On the game interpretation of a shadow price process in utility maximization problems under transaction costs
aa r X i v : . [ q -f i n . P M ] D ec ON THE GAME INTERPRETATION OF A SHADOW PRICEPROCESS IN UTILITY MAXIMIZATION PROBLEMS UNDERTRANSACTION COSTS
DMITRY B. ROKHLIN
Abstract.
To any utility maximization problem under transaction costs onecan assign a frictionless model with a price process S ∗ , lying in the bid/askprice interval [ S, S ]. Such process S ∗ is called a shadow price if it provides thesame optimal utility value as in the original model with bid-ask spread.We call S ∗ a generalized shadow price if the above property is true for the relaxed utility function in the frictionless model. This relaxation is defined asthe lower semicontinuous envelope of the original utility, considered as a func-tion on the set [ S, S ], equipped with some natural weak topology. We provethe existence of a generalized shadow price under rather weak assumptionsand mark its relation to a saddle point of the trader/market zero-sum game,determined by the relaxed utility function. The relation of the notion of ashadow price to its generalization is illustrated by several examples. Also, webriefly discuss the interpretation of shadow prices via Lagrange duality. Introduction
A possible approach to the analysis of optimization problems under transactioncosts consists in their reduction to the correspondent problems in frictionlessmodels. The main point of this approach is to determine a frictionless priceprocess S ∗ , called a shadow price, lying in the bid/ask price interval [ S, S ] andensuring the same optimal utility value. This method was successfully applied tosome continuous time portfolio optimization problems in the recent papers [17],[13], [14], [12]. Previously in the same context a shadow price process with suchinterpretation explicitly appeared in [20].In discrete time setting for the case of finite probability space the existence ofa shadow price in an investment/consumption optimization problem was estab-lished in [18]. Inspired by this result, we consider an optimal investment problemin discrete time model over general probability space. It should be mentionedthat very recently in the paper [3] the existence of a shadow price process wasestablished in general multi-currency continuous time market models under shortselling constraints.The main feature of the present paper is the game interpretation of a shadowprice process. As it was was mentioned in the cited papers, a shadow pricecan be interpreted as a least favourable frictionless price from trader’s point of
Mathematics Subject Classification.
Key words and phrases.
Transaction costs, utility maximization, shadow price process, lowersemicontinuous envelope, saddle point, duality. view. So, it is natural to consider a trader/market zero-sum game determinedby trader’s utility Ψ(
S, γ ), regarded as a function of frictionless price process S and an investment strategy γ . Moreover, one can expect that a pair ( γ ∗ , S ∗ ),composed of an optimal strategy γ ∗ and a shadow price S ∗ , corresponds to asaddle point of Ψ.However, an application of customary minimax theorems (see [28]) is notstraightforward. Firstly, usually Ψ is not convex or quasiconvex in S . Secondly,in general it is not lower semicontinuous in a topology, ensuring the compactnessof the set [ S, S ]. To overcome at least the second difficulty, for each γ we passto the lower semicontinuous envelope b Ψ of Ψ in some natural weak topologyon [
S, S ], and introduce the corresponding notion of a generalized shadow priceprocess S ∗ .The method, involving a consideration of the lower semicontinuous envelope(relaxation) of the objective functional is extensively used in analysis of varia-tional problems [9], [2], [7]. In the present context it appears that the relaxedproblem fits nicely into the framework of the intersection theorem, proved byHa [15] (see Theorem 2 below). Applying this result, in Section 2 we establishthe existence of a generalized shadow price and the minimax property of b Ψ un-der rather weak assumptions (Theorem 1). Moreover, if there exists an optimalsolution γ ∗ of the original utility maximization problem, then a pair ( γ ∗ , S ∗ ),where S ∗ is a generalized shadow price, is exactly the strategic saddle point ofthe game, determined by the relaxed utility function b Ψ( S, γ ) (Theorem 3).Thus, the advantage of passing to the relaxed problem is twofold: (1) theexistence of a generalized shadow price process S ∗ is guaranteed under weakassumptions, (2) the relaxed utility b Ψ has nice minimax and saddle-point prop-erties.The relation of the notion of a shadow price to its generalization is illustratedby several examples in Section 3. If the original utility function Ψ(
S, γ ) is alreadylower semicontinuous in S in an appropriate topology, the proposed approachgives the existence of a shadow price (Examples 1 and 2). Another interestingcase appears when b Ψ = Ψ but it is still possible to give a convenient analyticaldescription of the saddle points ( S ∗ , γ ∗ ) of b Ψ. If γ ∗ is an optimal solution ofthe original utility maximization problem under transaction costs, Ψ( S ∗ , γ ∗ ) = b Ψ( S ∗ , γ ∗ ) and the optimality of γ ∗ for the functions γ Ψ( S ∗ , γ ), γ b Ψ( S ∗ , γ )is characterized by identical conditions, then a generalized shadow price S ∗ is infact a shadow price (Example 3).Furthemore, we give an example of two-step model on a countable probabilityspace with linear utility functional such that a generalized shadow price existsand a shadow price is not (Example 4). Independently an example of the samenature in three-step model was constructed in [3]. In spite of the nonlinearityof the objective functional, the advantage of the latter example is the use oflogarithmic utility, while in Example 4 the utility contains a Banach limit. Wefind it interesting to test our approach on the example of [3]. It appears that the N THE GAME INTERPRETATION OF A SHADOW PRICE 3 ”unsuccessful candidate” for a shadow price, mentioned in [3], is a generalizedshadow price (see Example 5 of the present paper).It is worth mentioning that usually a ”shadow” or ”equilibrium” resourceprices are associated with an optimal solution of the Lagrange (or Fenchel) dualproblem. In Section 4 we trace this connection in the problem under consider-ation, confining ourselves to the case of finite probability space. We show thata shadow price is equal to the relation of equilibrium prices of stock and bond.The related calculations indicate quite explicitly that the zero duality gap andthe solvability of the Lagrange dual problem immediately imply the existence ofa shadow price process. This point seems promising for generalizations, concern-ing the existence of a shadow price. However, Examples 4 and 5 show that thisway is not so easy in the infinite-dimensional setting. See also the comments,concerning the papers [5], [6], in the introductory section of [3].2.
Main result
Consider a trader, who can distribute his wealth between a bond with zerointerest rate (and price 1) and a risky asset (stock). As usual, he acts in randomsetting, described by a probability space (Ω , F , P ) endowed with discrete-timefiltration F = ( F t ) Tt = − , F − = {∅ , Ω } . The stock can be sold at the bid price S t and purchased at the ask price S t at a time moment t . It is assumed that0 < S t ≤ S t and the processes S , S are F -adapted. A trading strategy isdetermined by an F -adapted portfolio process ( β t , γ t ) Tt = − , consisting of β t untisof bond (or cash) and γ t units of stock. A trading strategy is called self-financing if any change in risky position is compensated by the cash flow:∆ β t = S t (∆ γ t ) − − S t (∆ γ t ) + , t = 0 , . . . , T, where ∆ a t = a t − a t − , x + = max { x, } , x − = max {− x, } . To fix the values β − , γ − we assume that the trader starts from one unit of bond: β − = 1, γ − = 0. Moreover, at the terminal date T the asset holdings are converted tocash: γ T = 0. Hence, trader’s terminal wealth is given by X T ( γ ) = 1 + T X t =0 ( S t (∆ γ t ) − − S t (∆ γ t ) + ) , γ − = 0 , γ T = 0 . (2.1)For the frictionless model ( S = S = S ) this formula shapes to the customaryform: X T ( γ ) = 1 + ( γ ◦ S ) T := 1 + T X t =1 γ t − ∆ S t . Denote by L ( F t ) the set of equivalence classes of P -a.s. equal F t -measurablereal-valued random variables. The sets L p ( F t ), 1 ≤ p < ∞ and L ∞ ( F t ) consistof p -th power P -integrable and P -essentially bounded elements of L ( F t ) respec-tively. We equip L with the topology of convergence in probability, induced bythe metric ρ ( f, g ) = E | f − g | | f − g | . (2.2) DMITRY B. ROKHLIN
Unless otherwise stated, the sets L p , p ∈ [1 , ∞ ); L ∞ are considered as Banachspaces with the norms k f k p = ( E | f | p ) /p , k f k ∞ = ess sup | f | . We consider two possible choices of spaces, containing the portfolio strate-gies γ : γ t ∈ L s ( F t ), s ∈ { , ∞} . However, in each case we equip L s ( F t ) withthe topology τ t of convergence in probability. Denote by F the vector space Q T − t =0 L s ( F t ) with the product topology τ = Q T − t =0 τ t and let Y be a convexsubset of F . Since the values γ − = 0, γ T = 0 are fixed, in what follows γ is con-sidered as an element of F (except for Section 4). We allow portfolio constraintsof the form γ ∈ Y .Assume that S t , S t ∈ L q ( F t ) for some q ∈ [1 , ∞ ]. Put τ wt = σ ( L q ( F t ) , L p ( F t )),where 1 /p + 1 /q = 1. So, τ wt is the weak topology of L q for q ∈ [1 , ∞ ) and theweak-star topology of L ∞ . In any case the set[ S t , S t ] = { S t ∈ L q ( F t ) : S t ≤ S t ≤ S t } is τ wt -compact. Since the closedness of [ S t , S t ] is clear, this assertion follows fromthe τ wt -compactness of the unit ball for q ∈ (1 , ∞ ] and the uniform integrabilityof [ S t , S t ] for q = 1. Denote by E the vector space Q Tt =0 L q ( F t ) with the producttopology τ w = Q Tt =0 τ wt and put X = [ S, S ] := T Y t =0 [ S t , S t ] . A functional Φ : L r ( F T ) [ −∞ , ∞ ] , r = min { s, q } . Φ is called monotone if Φ( X ) ≥ Φ( Y ) whenever X ≥ Y , X, Y ∈ L r ( F T ) and quasiconcave if Φ( α X + α Y ) ≥ min { Φ( X ) , Φ( Y ) } for all X, Y ∈ L r ( F T ), α + α = 1, α i ≥
0. It is easy to see that Φ is quasiconcaveiff the upper level sets { X ∈ L r ( F T ) : Φ( X ) > β } are convex for all β ∈ R .We admit that trader’s preferences are represented by a monotone quasiconcavefunctional Φ. Recently such framework attracted a considerable attention inconnection with quasiconvex risk measures [4], [11].The optimal value of the utility maximization problem under transaction costsand portfolio constraints, represented by Y , is defined as follows λ = sup { Φ( X T ( γ )) : γ ∈ Y } . (2.3)Note that X T ( γ ) ∈ L r ( F T ) under the above notation.Along with (2.3) consider the optimization problem in a frictionless model,where the stock price is given by an adapted process S ∈ [ S, S ]: µ S = sup { Φ(1 + ( γ ◦ S ) T ) : γ ∈ Y } . (2.4)Following [18] we call an adapted process S ∈ [ S, S ] a shadow price if µ S = λ . N THE GAME INTERPRETATION OF A SHADOW PRICE 5
We are going to introduce a modification of the last notion. PutΨ(
S, γ ) = Φ(1 + ( γ ◦ S ) T )and denote by b Ψ( · , γ ) the τ w -lower semicontinuous envelope (relaxation) of Ψ( · , γ )as a function on [ S, S ] (see [21], Definition 2.1.13): b Ψ( S, γ ) = sup V ∈N ( S ) inf S ′ ∈ V Ψ( S ′ , γ ) , where N ( S ) is a local base of the topology τ w , restricted to [ S, S ]. As is known(see [21], Proposition 2.1.15), b Ψ( · , γ ) is the largest τ w -lower semicontinuous func-tion majorized by Ψ( · , γ ). Note thatΦ( X T ( γ )) ≤ b Ψ( S, γ ) ≤ Ψ( S, γ ) , S ∈ [ S, S ] (2.5)since X T ( γ ) ≤ γ ◦ S ′ ) T , S ≤ S ′ ≤ S and Φ is monotone.Consider instead of (2.4) the optimization problem for the relaxed functional b Ψ: b µ S = sup { b Ψ( S, γ ) : γ ∈ Y } . (2.6)We call S a generalized shadow price if b µ S = λ .Looking at (2.5), we immediately conclude that any shadow price S ∗ is ageneralized shadow price: λ = sup γ ∈Y Φ( X T ( γ )) ≤ sup γ ∈Y b Ψ( S ∗ , γ ) = b µ S ∗ ≤ sup γ ∈Y Ψ( S ∗ , γ ) = λ. If Φ is quaisiconcave then Ψ( S, · ), b Ψ( S, · ) are quasiconcave as well. Indeed, for α + α = 1, α i ≥ γ i ∈ Y we haveΨ( S, α γ + α γ ) = Φ( α (1 + ( γ ◦ S ) T ) + α (1 + ( γ ◦ S ) T )) ≥ min i =1 , Φ(1 + ( γ i ◦ S ) T ) = min i =1 , Ψ( S, γ i ) . Let b Ψ( S, γ i ) > β . Take V i ∈ N ( S ) such that inf S ′ ∈ V i Ψ( S ′ , γ i ) > β and put V = V ∩ V . The inequality b Ψ( S, α γ + α γ ) ≥ inf S ′ ∈ V Ψ( S ′ , α γ + α γ ) ≥ min i =1 , inf S ′ ∈ V Ψ( S ′ , γ i ) > β means that the upper level sets { γ ∈ Y : b Ψ( S, γ ) > β } are convex.Now we state the main result of the present paper. Theorem 1.
Let Φ be monotone and quasiconcave, S t , S t ∈ L q ( F t ) , t = 0 , . . . , T for some q ∈ [1 , ∞ ] . Then there exists a generalized shadow price S ∗ ∈ [ S, S ] and the following minimax relations hold true: λ = sup γ ∈Y Φ( X T ( γ )) = sup γ ∈Y inf S ∈ [ S,S ] b Ψ( S, γ ) = sup γ ∈Y inf S ∈ [ S,S ] Ψ( S, γ ) (2.7)= inf S ∈ [ S,S ] sup γ ∈Y b Ψ( S, γ ) = sup γ ∈Y b Ψ( S ∗ , γ ) = b µ S ∗ . In fact, Theorem 1 is a direct consequence of the following intersection theorem([15], Theorem 3). We formulate it in a slightly weaker form.
DMITRY B. ROKHLIN
Theorem 2 (Ha, 1980) . Let E , F be Hausdorff topological vector spaces, X ⊂ E be a convex compact set, Y ⊂ F be a convex set. Let B ⊂ A ⊂ X × Y be subsetssuch that (a) for each y ∈ Y the set { x ∈ X : ( x, y ) ∈ A } is closed; (b) for each x ∈ X the set { y ∈ Y : ( x, y ) A } is convex; (c) B is closed in X × Y and for each y ∈ Y the set { x ∈ X : ( x, y ) ∈ B } isnonempty and convex.Then there exists a point x ∗ ∈ X such that { x ∗ } × Y ⊂ A .Proof of Theorem 1 . The topological vector spaces ( E , τ w ), ( F , τ ) and sets X = [ S, S ], Y , introduced above, satisfy the conditions of Theorem 2. Put A = { ( S, γ ) ∈ X × Y : b Ψ( S, γ ) ≤ λ } . Condition (a) of Theorem 2 is satisfied since b Ψ( · , γ ) is τ w -lower semicontinuousand the validity of (b) follows from the quasiconcavity of b Ψ( S, · ): { γ ∈ Y : ( S, γ ) A } = { γ ∈ Y : b Ψ( S, γ ) > λ } . Furthermore, consider the set-valued mapping b B from Y to the power set of X , defined as follows b B ( γ ) = (cid:0) { S t } I { ∆ γ t < } + { S t } I { ∆ γ t > } + [ S t , S t ] I { ∆ γ t =0 } (cid:1) Tt =0 and denote by B the graph of b B : B = { ( S, γ ) ∈ X × Y : S ∈ b B ( γ ) } . For (
S, γ ) ∈ B we have X T ( γ ) = 1 + T X t =0 ( S t (∆ γ t ) − − S t (∆ γ t ) + ) = 1 − T X t =0 S t ∆ γ t = 1 + ( γ ◦ S ) T (2.8)and Φ( X T ( γ )) = Ψ( S, γ ) ≤ λ . Thus, b Ψ( S, γ ) ≤ λ and B ⊂ A .We claim that B satisfies condition (c) of Theorem 2. Cleary, the sets { S ∈X : ( S, γ ) ∈ B } = b B ( γ ) are nonempty and convex. It remains to prove that B is closed in X × Y . Let (
S, γ ) ∈ X × Y lie in the closure of B in the producttopology τ w × τ , restricted to X × Y . To prove that (
S, γ ) ∈ B it is sufficient toshow that S t I { ∆ γ t =0 } = S t I { ∆ γ t < } + S t I { ∆ γ t > } , t = 0 , . . . , T. (2.9)For any t ∈ { , . . . , T } , n ∈ N and g t ∈ L ∞ ( F t ) there exist γ n ∈ Y and S n ∈ [ S, S ] of the form S n = SI { ∆ γ n < } + SI { ∆ γ n > } + b S n I { ∆ γ n =0 } , S ≤ b S n ≤ S such that ρ ( γ nt , γ t ) < /n , | E ( S nt − S t ) g t I { ∆ γ t =0 } | < /n , where ρ is definedby (2.2). Passing to subsequences (still denoted by γ nt , S nt ), we may assumethat γ nt → γ t P -a.s. Here γ nt , γ t are understood as functions, taken from thecorrespondent equivalence class. N THE GAME INTERPRETATION OF A SHADOW PRICE 7
On the set { ∆ γ t = 0 } we have I { ∆ γ nt < } → I { ∆ γ t < } , I { ∆ γ nt > } → I { ∆ γ t > } , I { ∆ γ nt =0 } → P -a.s.From the the dominated convergence theorem it follows thatlim n →∞ E (cid:0) g t ( S t I { ∆ γ nt < } + S t I { ∆ γ nt > } ) I { ∆ γ t =0 } (cid:1) = E ( g t ( S t I { ∆ γ t < } + S t I { ∆ γ t > } ) , (cid:12)(cid:12)(cid:12) E (cid:16) g t b S nt I { ∆ γ nt =0 } I { ∆ γ t =0 } (cid:17)(cid:12)(cid:12)(cid:12) ≤ E (cid:0) | g t | S t I { ∆ γ nt =0 } I { ∆ γ t =0 } (cid:1) → . Hence, E (cid:0) g t S t I { ∆ γ t =0 } (cid:1) = lim n →∞ E (cid:0) g t S nt I { ∆ γ t =0 } (cid:1) = E (cid:0) g t ( S t I { ∆ γ t < } + S t I { ∆ γ t > } ) (cid:1) for any g t ∈ L ∞ ( F t ) and (2.9) is satisfied.Now we can apply Theorem 2 and take an element S ∗ ∈ X such that { S ∗ }×Y ⊂ A . That is, b Ψ( S ∗ , γ ) ≤ λ for all γ ∈ Y and b µ S ∗ = sup { b Ψ( S ∗ , γ ) : γ ∈ Y } ≤ λ. The reverse inequality is clear. Thus, S ∗ is a generalized shadow price.The equalities in the first line in (2.7) follow from (2.5) and (2.8). Furthermore,the inequalitiessup γ ∈Y inf S ∈ [ S,S ] b Ψ( S, γ ) ≤ inf S ∈ [ S,S ] sup γ ∈Y b Ψ( S, γ ) ≤ sup γ ∈Y b Ψ( S ∗ , γ )are evident and the equality λ = sup γ ∈Y b Ψ( S ∗ , γ ) is already proved. (cid:3) In the context of duality theory (see e.g. [25], section 1) the problemsmaximize f ( γ ) = inf S ∈ [ S,S ] b Ψ( S, γ ) over all γ ∈ Y , minimize g ( S ) = sup γ ∈Y b Ψ( S, γ ) over all S ∈ [ S, S ] (2.10)are said to be dual to each other and the common value (2.7) is called the saddle-value of b Ψ. The first of these problems coincides with (2.3).The function g is lower semicontinuous as the pointwise supremum of a familyof lower semicontinuous functions and the set [ S, S ] is compact. Hence, the dualproblem (2.10) is solvable. The equality g ( S ∗ ) = sup γ ∈Y b Ψ( S ∗ , γ ) = inf S ∈ [ S,S ] sup γ ∈Y b Ψ( S, γ ) = λ shows that generalized shadow prices are exactly the solutions of (2.10). Theorem 3.
A pair ( S ∗ , γ ∗ ) ∈ [ S, S ] × Y is a saddle point of the relaxed utilityfunction b Ψ : b Ψ( S ∗ , γ ) ≤ b Ψ( S ∗ , γ ∗ ) ≤ b Ψ( S, γ ∗ ) , ( S, γ ) ∈ [ S, S ] × Y , (2.11) if and only if γ ∗ is an optimal solution of (2.3): Φ( X T ( γ ∗ )) = λ and S ∗ is ageneralized shadow price. DMITRY B. ROKHLIN
Proof . Condition (2.11) can be reformulated as follows: g ( S ∗ ) = b Ψ( S ∗ , γ ∗ ) = f ( γ ∗ ) . (2.12)Since g ( S ) ≥ f ( γ ), ( S, γ ) ∈ [ S, S ] ×Y it follows that if ( S ∗ , γ ∗ ) is a saddle point of b Ψ then γ ∗ is an optimal solution of (2.3) and S ∗ is an optimal solution of (2.10)(or, equivalently, a generalized shadow price). Conversely, if γ ∗ is an optimalsolution of (2.3) and S ∗ is a generalized shadow price then λ = g ( S ∗ ) ≥ b Ψ( S ∗ , γ ∗ ) ≥ f ( γ ∗ ) = λ. Thus, (2.12) holds true and ( S ∗ , γ ∗ ) is a saddle point of b Ψ. (cid:3) The above arguments show that the existence of an optimal solution of (2.3)is equivalent to the existence of a saddle point of the relaxed utility function b Ψ.3.
Examples
In the first two examples given below Ψ( · , γ ) is τ w -lower semicontinuous on[ S, S ] and, hence, there exists a shadow price. Note that in these examples thetopological space ([
S, S ] , τ w ) is first countable and it is enough to show thatΨ( S, γ ) ≤ lim inf n →∞ Ψ( S n , γ ) , S ∈ [ S, S ] (3.1)for any sequence S n ∈ [ S, S ], converging to S in τ w , to check τ w -lower semicon-tinuity of Ψ( · , γ ).In Example 3 the relaxation b Ψ does not coincide with Ψ but S ∗ is a shadowprice iff it is a generalized shadow price. Examples 4 and 5 (the last one isborrowed from [3]) show that even in the case of countable probability space ashadow price need not exist, while the existence of a generalized shadow price isensured by Theorem 1.In all examples the utility functional Φ is concave and Y = F . Example 1.
Let Ω be finite and let P be striclty positive on the atoms of F T . Consider a monotone concave function U : R [ −∞ , ∞ ) such that U isfinite (and hence continuous) on the open half-line (0 , ∞ ) and U ( x ) = −∞ , x ∈ ( −∞ , E U ( X T ( γ )) over all γ ∈ T − Y t =0 L ( F t )where X T ( γ ) is defined by (2.1). The choice of q does not affect anything: q = 1for instance.If F t is generated by the partition ( D it ) m t i =1 then L ( F t ) is an m t -dimensionalspace. Put f it = f t ( ω ), ω ∈ D it for f t ∈ L ( F t ). All Hausdorff vector topologies ona finite dimensional space coincide ([1], Theorem 5.21). Thus, we can assume that τ wt is the topology of pointwise convergence with a local base at zero generatedby the sets { f ∈ L ( F t ) : | f it | < /n } , i = 1 , . . . , m t , n ∈ N . N THE GAME INTERPRETATION OF A SHADOW PRICE 9
To show that Ψ( · , γ ) is lower semicontinuous in the product topology τ w = Q Tt =0 τ wt it is enough to check that (3.1) is true when S nt → S t pointwise.If 1 + ( γ ◦ S ) T ( ω ) > t , ω then the same is true for 1 + ( γ ◦ S n ) T ( ω )for sufficiently large n . It follows thatlim n →∞ Ψ( S n , γ ) = lim n →∞ E U (1 + ( γ ◦ S n ) T ) = E U (1 + ( γ ◦ S ) T ) = Ψ( S, γ ) . On the other side, if 1 + ( γ ◦ S ) T ( ω ) ≤ t , ω then Ψ( S, γ ) = Φ(1 + ( γ ◦ S ) T ) = −∞ .The lower semicontinuity of Ψ( · , γ ) implies the existence of a shadow price. Arelated result was established in [18]. Example 2.
Let Ω be countable and let P be strictly positive on the atoms of F T . Assume that the processes S , S are bounded: S t , S t ∈ L ∞ ( F t ), and considerthe optimization problemmaximize E U ( X T ( γ )) over all γ ∈ T − Y t =0 L ∞ ( F t )with a monotone concave (and hence continuous) function U : R R . We put q = 1.If F t is generated by a partition ( D it ) i ∈ J t , J t ⊂ N , then for f t ∈ L ( F t ) we put f it = f t ( ω ), ω ∈ D it . Consider on L ( F t ) the topology τ pt of pointwise convergencewith a local base at zero generated by the sets { f ∈ L ( F t ) : | f it | < /n } , i ∈ J t , n ∈ N . The topologies τ wt = σ ( L ( F t ) , L ∞ ( F t )), τ pt are different on L ( F t ) if the set J t isinfinite, since τ p is first countable and τ w is not (see [1], Theorem 6.26). Clearly, τ pt ⊂ τ wt . It follows that they coincide on the set [ S t , S t ] which is τ wt -compactand τ pt -Hausdorff (see [27], section 3.8).Take a sequence S n ∈ [ S, S ], converging to S in the product topology τ p = Q Tt =0 τ pt . This amounts to the pointwise convergence S nt → S t . The correspon-dent sequence U (1 + ( γ ◦ S n ) T ) is uniformly bounded and Ψ( · , γ ) is τ p -continuouson [ S, S ]:lim n →∞ Ψ( S n , γ ) = lim n →∞ E U (1 + ( γ ◦ S n ) T ) = E U (1 + ( γ ◦ S ) T ) = Ψ( S, γ )due to the dominated convergence theorem. This implies the existence of ashadow price.A counterexample, given in [3] (see Example 5 below), indicates that the as-sumptions on boundedness of S t , S t and finiteness of U cannot be dropped si-multaneously. Example 3.
Let T = 1, Ω = [0 , F = {∅ , Ω } , and let F be the Borel σ -algebra of [0 ,
1] with the Lebesgue measure P ( dω ) = dω . Assume that S = S = S , S , S ∈ L ∞ ( F ) and S − S ≥ α > for some real number α >
0. Consider the optimization problemmaximize E U ( X ( γ )) over all γ ∈ R (3.2)with a monotone concave function U : R R . From (2.1) we get X ( γ ) = 1 + γ +0 ( S − S ) − γ − ( S − S )Put s = ∞ , q = 1. Thus, [ S , S ] is considered as a set in L ( F ) with theweak topology τ w of L ( F ). We look for the lower semicontinuous envelope ofthe functional S Ψ( S, γ ) = E U (1 + γ ( S − S ))defined on the set { S } × [ S , S ]. This problem reduces to relaxation of theintegral functional S F ( S ) = Z (cid:2) U (1 + γ ( S − S )) + δ ( S | [ S , S ]) (cid:3) dω, S ∈ L ( F ) , where δ ( x | A ) = 0, x ∈ A ; δ ( x | A ) = + ∞ , x A since F ( S ) = (cid:26) Ψ( S, γ ) , S ∈ [ S , S ] a.s. , + ∞ , otherwise . Furthermore, the function f ( ω, x ) = U (1 + γ ( x − S )) + δ ( x | [ S ( ω ) , S ( ω )])is Borel on [0 , × R and lower semicontinuous in x for each ω . Hence, f is a normal integrand (see [9], Chapter VIII, Definition 1.1), uniformly bounded frombelow. The relaxation of F is given by the formula b F ( S ) = Z b f ( ω, S ( ω )) dω, where b f ( ω, · ) is the largest convex lower semicontinuous minorant of f ( ω, · ) foreach ω : see [9], Chapter IX, Propositions 1.2 and 2.3 or [26], [16] (chapter 2,section 9) for more general results of this sort. Using the concavity of f ( ω, · ) on[ S ( ω ) , S ( ω )], we conclude that b f ( ω, · ) is linear on this interval: b f ( ω, S ) = f ( ω, S ) S − S S − S + f ( ω, S ) S − S S − S + δ ( S | [ S , S ]) . Thus, for S ∈ [ S , S ] we have b Ψ( S, γ ) = b F ( S ) = E (cid:18) U (1 + γ ( S − S )) S − S S − S + U (1 + γ ( S − S )) S − S S − S (cid:19) . Now assume that the function U is strictly increasing and differentiable andthere exists an optimal solution γ ∗ of (3.2). From Theorem 3 it follows that N THE GAME INTERPRETATION OF A SHADOW PRICE 11 S ∗ is a generalized shadow price iff ( S ∗ , γ ∗ ) is a saddle point of b Ψ. From therepresentation b Ψ( S, γ ) = = E (cid:18) U (1 + γ ( S − S )) − U (1 + γ ( S − S )) S − S S + S U (1 + γ ( S − S )) − S U (1 + γ ( S − S )) S − S (cid:19) . it is clear that the inequality b Ψ( S ∗ , γ ∗ ) ≤ b Ψ( S, γ ∗ ), S ∈ [ S , S ] is equivalent tothe condition S ∗ I { γ ∗ =0 } = S I { γ ∗ > } + S I { γ ∗ < } . (3.3)Furthermore, the inequality b Ψ( S ∗ , γ ) ≤ b Ψ( S ∗ , γ ∗ ), γ ∈ R reduces to thecondition ∂ b Ψ ∂γ ( S ∗ , γ ∗ ) = 0due to the concavity of b Ψ( S ∗ , · ). After elementary calculations we get ∂ b Ψ ∂γ ( S ∗ , γ ∗ ) = E (( S − S ) U ′ (1 + γ ∗ ( S − S ))) = 0 for γ ∗ > ,∂ b Ψ ∂γ ( S ∗ , γ ∗ ) = E (cid:0) ( S − S ) U ′ (1 + γ ∗ ( S − S )) (cid:1) = 0 for γ ∗ < ,∂ b Ψ ∂γ ( S ∗ , γ ∗ ) = U ′ (1) E ( S ∗ − S ) = 0 for γ ∗ = 0 . Taking into account (3.3), we conclude that the last three equalities are equivalentto the following one: ∂ b Ψ ∂γ ( S ∗ , γ ∗ ) = E (( S ∗ − S ∗ ) U ′ (1 + γ ∗ ( S ∗ − S ∗ ))) = 0 . (3.4)Thus, ( S ∗ , γ ∗ ) is a saddle point of b Ψ iff the relations (3.3), (3.4) hold true.From this observation it follows that any generalized shadow price is a shadowprice . Indeed, condition (3.4) ensures that γ ∗ is an optimal solution in thefrictionless model with the price process S ∗ : E U (1 + γ ( S ∗ − S ∗ )) ≤ E U (1 + γ ∗ ( S ∗ − S ∗ )) . Moreover, in view of (3.3) we have E U (1 + γ ∗ ( S ∗ − S ∗ )) = E U (1 + ( γ ∗ ) + ( S − S ) − ( γ ∗ ) − ( S − S )) = b Ψ( S ∗ , γ ∗ ) = λ. Hence, although b Ψ( S, γ ) = Ψ( S, γ ) = E U (1 + γ ( S − S ))(if, e.g., U is strictly concave), in this example a process S ∗ is a generalizedshadow price iff it is a shadow price. Example 4.
Let Ω = N , T = 1, F is generated by the atoms D n = { n − , n } , n ∈ N and F coincides with the power set of N . The probability measure isdefined by P ( { n } ) = 2 − n . Put S = 1, S = 4, S = S = S = ∞ X n =1 (cid:0) I { n } + I { n − } (cid:1) . Since S t is bounded we can put s = q = r = ∞ . From the definitions of X ( γ )and S it follows that X ( γ ) = 1 + γ +0 ( S − S ) − γ − ( S − S ) ≤ . and γ = 0 is an optimal trading strategy for any monotone functional Φ on L ∞ ( F ).Denote by LIM : L ∞ ( F ) R a Banach limit (see e.g. [8], Chapter II,Exercise 22) and put Φ( X ) = E X + LIM ( X ) . Clearly, Φ is a linear monotone functional on L ∞ ( F ). We have λ = sup { Φ( X ( γ )) : γ ∈ L ∞ ( F ) } = 1 . We show that there is no shadow price in this model. Assume first that S isa shadow price which is not equal to the conditional expectation E ( S |F ) = ∞ X n =1 E ( S I D n ) P ( D n ) I D n = ∞ X n =1 P (2 n ) + P (2 n − P (2 n ) + P (2 n − I D n = 2 . If S = 2 on D n , then putting γ ( i ) = 0, i D n ; γ ( i ) = δ , i ∈ D n we getΦ(1 + ( γ ◦ S ) ) = E (1 + γ ∆ S ) = 1 + E ( γ (2 − S )) = 1 + δ (2 − S ) P ( D n ) . It follows that µ S = sup { Φ(1 + ( γ ◦ S ) ) : γ ∈ L ∞ ( F ) } = + ∞ . Now assume that S = E ( S |F ) = 2. For γ = 1 we haveΦ(1 + ( γ ◦ S ) ) = LIM (1 + ∆ S ) = 32 . For computing the value of the Banach limit in the last equality we have usedits shift-invariance property:2LIM (∆ S ) = LIM ∞ X n =1 (2 I { n } − I { n − } ) + LIM ∞ X n =1 (2 I { n − } − I { n } )= LIM (1) = 1 . Thus, µ S ≥ / > λ = 1 and S = 2 is not a shadow price.The existence of a generalized shadow price S ∗ is guaranteed by Theorem 1.Let us show that S ∗ = 2. For S = 2 we haveΨ( S, γ ) = Φ(1 + ( γ ◦ S ) ) = LIM (1 + γ ( S − S )) . N THE GAME INTERPRETATION OF A SHADOW PRICE 13
Consider a neighbourhood U of S in the topology τ w = σ ( L ∞ ( F ) , L ( F )),restricted to [ S , S ]: U = { S ′ ∈ [ S , S ] : | E g i ( S ′ − S ) | < ε, i = 1 , . . . , m } , g i ∈ L ( F ) , ε > . The set U n = ( S ′ ∈ [ S , S ] : S ′ = S on n [ j =1 D j ) is contained in U for sufficiently large n . Indeed, for S ′ ∈ U n we have | E g i ( S ′ − S ) | ≤ ∞ X j = n +1 | g ji || S j − S j | P ( D j ) , where g ji = g i ( ω ), ω ∈ D j and S j , S j are defined similarly. The right-hand sideof the last inequality can be made arbitrary small by an appropriate choice of n .Take S ′ ∈ U n ⊂ U such that1 + γ ∆ S ′ = 1 + γ +0 ( S − S ) − γ − ( S − S ) = X ( γ ) on [ j ≥ n +1 D j . Clearly, Ψ( S ′ , γ ) = 1 + LIM ( γ ∆ S ′ ) = LIM ( X ( γ )) . It follows that b Ψ( S, γ ) ≤ LIM ( X ( γ )) ≤ S ∗ = 2 determines a generalizedshadow price. Example 5.
Let us reproduce the counterexample of [3]. Put Ω = N , T = 2, F = {∅ , Ω } . Let F be generated generated by the atoms D k = { k + 1 , k + 2 } , k ∈ N := { } ∪ N , and let F be the power set of N .Assume that the stock bid prices are falling deterministically: S = 3, S = 2 S = 1 and the ask prices are defined as follows: S = 3, S = 2 + k on D k , k ∈ N ,S ( ω ) = 1 for ω = 2 k + 1 , k ∈ N ,S ( ω ) = 3 + k for ω = 2 k + 2 , k ∈ N . The probability measure is defined as follows: P ( D ) = 1 − − n , P ( D k ) = 2 − n − k ; P ( { k + 1 } ) = (1 − − n − k ) P ( D k ) , P ( { k + 2 } ) = 2 − n − k P ( D k ) , k ∈ N , where n ∈ N is fixed sufficiently large to make E ( S − S |F ) < X ( γ )) = E ln( X ( γ )) (we put ln x = −∞ for x ≤ E f = lim M → + ∞ E ( f ∧ M ) . for any measurable function f with values in the extended real line R ∪ {±∞} .Particularly, E f = −∞ if E f − = + ∞ . The picture and clear economical argumentation, given in [3], show that it isoptimal not to trade at step 0 ( γ ∗ = 0) and to go short at step 1 ( γ ∗ < X ( γ ) = 1 + X t =0 (cid:0) S t (∆ γ t ) − − S t (∆ γ t ) + (cid:1) = 1 − S γ + S ( γ − γ ) − − S ( γ − γ ) + + S γ +1 − S γ − , (3.5)where S = S = S . It is easy to see that γ < X ( γ ) for some ω = 2 k . Assuming that γ ≥
0, it is not optimal to posses apositive amount of stock at step 1: X ( γ ) = 1 + ( S − S ) γ for γ = 0; X ( γ ) = 1 + ( S − S ) γ + ( S − S ) γ ≤ S − S ) γ for γ ∈ (0 , γ ); X ( γ ) = 1 + ( S − S ) γ + ( S − S ) γ ≤ S − S ) γ for γ ≥ γ , since ( S − S ) γ ≤ ( S − S ) γ ≤ ( S − S ) γ for γ ≥ γ .Under the assumptions γ ≥ γ ≤ X ( γ ) = 1 + ( S − S ) γ + ( S − S ) γ ≤ S − S ) γ . It follows that the maximization of E ln( X ( γ )) can be carried over the set { γ =0 , γ ≤ } : λ = sup { E ln( X ( γ )) : γ t ∈ L ( F t ) , t = 0 , } = sup { E ln(1 + ( S − S ) γ ) : γ ∈ L ( F , − R + ) } . (3.6)Moreover, since E ( S − S |F ) <
0, it is not optimal to do nothing. Denoteby γ k the value of γ on D k . We have E ln(1 + ( S − S ) γ ) = ∞ X k =0 (cid:18) (1 − − n − k ) ln(1 − γ k )+ 2 − n − k ln (cid:0) k ) γ k (cid:1)(cid:19) P ( D k ) . (3.7)The optimal portfolio γ ∗ ,k < γ ∗ ,k ∈ ( − (1 + k ) − , − − n − k ) ln(1 − γ k ) + 2 − n − k ln (cid:0) k ) γ k (cid:1) ≤ (1 − − n − k )( − γ k ) ≤ − − n − k k + 1which shows that the optimal utility value λ is finite.Since the optimal strategy γ ∗ < γ ∗ = − γ ∗ is active, shadow prices S ∗ , S ∗ should coincide with S , S . Otherwise, the same strategy would give strictlyhigher utility value in the frictionless market with stock price S ∗ . But in thisfrictionless market the optimal utility value µ S ∗ is infinite since1 + γ ∆ S ∗ = 1 − γ → + ∞ , γ → −∞ . N THE GAME INTERPRETATION OF A SHADOW PRICE 15
Thus, there is no shadow price in this model.However, as we will see shortly, the process S ∗ = ( S , S , S ) is a generalized shadow price. The point is that in the relaxed problem short selling at step0 is automatically prohibited. Put s = 0, q = 1 in the notation of Section 2.By the same reasons as in Example 2, the topology τ pt of pointwise convergencecoincides with the weak topology τ wt = σ ( L ( F t ) , L ∞ ( F t )) on the set [ S t , S t ]. Forany Q t =0 τ pt -neighbourhood U of S ∗ there exist sufficiently large k and S ′ ∈ U such that S ′ = S , S ′ = S on D k . We have1 + ( γ ◦ S ′ ) = 1 + ( S − S ) γ + ( S − S ) γ = ( k − γ + ( I { k +2 } − ( k + 1) I { k +1 } ) γ k on D k . If γ <
0, then 1 + ( γ ◦ S ′ ) (2 k + 2) < k . Thus,Ψ( S ′ , γ ) = E ln(1 + γ ◦ S ′ ) = −∞ and b Ψ( S ∗ , γ ) = −∞ for any γ < γ ◦ S ∗ ) = 1 + ( S − S ) γ + ( S − S ) γ ≤ S − S ) γ for γ ≥ . It follows that b Ψ( S ∗ , ( γ , γ )) ≤ b Ψ( S ∗ , (0 , γ )) and one can assume γ = 0 in therelaxed utility maximization problem (2.6): b µ S ∗ = sup { b Ψ( S ∗ , γ ) : γ = 0 , γ ∈ L ( F ) } . To prove that µ S ∗ = λ , we go back to the ”unrelaxed” frictionless problem: b µ S ∗ ≤ sup { Ψ( S ∗ , γ ) : γ = 0 , γ ∈ L ( F } = sup { E ln (cid:0) S − S ) γ (cid:1) : γ ∈ L ( F ) } . Looking again at (3.7), we conclude that optimal values γ ∗ ,k are negative. Com-paring the last expression with (3.6), we obtain the inequality b µ S ∗ ≤ λ . Thereverse inequality is evident.4. Shadow prices via Lagrange duality
As is known, in mathematical economics shadow resource prices are associatedwith the optimal solution of the dual problem: see e.g. [19], [10] (Chapter 5).To avoid conflicts with the terminology of the present paper we, following [24],use the term ”equilibrium prices” instead. These prices are introduced along thefollowing lines. Let x = ( x , . . . , x n ) represent activities of a firm and let f ( x )be the cost of the corresponding operation. The activities are subject to theresource constraints g i ( x ) ≤ i = 1 , . . . , m . Put ϕ ( u ) = inf { f ( x ) : g i ( x ) ≤ u i , i = 1 , . . . , m } . The components of a vector λ ∗ = ( λ ∗ , . . . , λ ∗ m ) are called equilibrium resourceprices if the firm cannot reduce the optimal cost of the operation by buying or selling resources at these prices: ϕ ( u ) + m X i =1 λ ∗ i u i ≥ ϕ (0) , u ∈ R m . For a convex problem vectors λ ∗ of equilibrium prices are exactly the optimalsolutions of the dual problem (see [24], Theorem 28.2 and Corollary 28.4.1 forthe precise statement).In the problem under consideration the trader has two resources at his disposal:bonds and stocks. It is natural to expect that the equilibrium prices of theseresources are related to the shadow price process introduced above.Assume that Ω is finite, F T coincide with the power set of Ω and P ( ω ) > ω ∈ Ω. First of all we rewrite the self-financing condition, separating the ”resourceconstraints”: (∆ β t − L t S t + M t S t )( ω ) ≤ , t ∈ , . . . , T, ω ∈ Ω; (4.1)(∆ γ t + L t − M t )( ω ) ≤ , t ∈ , . . . , T, ω ∈ Ω; (4.2) − L t ( ω ) ≤ , − M t ( ω ) ≤ , t ∈ , . . . , T, ω ∈ Ω . (4.3)Here, as above, β − = 1, γ − = 0. By L t (respectively, M t ) we denote the numberof stocks sold (respectively, purchased) at time t at price S t (respectively, S t ).Clearly, passing to the inequality constraints (corresponding to the possibility ofconsumption) and allowing the simultaneous transfers from bonds to stocks andback: L t M t = 0 do not increase trader’s monotone utility. We should also takeinto account the ”boundary condition”: γ T ( ω ) = 0 , ω ∈ Ω (4.4)and the ”information constraints”:( β t , γ t , L t , M t ) ∈ L ( F t , R ) , t ∈ , . . . , T. (4.5)Consider a concave utility function U as in Example 1: U is finite on (0 , ∞ )and U ( x ) = −∞ , x ≤ C the set of processes ( β, γ, L, M ),satisfying (4.5) and such that β T >
0. The problem is to minimize − E U ( β T ) (4.6)over the set C under the constraints (4.1) – (4.4). Formally, this is an ordinaryconvex optimization program ([24], Section 28).Consider the Lagrange function L = − E U ( β T ) + T X t =0 E ( Z t (∆ β t − L t S t + M t S t )) + T X t =0 E ( Z t (∆ γ t + L t − M t )) − T X t =0 E ( Z t L t ) − T X t =0 E ( Z t M t ) + E ( ν T γ T ) for ( β, γ, L, M ) ∈ C. (4.7)The Lagrange multipliers are represented by a process Z t = ( Z t , Z t , Z t , Z t )with non-negative components: Z t ∈ L ( F t , R ), t = 0 , . . . , T and ν T ∈ L ( F T ).Note that the process Z may be assumed adapted since for adapted processes N THE GAME INTERPRETATION OF A SHADOW PRICE 17 ( β, γ, L, M ) ∈ C the number of constraints in (4.1) – (4.3) for fixed t coincideswith the number of atoms of F t and Z t can be taken constant on these atoms.To complete the definition of L , we put in accordance to the general schemeof [24] (Section 28) L = + ∞ , if ( β, γ, L, M ) C ; L = −∞ , if ( β, γ, L, M ) ∈ C, Z t L ( F t , R ) for some t. Collecting terms, containing the same elements β t , γ t , L t , M t , we rewrite (4.7)in the following way: L = L + L + L , L = E (cid:0) − U ( β T ) + Z T β T (cid:1) − T − X t =0 E β t ∆ Z t +1 − E Z , L = E γ T ( Z T + ν T ) − T − X t =0 E γ t ∆ Z t +1 , L = T X t =0 E L t ( Z t − Z t S t − Z t ) + T X t =0 E M t ( − Z t + Z t S t − Z t ) . The objective function of the dual problem is given by g ( Z, ν T ) = inf {L : ( β, γ, L, M ) ∈ C } . Put V ( x ) = inf y ( − U ( y ) + xy ). After simple calculations we get g ( Z, ν T ) = E V ( Z T ) − E Z if Z t ∈ L ( F t , R ), t = 0 , . . . , T and the following conditions hold true E (∆ Z t +1 |F t ) = 0 , E (∆ Z t +1 |F t ) = 0 , t = 0 , . . . , T − Z t − Z t S t = Z t , − Z t + Z t S t = Z t , t = 0 , . . . , T ; ν T = − Z T . Otherwise, g ( Z, ν T ) = −∞ .It readily follows that the optimal value of the dual problem can be representedas sup { E ( V ( Z T ) − Z ) : Z ∈ D } , (4.8) D = { Z ∈ M + : Z t S t ≤ Z t ≤ Z t S t for some Z ∈ M + } , where M + is the set of non-negative P -martingales. The representations ofthis sort are well known: see [5] for continuous time case and [22], [23] forgeneralizations in discrete time.The objective function (4.6) of the primal problem is finite on C and the point( β, γ ), where β t = 1, γ t = 0, t = 0 , . . . T , belongs to the relative interior of C andsatisfies the constraints (4.1) – (4.4), which are affine. If the optimal value − λ of the primal problem is finite then there is no duality gap and the dual problemis solvable ([24], Theorem 28.2 and Corollary 28.4.1). That is, − λ = sup { E ( V ( Z T ) − Z ) : Z ∈ D } = E ( V ( b Z T ) − b Z )fo some b Z ∈ D . Let us introduce an adapted process S ∗ t ∈ [ S t , S t ] such that S ∗ t b Z t = b Z t . (4.9)On the atoms of F t with b Z t = 0 , b Z t = 0 the values S ∗ t ∈ [ S t , S t ] are chosenarbitrary. Put − µ S ∗ = sup { E ( V ( Z T ) − Z ) : Z ∈ D ( S ∗ ) } , (4.10) D ( S ∗ ) = { Z ∈ M + : Z = Z S ∗ ∈ M + } . The maximization in (4.10) is carried over smaller set as compared to (4.8),and the objective functions are the same. Hence, − λ ≥ − µ S ∗ . On the otherhand, the optimal solution b Z of (4.8) is feasible for (4.10): b Z ∈ D ( S ∗ ) since b Z = S ∗ t b Z t ∈ M + . It follows that λ = µ S ∗ . 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