( x ( ω ) = x and y ( ω ) ≤ y ))= P m ( x ( ω ) < x ) + P m ( x ( ω ) = x ) P m ( y ( ω ) ≤ y )= lim x → x − F m ( x ) + y ( lim x → x + F m ( x ) − lim x → x − F m ( x )) = z. (27)We deduce first that U m is measurable since its sublevel sets are measurable.We then deduce that U m is a uniform random variable as (27) is the definingproperty of uniform random variables.Property (21) of rearrangement follows immediately from the fact that U m is uniform and from the definition of rearrangement.We note that for α ∈ (0 , { x ∈ R | F X ( x ) ≥ α } ≤ k = ⇒ F X ( k ) ≥ α. So from the definition of rearrangement P ( R + m ( X )( ω ) ≤ k ) = P ( F − X ( U m ( ω )) ≤ k ) = P (inf { x ∈ R | F X ( x ) ≥ U m ( ω ) } ≤ k ) ≤ P ( F X ( k ) ≥ U m ( ω )) = F X ( k ) . The last step uses (27). We have established (22). Property (23) is now obvious.29rom the definition of U m , if x ( ω ) ≤ x ( ω ′ ) then U m ( ω ) ≤ U m ( ω ′ ). F X isincreasing and x is equal to the Radon–Nikodym derivative d Q m d P m . Hence (24)follows.From the definition of U m , U m ( x, y ) is independent of y when F m is contin-uous at x . Hence R ± m ( X )( x, y ) is also independent of y . Note that F m = F d Q m d P m .This establishes (26).To establish (25) let us suppose d P m ( X ) (cid:22) d P m ( Y ). This means that F X ( k ) ≥ F Y ( k ) ∀ k ∈ R where F X and F Y are the P m -measure distribution functions of X and Y . Hence F − X ( p ) ≤ F − Y ( p ) ∀ p ∈ [0 , . (28)We then find Q ( R + m ( X ) ≤ k ) = E m ( x ( R + m ( X ) ≤ k ) ) = E m ( x ( F − X ◦ U m ≤ k ) ) ≥ E m ( x ( F − Y ◦ U m ≤ k ) ) by (28)= Q ( R + m ( Y ) ≤ k ) . So d Q m ( R + m ( X )) (cid:22) d Q m ( R + m ( Y )) as claimed. Lemma A.6. If (Ω , F , P ) is a probability space, X and Y are real randomvariables and Z is an R k random variable satisfying d P ( X | Z ) (cid:22) d P ( Y | Z ) then d P ( X ) (cid:22) d P ( Y ) .Proof. P ( X ≤ k ) = R R k P ( X ≤ k | Z ) d Z ≤ R R k P ( Y ≤ k | Z ) d Z = P ( Y ≤ k ) . Proof of Theorem 2.9.
By Theorem 2.4, we only need consider the case whenΩ = (0 , ∞ ) n equipped with a measure µ satisfying E µ ( x i ) = 1 for each coordi-nate function x i .Given an integer j , 1 ≤ j ≤ n , we define a random n − q j ( ω )consisting of all the components of q except the j th. We write ˆ µ j for themeasure induced on (0 , ∞ ) n − by q ˆ j . We write q j for the j th component of q ,and write µ j for the measure on (0 , ∞ ) induced by q j .Given a random variable X on (0 , ∞ ) n × [0 ,
1) and a value Q ∈ (0 , ∞ ) n − we may define X j,Q : (0 , ∞ ) × [0 , → R by X j,Q ( x, y ) = X ( Q ⊕ j x, y ) , where Q ⊕ j x is the vector obtained by inserting a new component with value x at the j th index of the vector Q . X j,Q is ˆ µ j -almost-surely measurable.Let y denote the final coordinate function on (0 , ∞ ) n × [0 , conditional rearrangements R + j and R − j as follows R ± j ( X )( ω ) := R ± µ j ( X j, ˆ q j ( ω ) ) ( q j ( ω ) , y ( ω )) .
30e define R j = R + j if sign j = 1, and R j = R − j otherwise. Since X j,Q isˆ µ j -almost-surely measurable, R ± j is well-defined mod 0.We need to check that R ± j is measurable. We note that F − X j, ˆ qj ( ω ) ( p ) = inf { z ∈ R | F X j, ˆ qj ( ω ) ( z ) ≥ p } = inf { z ∈ Q | F X j, ˆ qj ( ω ) ( z ) ≥ p } using the monotonicity of distribution functions. Define f ( z, ω, p ) = ( z F X j, ˆ qj ( ω ) ( z ) ≥ p, ∞ otherwise . It is obvious from chasing through the definitions that f is measurable. Theinfimum of a countable sequence of measurable functions is measurable. Hence F − X j, ˆ qj ( ω ) ( p ) is measurable as a function of the pair ( ω, p ). By definition R + µ j ( X j, ˆ q j ( ω ) )( x, y ) = F − X j, ˆ qj ( ω ) ( U µ j ( x, y )) , so this quantity is measurable as a function of ( ω, x, y ). The measurability of R ± j ( X ) is now immediate.We inductively define R ∗ ( X ) = X and R ∗ j ( X ) = R j ( R ∗ j − ( X )) for 1 ≤ j ≤ n .We define R ( X ) = R ∗ n ( X ).Let us suppose as induction hypothesis that we have established for some j < n that d P i ( X ) = d P i ( R ∗ j ( X )) if i = 0 or i > j,d P i ((sign j ) X ) (cid:22) d P i ( R ∗ j ((sign j ) X )) otherwise . (29)We may then apply equations (21), (22), (23) and (25) to find d P i ( X | ˆ q j +1 ) = d P i ( R ∗ j +1 ( X )) | ˆ q j +1 ) if i = 0 or i > j + 1 ,d P i ( R ∗ j +1 ((sign j ) X ) | ˆ q j +1 ) (cid:22) d P i ( R ∗ j +1 ((sign j ) X ) | ˆ q j +1 ) otherwise . (30)Applying Lemma A.6 below, we may deduce from equations (30) that our in-duction hypothesis (29) will also hold when j → j + 1. We deduce that (29)holds for 0 ≤ j ≤ n . This establishes properties (i) and (ii) of R ( X ).For each i (0 ≤ i ≤ n ), define a partial order (cid:22) i on R n by x (cid:22) i y ⇐⇒ ( (sign j ) x j ≤ (sign j ) y j ≤ j ≤ ix j = y j i < j ≤ n. We suppose as induction hypothesis that for some 1 ≤ i ≤ n − R ∗ i − ( X )( ω ) ≤ R ∗ i − ( X )( ω ′ ) if q ( ω ) ≺ i q ( ω ′ ) . (31)31rite q a ( ω ) for the vector containing the first ( i −
1) components of q ( ω ), q b ( ω )for the i th component of q ( ω ) and q c ( ω ) for the remaining components. So q ( ω ) = q a ( ω ) ⊕ q b ( ω ) ⊕ q c ( ω ).Suppose that q ( ω ) ≺ i +1 q ( ω ′ ) then q a ( ω ) (cid:22) q a ( ω ′ ), q b ( ω ) ≤ q b ( ω ′ ), q c ( ω ) = q c ( ω ′ ). We also have either: (a) q a ( ω ) ≺ q a ( ω ′ ) and q b ( ω ) = q b ( ω ′ ); (b) q a ( ω ) = q a ( ω ′ ) and q b ( ω ) < q b ( ω ′ ); or (c) q a ( ω ) ≺ q a ( ω ′ ) and q b ( ω ) < q b ( ω ′ ).In case (a), our induction hypothesis (31) tells us that R ∗ i − ( X )( ω ) ≤ R ∗ i − ( X )( ω ′ ) . Hence by property (25) of rearrangement R ∗ i ( X )( ω ) = R i ( R ∗ i − ( X ))( ω ) ≤ R i ( R ∗ i − ( X ))( ω ′ ) = R ∗ i ( X )( ω ′ ) . In case (b), we may apply (24) to the rearrangement R i of the randomvariable R ∗ i − ( X ) to find that R ∗ i ( X )( ω ) ≤ R ∗ i ( X )( ω ′ ). In case (c) we applyour results for case (a) and case (b) in succession and use the transitivity of ≤ to again find that R ∗ i ( X )( ω ) ≤ R ∗ i ( X )( ω ′ ). Thus (31) remains true when wechange ( i − → i .The induction hypothesis (31) is trivially true when i = 1, so claim (iii)follows. Proof of Corollary 2.10.
Let X ∈ L (Ω). We define ˜ X ∈ L (Ω × [0 ,
1) by˜ X ( ω, y ) = X ( ω ). This will satisfy d P i ( X ) = d P i ( ˜ X ) for all i .Consider case (b) of our claim. By property (26) of rearrangement, R j , andhence R , only depends upon q . So we may write R ( ˜ X ) = ˆ X ( q ) for some ˆ X . Wedefine ˜ R ( X ) = ˆ X ( q ), and it will satisfy all the desired properties.Now consider case (a) of our claim. Let us write { x n } for the countable set ofdiscontinuities of F q . We define a set ∆ n := ( q ) − ( x n ). Since the probabilityspace is standard and atomless, there is a mod 0 isomorphism φ n from the set∆ n to the set { x n } × I. We write ∆ = S ∆ n . Property (26) tells us that therearrangement R ( ˜ X )( ω, y ) only depends upon y if x ∈ Ω \ ∆. So we may definea function ˆ X on (0 , ∞ ) \ { x n } by ˆ X ( q ) = R ( ˜ X ) on Ω \ ∆. We now define˜ R ( X )( ω ) = ( ˆ X ( q ( ω )) ω ∈ Ω \ ∆ ,R ( ˜ X )( φ ( X )) otherwise . Since each φ n is a mod 0 isomorphism on ∆ n and preserves the Radon–Nikodymderivatives, we see that d P i ( ˜ R ( X )) = d P i ( R ( ˜ X ))for i = 0 ,
1. The result follows.
A.3 Proofs for Section 3
Let us briefly review how the measure Q is constructed. Suppose that furtherto the assumptions of Definition 3.3, we may define a process Z t by Z t = Z t ( σ − ( r X s − µ )) · d W s (32)32here · denotes the usual inner product of vectors. We have suppressed theparameters ( X s , s ) of the functions σ and µ to keep our expressions readable,and will do this throughout this section. We then define q t to be the Dol´eans-Dade exponential of Z t , q t = exp (cid:18) Z t −
12 [
Z, Z ] t (cid:19) , (33)so that q is a positive process and a local P -martingale. If q t is a P -martingale,then the measure Q can be defined by Q ( A ) = E P ( q T A ) (34)for a measurable set A ⊂ Ω. Proof of Theorem 3.8.
Applying Itˆo’s Lemma to the defining equation for theDol´eans-Dade exponential we compute that Z t q s d[ q, q ] s = [ Z, Z ] t . Hence by (32) Z t q s d[ q, q ] s = Z t | σ − ( r X s − µ ) | d s. (35)Since q t = d Q d P (cid:12)(cid:12)(cid:12) F t ,q t is manifestly an invariantly-defined stochastic process (for the obvious choiceof functor). Hence the left-hand side of equation (35) is manifestly an invariantly-defined stochastic process. We can characterise the process | σ − ( r X t − µ ) | asthe unique non-negative element in A t ∈ L (Ω × [0 , T ] , P × λ ) satisfying Z t q s d[ q, q ] s = Z t A s d s.A t defined in this way is manifestly invariantly defined, so the absolute marketprice of risk is also invariantly defined. Proof of Theorem 3.11.
Suppose for a contradiction that n -dimensional Wienerspace, Ω n , is isomorphic to m -dimensional Wiener space with m > n . Usingthis isomorphism we may find m independent standard Brownian motions onΩ n , ˜ W jt (1 ≤ j ≤ m ). By the martingale representation theorem, there areunique, predictable processes α ijt (1 ≤ i ≤ n , 1 ≤ i ≤ m ) such that˜ W jt = Z t n X a =1 α ajs d W as . Let α t be the n × m matrix with components α ij and let id m denote the identitymatrix of dimension m . We compute the quadratic-covariation matrix of eachside in the above expression to obtain id m = ( α t )( α t ) ⊤ . Since α t has rank lessthan or equal to n , and id m has rank m we obtain the desired contradiction.33 roof of Theorem 3.12. Given such a complete market, let Q t be defined as in(13) and let ˜ Z t = log Q t + 12 Z t A ( s ) d s. (36)We computed ˜ Z t = d(log Q t ) + 12 A ( t ) d t = 1 Q t d Q t − Q d[ Q, Q ] t + 12 A ( t ) d t = 1 Q t d Q t . Hence ˜ Z t is a continuous local martingale. We now define˜ W t = − Z t A ( s ) d ˜ Z s . (37) W t is a continuous local martingale by our assumptions on A ( t ). We computeits quadratic variation.[ ˜ W , ˜ W ] t = Z t A ( s ) d[ ˜ Z, ˜ Z ] s = Z t A ( s ) d[log Q, log Q ] s = Z t Q s A ( s ) d[ Q, Q ] s = Z t d s = t by (37), (36), Itˆo’s Lemma and (12). It follows by L´evy’s characterisation ofBrownian motion that ˜ W t is Brownian motion.We may now find additional Brownian motions, ˜ W it for 2 ≤ i ≤ n , suchthat the vector process ˜ W t with components ˜ W it is a standard n -dimensionalBrownian motion.To see this, we use the fact that Ω is assumed to be an n -dimensional Wienerspace, so admits an n -dimensional standard Brownian motion ˆ W t . Using themartingale representation theorem, we may write ˜ W t = R t α s · d ˆ W s for a pre-dictable vector process α t of norm 1. Given a vector v ∈ R n of norm 1, wedefine a number i k for each 2 ≤ k ≤ n by i k = inf { i | dim h v, e , e , . . . , e i i ≥ k } .Then { v, e i , e i , . . . , e i n } is a basis of R n . Applying the Gram–Schmidt processto this basis yields an orthonormal basis { v i } for R n with v = v and which isdetermined entirely by v . Applying this construction with v = α s we obtain apredictable orthonormal basis { α it } . We now define˜ W it = Z t α is · d W s . The process ˜ W t is a continuous semi-martingale and its quadratic-covariationmatrix has components[ ˜ W i , ˜ W j ] t = Z t α is · α js d s = t δ ij . Hence by L´evy’s characterisation this is indeed n -dimensional Brownian motion.We now define a stochastic process X t byd X t = ( r X t + A ( t ) e )d t + d ˜ W t . (38)34ere we use the boundedness and measurability of A to ensure existence anduniqueness of the solution to this SDE. The continuous-time market associatedto (38) has Z t given by formula (32), sod Z t = − A ( t ) d ˜ W t . (39)In particular d[ Z, Z ] t = A ( t ) d t , so equation (33) becomeslog( q t ) = Z T − Z t A ( s ) d s. So we findd(log q t ) = d Z T − A ( t ) d t = − A ( t )d ˜ W t − A ( t ) d t, by (39) . On the other hand we compute from (36) and (37) thatd(log Q t ) = d ˜ Z t − A ( t ) d t = − A ( t )d ˜ W t − A ( t ) d t. Since we also have q = Q = 1, we see that Q t = q t .Prices in M are, by definition, given by c t ( X ) = E ( e − r ( T − t ) QX | F t ) = E ( e − r ( T − t ) Q t X t ) . Prices in the complete market associated with (38) are given by the same for-mulae with Q replaced by q . Hence the costs are the same in both markets,showing that we have identified a market isomorphism. Proof of Theorem 3.15.
Without loss of generality our market is a canonicalBachelier market. Let A be an invariant convex set of martingales. Let Y bean element of A . By the martingale representation theorem Y t = Y + n X i =1 Z t a is d W is for some predictable processes a is . By invariance of A , we see that Y t = Y + Z t a s d W s − n X i =2 Z t a is d W is is also in A , as flipping the signs of the Brownian motions W kt for 2 ≤ k ≤ n induces an isomorphism of the canonical Bachelier model.By the convexity of A , Y t = Y + Z t a s d W is A . Hence by the theory of [14], the martingale Y t can be replicated usinga predictable self-financing trading strategy using only the asset W t and therisk-free asset. A second application of the martingale representation theoremshows that the asset W t may itself be replicated by a trading strategy usingonly the assets X it . The hedging portfolio obtained in this way gives rise to theportfolio referred to in the statement of the theorem.We wish to compute this portfolio explicitly in the case of markets of theform (9).We may read off from (32) and (37) thatd ˜ W t = − A ( s ) σ − ( r X s − µ ) · d W s . From (9) we may writed ˜ W t = − A ( t ) σ − ( r X t − µ ) · ( σ − (d X t − µ d t )) = − A ( t ) ( σσ ⊤ ) − ( r X t − µ ) · (d X t − µ d t ) . We can now read off that the portfolio of risky assets one should hold in orderto replicate W t is proportional to ( σσ ⊤ ) − ( r X s − µ ) . Appendix B Basic concepts of category theory
In this section, we review the concepts from category theory required for thispaper.