Path-dependent Kyle equilibrium model
aa r X i v : . [ q -f i n . T R ] J un Path-dependent Kyle equilibrium model.
José Manuel Corcuera ∗ and Giulia Di Nunno † June 10th, 2020
Abstract
We consider an auction type equilibrium model with an insider in line with the one originally introducedby Kyle in 1985 and then extended to the continuous time setting by Back in 1992. The novelty introducedwith this paper is that we deal with a general price functional depending on the whole past of the aggregatedemand, i.e. we work with path-dependency. By using the functional Itô calculus, we provide necessaryand sufficient conditions for the existence of an equilibrium. Furthermore, we consider both the cases ofa risk-neutral and a risk-averse insider.
Key words : Kyle model, market microstructure, equilibrium, insider trading, stochastic control, semi-martingales, functional Itô calculus.
JEL-Classification
C61 · D43 · D44 · D53 · G11 · G12 · G14
MS-Classification 2020 : 60G35, 62M20, 91B50, 93E03
It is well known that privileged information and informational asymmetries are everywhere in the realeconomy. In his pioneering work, Kyle (1985) constructed a model in a discrete time setting with marketmakers, uninformed traders and one insider, who knows the fundamental value of an asset at a certain fixedreleased time. Also, the model included a price functional relating market prices and the total demand. Back(1992) extends Kyle’s model to the continuous time case. Since these worked appeared, several generalisationsand extensions have been produced. To mention some, we have Back and Pedersen (1998), who considera dynamic fundamental price and Gaussian noises with time varying volatility; Cho (2003), who considerspricing functions depending on the path of the demand process and also studies the case when the informedtrader is risk-averse; Lasserre (2004), who considers a multivariate setting; Back and Baruch (2004) wherethe market depth (i.e. the marginal effect on price of the volume traded) depends on the market price ofthe stock; Aase et al. (2012a), (2012b), who put emphasis on filtering techniques to solve the equilibriumproblem; Campi and Çetin (2007), who consider a defaultable bond instead of a stock as in the Kyle-Backmodel and also consider the knowledge of the default time as the insider’s privileged information; Danilova(2010), who deals with non-regular pricing rules; Caldentey and Stacchetti (2010) who take a random releasetime into account; Campi et al. (2013), who consider again a defaultable bond, but this time they considerthe privileged information to be represented by some dynamic signal related with the default time; andCollin-Dufresne and Fos (2016) where the market depth depends on the (random) volatility of the noisein the market. In Corcuera and Di Nunno (2018), the authors propose a general framework to include allthe particular extensions mentioned above and study the general characteristics of the equilibria. RecentlyCorcuera, Di Nunno and Fajardo (2019) have also considered the same general situation, but with a randomprice pressure and a random release time of information.In this paper we propose a step even further and we consider a price functional that depends on the wholepath of the aggregate demand. We study the properties of the equilibrium and sufficient and necessary ∗ Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain.
E-mail: [email protected] work of J. M. Corcuera is supported by the Spanish grant MTM2016-76420-P. † University of Oslo, Department of Mathematics, P.O. Box 1053 Blindern NO-0316 Oslo, and NHH, Department of Businessand Management Science, Helleveien 30, 5045 Bergen, Norway.
E-mail: [email protected]. This work has received supportfrom the Research Council of Norway via the project
STORM: Stochastics for Time-Space Risk Models (nr. 274410).
We consider a market with two assets, a stock and a bank account with interest rate r equal to zero for thesake of simplicity. The trading is continuous in time over the period [0 , ∞ ) and it is order driven. Thereis a (possibly random) release time of information τ < ∞ a.s., when the fundamental value of the stock isrevealed. The fundamental value process represents the actual value of the asset, which would be the sameas the market price of the asset only if all the information was public. We could say, with Malkiel (2007),that the fundamental value is the intrinsic value of a stock, via an analysis of the balance sheet, the expectedfuture dividends, and the growth prospects of a company. The fundamental value process is denoted by V .We shall denote the market price of the stock at time t by P t . This represents the market evaluation of theasset. Just after the revelation time τ, the price of the stock coincides with the fundamental value. Then weconsider P t defined only on t ≤ τ . Obviously, it is possible that P t = V t for t ≤ τ .We assume that all the random variables and processes mentioned are defined in the same complete filteredprobability space (Ω , F , H , P ) where the filtration H and any other filtration considered in this present workare complete and right-continuous by taking, when necessary, the usual augmentation.There are three kinds of traders. A number of liquidity traders , who trade for liquidity or hedging reasons,an informed trader or insider , who has privileged information about the firm and can deduce its fundamentalvalue, and the market makers , who set the market price and clear the market. At time t , the insider’s information is the full information H t and her flow of information is represented by thefiltration H = ( H t ) t ≥ . Since this is also the filtration with respect to which all the processes considered inthe present work are adapted, we shall omit to write it in the notation. A random release time of information τ is considered from insider’s perspective to be of one of these types: • τ it is bounded and predictable, • τ it is not a predictable stopping time, but it is independent of the observable variables.We assume that the fundamental value V is a continuous martingale such that σ V ( t ) := d[ V,V ] t d t is welldefined.Hereafter we describe in detail the three types of agents involved in this market model, namely their role, theirdemand process, and their information. Let Z be the aggregate demand process of the liquidity traders. We2ecall that these are a large number of traders motivated by liquidity or hedging reasons. They are perceivedby the insider as constituting noise in the market, thus also called noise traders. It is assumed that Z is a continuous, square integrable, martingale, starting at zero, independent of V , and such that σ Z ( t ) := d[ Z,Z ] t d t is well defined. As it is shown in Corcuera et al. (2010), if Z had jumps, an equilibrium would not bepossible. Remark 1
In this equilibrium model, the time τ and the processes V and Z are exogenously given. Market makers clear the market giving the market prices. They rely on the information given by the totalaggregate demand Y , which they observe, and the release time τ , that is a stopping time for them. Hence,their information flow is: F = ( F t ) t ≥ , where F t = ¯ σ ( Y s , τ ∧ s, ≤ s ≤ t ) . Here ¯ σ denotes the σ -fieldcorresponding to the usual augmentation of the natural filtration.The total aggregate demand is defined as Y := X + Z , where X denotes the insider demand process, whichis naturally assumed to be a predictable process and also a càdlàg semimartingale: (A1) X t = M t + A t + Z t θ s d s, t ≥ , where M is a continuous martingale with M = 0 , A a bounded variation predictable process with A t = X
We say that P is strictly increasing as a functional of { Y s , ≤ s ≤ t } if, for any t ≤ τ , andfor any pair { Y s , Y ′ s , ≤ s ≤ t } for which there exits some time u ∈ [0 , t ] such that, fixed ω , Y s ( ω ) = Y ′ s ( ω ) , ≤ s ≤ u, and Y s ( ω ) < Y ′ s ( ω ) for u < s ≤ t, a.s.or Y s ( ω ) = Y ′ s ( ω ) , ≤ s < u, and Y s ( ω ) < Y ′ s ( ω ) for u ≤ s ≤ t, a.s.we have that P ′ t = P t ( Y ′ s , ≤ s ≤ t ) > P t = P t ( Y s , ≤ s ≤ t ) . Remark 2
We also could say that P is strictly increasing at the path { Y s , ≤ s ≤ t } if it is strictly in-creasing when we consider all the paths { Y s ∧ u , ≤ u ≤ t } , s ∈ [0 , t ] . From the economic point of view, due to the competition among market makers, the market prices are competitive , in the sense that P t = E ( V t |F t ) , ≤ t ≤ τ. (1)Therefore ( P t ) ≤ t ≤ τ is an F -martingale. Definition 2
The couple ( P, X ) is an equilibrium if market prices admit a pricing rule (i.e. a functionalof Y ), that we shall name equilibrium pricing rule , P t = P t ( Y s , ≤ s ≤ t ) , ≤ t ≤ τ. such that, at the same time, the market prices P are competitive given X , i.e. P t = E ( V t |F t ) , ≤ t ≤ τ, and the strategy X is optimal for the insider given the prices P . W be the wealth process corresponding to theinsider’s portfolio X . To obtain the formula for the insider’s wealth assume that trades occur at times ≤ t ≤ t ≤ ... ≤ t N = τ. If at time t i − there is an order to buy X t i − X t i − shares, its cost will be P t i × ( X t i − X t i − ) , so there is a change in the insider’s bank account given by − P t i × ( X t i − X t i − ) = − P t i − × ( X t i − X t i − ) − (cid:0) P t i − P t i − (cid:1) × ( X t i − X t i − ) , where the second term in the right-hand side accounts for the impact of the demand on the current price.Due to the fact that the price of the asset equals its fundamental value at the release time τ , there is, inaddition, the extra income X τ V τ . Then the total wealth at τ is given by W τ = − N X i =1 P t i − × ( X t i − X t i − ) − N X i =1 (cid:0) P t i − P t i − (cid:1) × ( X t i − X t i − ) + X τ V τ , so taking the limit with the time between trades going to zero, we have W τ = − Z τ P t − d X t − [ P, X ] τ + X τ V τ where (here and throughout the whole article) P t − := lim s ↑ t P s a.s.Then the informed trader aims at maximising E ( U ( W τ )) = E (cid:18) U (cid:18) − Z τ P t − d X t − [ P, X ] τ + X τ V τ (cid:19)(cid:19) (2)for a given utility function U , that is, a strictly increasing and concave function satisfying the Inada condi-tions. The case when U is the identity function corresponds to the so called risk-neutral case. The insider’sstrategy X of type (A.1) providing the maximum is called optimal . Trading is developed in the context of imperfect competition , in the sense that prices are affected by thedemand, that is P t = P t ( Y s , ≤ s ≤ t ) . Here and in the sequel, we shall write Y · t to indicate the path of theprocess Y from zero to t : Y · t ( s ) := Y s , ≤ s ≤ t. Notice that we also can look Y · t as the process Y stopped at t , in such a way that Y · t ( s ) := (cid:26) Y s for ≤ s ≤ tY t for t ≤ s ≤ τ Therefore we can write, alternatively, P t = P t ( Y · t ) and to consider P t as a functional of the process Y stoppedat t . We recall that this functional is assumed to be strictly increasing in the total aggregate demand Y , seeDefinition 1. We shall also add some regularity on the functionals we are going to consider when needed.We shall also consider the following perturbation of a process Y . For h ∈ R , we define Y h · t ( s ) := (cid:26) Y s for ≤ s < tY t + h for t ≤ s ≤ τ . In the sequel, we are going to use a functional version of the Itô formula obtained by functional Itô calculus.We give here some definitions and the results necessary to our scopes as can be found in Cont and Fournié(2013).Now we can define the continuity of the functionals with respect to time and to the path of the process.Given two processes Y and Z , we consider a distance between the two stopped processes Y · t and Z · t ´ definedby d ∞ ( Y · t , Z · t ´ ) := k Y · t − Z · t ´ k ∞ + | t − t ´ | . efinition 3 A functional P is said to be left-continuous at ( t, Y · t ) if for all ε > there exist η > , suchthat, for all ≤ t ´ ≤ t ≤ τ d ∞ ( Y · t , Z · t ′ ) < η ⇒ | P t ( Y · t ) − P t ´ ( Z · t ´ ) | < ε.P is said to be left-continuous if it is left-continuous at any ( t, Y · t ) . Right-continuity is defined analogously.
Continuity means that left- and right-continuity occur at the same time. If, in the previous definition, weconsider only times t ´= t then we say that the functional is said to be continuous at fixed times . It can beseen that continuity at fixed times implies that the process ( P t ( Y · t )) ≤ t ≤ T is adapted if Y is adapted.Since the space of càdlàg functions is not separable under the sup-norm, we need the following additionalregularity, even for the continuous functionals defined above. Definition 4
A functional P is said to be boundedness preserving if for every K there exists a constant C K such that: ∀ t ≤ T, k Y · t k ∞ < K ⇒ | P t ( Y · t ) | < C K . Definition 5
We call horizontal derivative of the functional P at ( t, Y · t ) , the limit given by D t P t := lim ∆ t ↓ P t +∆ t ( Y · t ) − P t ( Y · t )∆ t , provided it exists. Example 1
From the definition, it is easy to see that D t G ( Y t ) = 0 , D t F ( t, Y t ) = ∂ t F ( t, Y t ) , for any smooth functions G, F . Also, for any integrable h , we have D t Z t h ( Y s )d s = h ( Y t ) . Furthermore, for a smooth integrable function f and a Brownian motion W , we obtain D t (cid:18)Z t f ( s, W s )d W s (cid:19) = − ∂ y f ( t, W t ) . A justification of the last statement comes from the usual Itô formula assuming that f ( · , · ) = ∂ y F ( · , · ) . Infact, it is enough to take horizontal derivatives in both sides of Z t f ( s, W s )d W s = F ( t, W t ) − F (0 , W ) − Z t ∂ s F ( s, W s )d s − Z t ∂ y f ( s, W s )d s, to obtain the result. Definition 6
We call vertical derivative of the functional P at ( t, Y · t ) the limit, provided it exists, given by ∇ Y P t := lim h → P t ( Y h · t ) − P t ( Y · t ) h . Example 2
If, as above, we consider some smooth functions
G, F , we have ∇ Y G ( Y t ) = dd y G ( Y t ) , ∇ Y F ( t, Y t ) = ∂ y F ( t, Y t ) and for an integrable h we have ∇ Y Z t h ( Y s )d s = 0 . oreover, with integrability assumptions on f and for the Brownian motion W we have ∇ W (cid:18)Z t f ( s, W s )d W s (cid:19) = f ( s, W s ) . In fact, assuming as in the previous example that f ( · , · ) = ∂ y F ( · , · ) , the classical Itô formula gives Z t f ( s, W s )d W s = F ( t, W t ) − F (0 , W ) − Z t ∂ s F ( s, W s )d s − Z t ∂ y f ( s, W s )d s, then, taking the vertical derivative on both sides, we have ∇ W (cid:18)Z t f ( s, W s )d W s (cid:19) = ∂ y F ( t, W t ) = f ( t, W t ) . Remark 3
In general D and ∇ do not commute. If we set L t := [ D t , ▽ W ] = D t ▽ W − ▽ W D t , then we have that L t (cid:18)Z t f ( s, W s )d W s (cid:19) = [ D t , ▽ W ] (cid:18)Z t f ( s, W s )d W s (cid:19) = D t ▽ W (cid:18)Z t f ( s, W s )d W s (cid:19) − ▽ W D t (cid:18)Z t f ( s, W s )d W s (cid:19) = D t f ( t, W t ) + 12 ▽ W f ( t, W t ) = 0 , if ( f ( t, W t )) t ≥ is not a local martingale. Definition 7
We say that a left-continuous functional belongs to C j,kb if it is j -times horizontally differ-entiable with derivatives continuous at fixed points and boundedness preserving, and it is k -times verticallydifferentiable with left-continuous and boundedness preserving derivatives. Theorem 1 (Functional Itô formula) . If Y is a continuous semimartingale and P is C , b , then P t ( Y · t ) = P ( Y ) + Z t D s P s d s + Z t ∇ Y P s d Y s + 12 Z t ∇ Y P s d[ Y, Y ] s , P -a.s. ≤ t ≤ τ. Proof.
See Theorem 4.1 in Cont and Fournié (2013).
In this section we present necessary and sufficient conditions for the existence of an equilibrium when therelease time τ and the pricing functional satisfy some conditions. The nature of these conditions will befurther studied in the next section. In this analysis we shall consider both a risk-neutral insider and arisk-averse insider with exponential utility function.First, we have the following result that reduces the set of strategies in which we find the optimum in theoptimisation problem here above considered. An analogous result is given in Corcuera et al. (2019). Proposition 1
Assume that the functional P is left-continuous, bounded preserving and strictly increasing,then admissible strategies X with a continuous martingale part or jumps are suboptimal in the class of allpredictable semimartingale strategies. roof. Since we assume that the functional P is strictly increasing with Y = X + Z, we have that [ P, X ] > for strategies with a continuous martingale part. Now if we approximate such strategy X by a polygonal approximation, say ˜ X , since (cid:12)(cid:12)(cid:12)(cid:12)Z τ ˜ P t − d ˜ X t − Z τ P t − d X t (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)Z τ ˜ P t − d ˜ X t − Z τ ˜ P t − d X t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z τ (cid:16) ˜ P t − − P t − (cid:17) d X t (cid:12)(cid:12)(cid:12)(cid:12) , (3)we can approximate R τ P t − d X t by R τ ˜ P t − d ˜ X t , where ˜ P t = P t (cid:0) ˜ X · t (cid:1) . In fact, fixed ω, if we consider a dyadicpartition Π n of [0 , τ ] and ˜ X the polygonal approximation to X, for all ε > , we can take n such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ ˜ P t − d ˜ X t − n X i =1 ˜ P t i − ( ˜ X t i − ˜ X t i − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ ˜ P t − d ˜ X t − n X i =1 ˜ P t i − ( X t i − X t i − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ ˜ P t − d X t − n X i =1 ˜ P t i − ( X t i − X t i − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε a.s.As for the second summand of the right-hand side of (3) we can apply the dominated convergence theoremfor stochastic integrals. As consequence, we can eliminate the negative contribution of [ P, X ] τ and approx-imate P as much as we want by continuity of the price functional. Therefore we can increase the valueof − R τ P t − d X t − [ P, X ] τ + X τ V τ if we take ˜ X instead of X . A similar argument applies for the case ofstrategies with jumps.Since we are going to consider left-continuous boundedness preserving functionals in the sequel, the set of admissible insider’s strategies (A1) can be reduced to those strategies X satisfying ( A1 ′ ) X t = Z t θ s d s, for all t ≥ , where θ is a càdlàg adapted process.Furthermore, the goal of the insider becomes to maximise the performance J ( X ) := E ( U ( W τ )) = E (cid:18) U (cid:18)Z τ ( V τ − P t )d X t (cid:19)(cid:19) (4)over the set of admissible strategies X satisfying ( A1 ′ ) . Remark 4
Observe that, in view of ( A1 ′ ) , we have d[ Y, Y ] t = σ Z ( t )d t. We also have a general result in the case when τ is a predictable stopping time for the insider. The sameresult is given in Corcuera et al. (2019). Proposition 2 If τ is a predictable stopping time for the insider and X is an optimal strategy then V τ = P τ a.s. Proof.
If the insider’s strategy is such that V τ − − P τ − = 0 then it is suboptimal since the insider couldapproximate a jump at τ with the same sign of V τ − − P τ − by an absolutely continuous strategy and improvingher wealth, as it is shown in the previous proposition. Remark 5
From the economic point of view, due to Bertrand’s type competition among market makers, inthe equilibrium market prices are rational , or competitive , in the sense that the competitive price is a pricesuch that the expectation of the market maker’s profit equals zero. In fact, the total final wealth W Mτ of themarket makers is given by W Mτ := − Y τ ( V τ − P τ ) − Z τ Y t d P t , then, if P t = E ( V t |F t ) , ≤ t ≤ τ see (1) , under the assumption that E (cid:0)R τ Y t d[ P, P ] t (cid:1) < ∞ , we have that E (cid:0) W Mτ (cid:1) = 0 . .1 Main results First, we consider the risk-neutral case.
Theorem 2
Suppose that τ = T and that for all t < T , the price functional P is C , b and such that D t P t + 12 ▽ Y P t σ Z ( t ) = 0 , (5) with L P t := [ D t , ▽ W ] = 0 (6) and ▽ Y P t = G ( t, P t ) , (7) where G ( t, y ) ≥ C > is C , . Assume that E (cid:16)R T ( P t − V t ) (cid:0) σ Z ( t ) + σ V ( t ) (cid:1) d t (cid:17) < ∞ . Then there is an equilibrium in the risk-neutral case if and only if ( i ) P T = V T , ( ii ) Y is an F -martingale Proof.
Firstly, we prove that (i) and (ii) are sufficient conditions. Set I ( t, y, v ) := Z yv z − vG ( t, z ) d z. Then, by the Itô formula, we have I ( T, P T , V T ) = I (0 , P , V ) + Z T ∂ I d t + Z T ∂ I d P t + Z T ∂ I d V t + Z T ∂ I d[ V, P ] t + 12 Z T ∂ I d[ P, P ] t + 12 Z T ∂ I d[ V, V ] t , where ∂ i means the partial derivative with respect to the i -th argument. Now, [ V, P ] ≡ , since V and Z areindependent and we consider only absolutely continuous strategies, see ( A1 ′ ) . Also, we have that d[ P, P ] t = G ( t, P t )d[ Y, Y ] t = G ( t, P t ) σ Z ( t )d t∂ I ( t, y, v ) = y − vG ( t, y ) , (8)so, by (5) ∂ I ( t, P t , V t )d P t = P t − V t ▽ Y P t ▽ Y P t d Y t = ( P t − V t ) d Y t . Moreover ∂ I ( t, y, v ) = Z yv − z − vG ( t, z ) ∂ G ( t, z )d z,∂ I ( t, y, v ) G ( t, y ) = G ( t, y ) − ( y − v ) ∂ G ( t, y ) (9)and ∂ (cid:0) ∂ I ( t, y, v ) G ( t, y ) σ Z (cid:1) = − ( y − v ) ∂ G ( t, y ) σ Z ( t ) , so ∂ I ( t, y, v ) + 12 ∂ I ( t, y, v ) G ( t, y ) σ Z = Z yv (cid:18) − ( z − v ) (cid:18) ∂ G ( t, z ) G ( t, z ) + 12 ∂ G ( t, z ) σ Z ( t ) (cid:19)(cid:19) d z + G ( t, v ) . ▽ Y D t P t + 12 ▽ Y P t σ Z ( t )= D t ▽ Y P t + 12 ▽ Y P t σ Z ( t )= D t G ( t, P t ) + 12 ▽ Y G ( t, P t ) σ Z ( t )= ∂ G ( t, P t ) + ∂ G ( t, P t ) D t P t + 12 ▽ Y ( ∂ G ( t, P t ) ▽ Y P t ) σ Z ( t )= ∂ G ( t, P t ) + ∂ G ( t, P t ) (cid:18) D t P t + 12 ▽ Y P t (cid:19) σ Z ( t ) + 12 ∂ G ( t, P t ) ( ▽ Y P t ) σ Z ( t )= ∂ G ( t, P t ) + 12 ∂ G ( t, P t ) G ( t, P t ) σ Z ( t ) . (10)Therefore, ∂ I ( t, y, v ) + 12 ∂ I ( t, y, v ) G ( t, y ) σ Z ( t ) = G ( t, v ) and Z T ( V t − P t ) d X t − I (0 , P , V ) + 12 Z T ∂ I d[ V, V ] t + Z T G ( t, V t )d t ! = − I ( T, P T , V T ) + Z T ( P t − V t ) d Z t + Z T ∂ I d V t . Consequently, E Z T ( V t − P t ) d X t ! − b = − E ( I ( T, P T , V T )) , where b is a constant that only depends on V . In fact, E (cid:16)R T ( P t − V t ) d Z t (cid:17) = 0 since E (cid:16)R T ( P t − V t ) σ Z ( t )d t (cid:17) < ∞ . Furthermore, we have that | ∂ I ( t, y, v ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z yv − G ( t, z ) d z (cid:12)(cid:12)(cid:12)(cid:12) < | y − v | C , therefore E Z T ( ∂ I ) σ V d t ! < C E Z T ( P t − V t ) σ V ( t )d t ! < ∞ , and consequently E (cid:16)R T ∂ I d V t (cid:17) = 0 . Finally, ∂ I ( t, y, v ) = 1 G ( t, v ) < C in a way that Z T ∂ I d[ V, V ] t , only depends on V. By (8) and ( i ) , we have ∂ I ( T, P T , V T ) = P T − V T G ( T, P T ) = 0 and, by (9) and ( i ) , we obtain ∂ I ( T, P T , V T ) = 1 G ( T, P T ) − ( P T − V T ) ∂ G ( T, P T ) = 1 G ( T, P T ) > .
9o we have a maximum of − E ( I ( T, P T , V T )) . We also have that P is an F -martingale from (5), the integra-bility condition and ( ii ) . Notice that R · V t d Z t is a martingale since V and Z are independent. Therefore by ( i ) and since V is an H -martingale, we obtain that P t = E ( P T |F t ) = E ( V T |F t ) = E ( E ( V T |H t ) |F t ) = E ( V t |F t ) . Now we show that ( i ) and ( ii ) are necessary conditions. In fact ( i ) is necessary by Proposition 2. By (5)and the functional Itô’s formula, we have d P t = ▽ Y P t d Y t , then the result follows from the fact that P is an F -martingale and ▽ Y P t ≥ C > .We can obtain an analogous result to Theorem 2 for the non-risk-neutral case when the utility function is U ( x ) = γe γx , γ < and when V t ≡ V. Theorem 3
Let V t ≡ V and τ = T. For any t < T , let P ∈ C , b be a price functional such that D t P t + 12 ▽ Y P t σ Z ( t ) = 0 , (11) with L P t := D t ▽ Y P t − ▽ Y D t P t = γσ Z ( t ) ( ▽ Y P t ) (12) and ▽ Y P t = G ( t, P t ) , where G ( t, y ) ≥ C > is C , . Assume that E (cid:16) exp n γ R T ( P t − V ) σ Z ( t )d t o(cid:17) < ∞ .Then there is an equilibrium in the non-risk-neutral case, with utility function U ( x ) = γe γx , if and only if ( i ) P T = V, ( ii ) Y is an F -martingale. Proof.
As in the previous proof, let I ( t, y, v ) := Z yv z − vG ( t, z ) d z, by the Itô’s formula, we have I ( T, P T , V ) = I (0 , P , V ) + Z T ∂ I d t + Z T ∂ I d P t + 12 Z T ∂ I d[ P, P ] t , and we can otain ∂ I ( t, y, v ) + 12 ∂ I ( t, y, v ) G ( t, y ) σ Z = Z yv (cid:18) − ( z − v ) (cid:18) ∂ G ( t, z ) G ( t, z ) + 12 ∂ G ( t, z ) σ Z ( t ) (cid:19)(cid:19) d z + G ( t, v ) Now, by (11) and (12) ▽ Y D t P t + 12 ▽ Y P t σ Z ( t )= D t ▽ Y P t − γσ Z ( ▽ Y P t ) + 12 ▽ Y P t σ Z ( t )= D t G ( t, P t ) + 12 ▽ Y G ( t, P t ) σ Z − γσ Z ( ▽ Y P t ) = ∂ G ( t, P t ) + ∂ G ( t, P t ) D t P t + 12 ▽ Y ( ∂ G ( t, P t ) ▽ Y P t ) σ Z ( t ) − γσ Z ( t ) ( ▽ Y P t ) = ∂ G ( t, P t ) + ∂ G ( t, P t ) (cid:18) D t P t + 12 ▽ Y P t (cid:19) σ Z ( t ) + (cid:18) ∂ G ( t, P t ) (cid:19) σ Z ( t ) ( ▽ Y P t ) − γσ Z ( t ) ( ▽ Y P t ) = ∂ G ( t, P t ) + (cid:18) ∂ G ( t, P t ) − γ (cid:19) σ Z ( t ) G ( t, P t ) . ∂ I ( t, y, v ) + 12 ∂ I ( t, y, v ) G ( t, y ) σ Z = γ ( y − v ) σ Z ( t ) + G ( t, v ) and Z T ( V − P t ) d X t − I (0 , P , V ) + Z T G ( t, V )d t ! = − I ( T, P T , V ) + Z T ( P t − V ) d Z t − γ Z T ( P t − V ) σ Z ( t )d t. Therefore, γ exp ( γ Z T ( V − P t ) d X t ) exp ( − γI (0 , P , V ) − γ Z T G ( t, V )d t ) = γ exp {− γI ( T, P T , V ) } exp ( γ Z T ( P t − V ) d Z t − γ Z T ( P t − V ) σ Z ( t )d t ) . Since ∂ I ( T, P T , V ) = P T − VG ( T, P T ) = 0 and ∂ I ( T, P T , V ) = 1 G ( T, P T ) − ( P T − V ) ∂ G ( T, P T ) = 1 G ( T, P T ) > . So the minimum value of I ( T, P T , V ) is when P T = V and its value is I ( T, P T , V ) := R P T V z − VG ( t,z ) d z = 0 . Then, since γ < , E γ exp ( γ Z T ( V − P t ) d X t ) exp ( − γI (0 , P , V ) − γ Z T G ( t, V )d t )! ≤ γ E exp ( γ Z T ( P t − V ) d Z t − γ Z T ( P t − V ) σ Z ( t )d t )! = γ. And we get the maximum value of E (cid:16) γ exp n γ R T ( V − P t ) d X t o(cid:17) when P T = V. The rest of the proof isanalogous to the one of the previous theorem.
In this section we study general necessary conditions to obtain an equilibrium and we see that the classes ofprice functionals of the previous section, characterised by the relationships (5) and (6) for the risk-neutralinsider and 11) and (12) for the risk-averse one, are actually justified by the arguments that follow. Notethat in this section, the release time of information τ is assumed predictable and bounded. A remark at theend of the session deals with the case of τ independent of the observable variables.Here below we study the effect of an ε -perturbation of the insider strategies: d X ( ε ) t := d X t + εβ t d t, where β is a bounded adapted processes, in the prices P t = P t ( Z · t + X · t ) . From now on, we are going to assume that there exist a strictly positive B ( R + ) ⊗ P F -measurable function K ( s, t )( ω ) , ≤ s ≤ t ≤ τ , ω ∈ Ω , continuous for all ≤ s ≤ t ≤ τ , such that, for a.a. t , ( R ) P ( ε ) t − P t = ε Z t K ( s, t ) β s d s + o ( ε ) R t , P F denotes the F -predictable σ -field. ε -perturbation of the strategies. Here above P ( ε ) t := P t ( Z · t + X ( ε ) · t ) , and R is a boundedprogressively measurable process. Observe that the random variables K ( s, t ) are strictly positive because P t = P t ( Y · t ) is a strictly increasing functional, see Definition (1). Note that, as a consequence of ( R ) , wehave that lim ε → P ( ε ) t − P t ε = Z t K ( s, t ) β s d s. Proposition 3
Assume that, for any bounded adapted process β , ( R ) holds by means of the kernels K described above. Then ∇ Y P t = K ( t, t ) . Proof.
Set β ( t ) s := h t − r [ t,r ] ( s ) . Taking limits in ( R ) we have that, a.s. P ⊗ Leb,P t ( Y h · t ) − P t ( Y · t ) = εK ( t, t ) + o ( ε ) lim sup r ↓ t R r By this we can conclude.The next result presents a factorisation property of the kernel and a sufficient condition to obtain it.
Proposition 4
Let G ) and F be C , . Assume that ∇ Y P t = G ( t, P t ) , (13) and D t P t + 12 ▽ Y P t σ Z ( t ) = F ( t, P t ) (14) hold. Then the kernel K admits factorisation K ( s, t ) = K ( s ) K ( t ) , (15) with K ( t ) = E (cid:18)Z t ∂ G ( s, P s )d Y s (cid:19) exp (cid:18)Z t ∂ F ( s, P s )d s (cid:19) , where E is the stochastic exponential, and K ( t ) = G ( t, P t ) K ( t ) . Moreover [ K , K ] ≡ . Proof.
Since P t = P + ∇ Y P s d Y s + Z t (cid:18) D s P s + 12 ▽ Y P s σ Z ( s ) (cid:19) d s we have that , d P ( ε ) t d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 = Z t ∂ G d P ( ε ) s d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 d Y s + Z t Gβ s d s + Z t ∂ F d P ( ε ) s d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 d s. (16)Therefore d P ( ε ) t d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 = E (cid:18)Z t ∂ G ( s, P s )d Y s (cid:19) exp (cid:18)Z t ∂ F ( s, P s )d s (cid:19) × Z t G ( s, P s ) β s E (cid:0)R s ∂ G ( u, P u )d Y u (cid:1) exp (cid:0)R s ∂ F ( u, P u ) (cid:1) d s. (17)12his is easy to be verified by showing that the differentials and the values at t = 0 of d P ( ε ) t d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 in (16) and(17) are the same. Finally, by a uniqueness argument, we have that d P ( ε ) t d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 = Z t K ( s ) K ( t ) β s d s, with K ( t ) = E (cid:18)Z t ∂ G ( s, P s )d Y s (cid:19) exp (cid:18)Z t ∂ F ( s, P s )d s (cid:19) , and K ( t ) = G ( t, P t ) E (cid:16)R t ∂ G ( u, P u )d Y u (cid:17) exp (cid:16)R t ∂ F ( u, P u ) (cid:17) . Finally it is easy to see that d K ( t ) = ∂ G + G ∂ Gσ Z ( t ) K ( t ) d t + ∂ G D t P t + ▽ Y P t K ( t ) d t − G∂ FK ( t ) d t. (18)In particular we obtain the following Proposition 5
Let P be a price functional such that (13) holds and (14) holds for F ≡ , i.e. D t P t + 12 ▽ Y P t σ Z ( t ) = 0 . (19) Then L P t = K ( t ) dd t K ( t ) . Proof.
By (19) we have L P t = D t ▽ Y P t + 12 ▽ Y (cid:0) ▽ Y P t σ Z ( t ) (cid:1) = ∂ G + 12 G ∂ Gσ Z ( t ) + 12 ∂ G (cid:18) D t P t + 12 ▽ Y P t (cid:19) = ∂ G + 12 G ∂ Gσ Z ( t ) . Now by (18) and since F ≡ , we have that L P t = K ( t ) d K ( t )d t . We have obtain a general result with a necessary condition for the an optimal strategy.
Theorem 4
Assume that P ∈ C , b , (13) holds for G in C , and that for all β bounded ( R ) hods in termsof the kernel K as above. If X is optimal, then we have [0 ,τ ) ( t ) E ( U ′ ( W τ ) ( V τ − P t ) | H t ) − E (cid:18) Z τt ∧ τ E ( U ′ ( W τ ) | H s ) K ( t, s )d X s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = 0 , a.s.- P ⊗ Leb (20)13 roof.
Take d X ( ε ) t := d X t + εβ t d t, where β is a bounded adapted processes, then, E (cid:16) U ( W ( ε ) τ ) − U ( W τ ) (cid:17) = E (cid:18) U (cid:18)Z τ (cid:16) V τ − P ( ε ) t (cid:17) d X ( ε ) t (cid:19) − U ( W τ )) (cid:19) = ε E (cid:18) U ′ ( W τ ) (cid:18)Z τ ( V τ − P t ) β t d t − Z τ (cid:18)Z t K ( s, t ) β s d s (cid:19) d X t (cid:19)(cid:19) + o ( ε )= ε E (cid:18) U ′ ( W τ ) (cid:18)Z τ (cid:18) V τ − P t − Z τt K ( t, s )d X s (cid:19) β t d t (cid:19)(cid:19) + o ( ε ) . Note that, by Fubini’s theorem, Z τ (cid:18)Z t K ( s, t ) β s d s (cid:19) d X t = E (cid:18)Z τ (cid:18)Z τt K ( t, s )d X s (cid:19) β t d t (cid:19) . Then d E (cid:16) U (cid:16) W ( ε ) τ (cid:17)(cid:17) d ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 = 0 implies that E (cid:18)Z τ U ′ ( W τ ) (cid:18) V τ − P t − Z τt K ( t, s )d X s (cid:19) β t d t (cid:19) = 0 . Since we can take β t = α u ( u,u + h ] ( t ) , with α u measurable and bounded and τ is a stopping time, we havethat [0 ,τ ) ( t ) E (cid:18) U ′ ( W τ ) (cid:18) V τ − P t − Z τt K ( t, s )d X s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = 0 , a.s.- P ⊗ Leb . And, from the Law of Iterated Expectations [0 ,τ ) ( t ) E ( U ′ ( W τ ) ( V τ − P t ) | H t ) − E (cid:18) Z τt E ( U ′ ( W τ ) | H s ) K ( t, s )d X s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = 0 The result above allows us to give some necessary conditions for an equilibrium.
Proposition 6
In the conditions of the Theorem 4 and assuming that the factorisation (15) holds, we havethat if ( P, X ) is an equilibrium, then E ( U ′ ( W τ ) ( V τ − P t ) | H t ) dd t (cid:18) K ( t ) (cid:19) − E ( U ′ ( W τ ) | H t ) K ( t ) (cid:18) D t P t + 12 ▽ P t σ Z ( t ) (cid:19) − K ( · ) dd t [ P, E ( U ′ ( W τ ) | H · )] t + dd t (cid:20) E ( U ′ ( W τ ) ( V τ − P · ) | H · ) , K ( · ) (cid:21) t . (21) Proof.
Thanks to ( A1 ′ ) , ( R ) , the factorisation property (15), and by means of Theorem 1 and Proposition3, we have that E (cid:18) Z τt E ( U ′ ( W τ ) | H s ) K ( t, s )d X s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = K ( t ) E (cid:18) Z τt K ( s ) E ( U ′ ( W τ ) | H s ) K ( s, s )d Y s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = K ( t ) E (cid:18) Z τt K ( s ) E ( U ′ ( W τ ) | H s ) (cid:18) d P s − (cid:18) D s P s + 12 ▽ Y P s σ Z ( s ) (cid:19) d s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = K ( t ) E (cid:18) Z τt K ( s ) E ( U ′ ( W τ ) | H s ) d P s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) − K ( t ) E (cid:18) Z τt E ( U ′ ( W τ ) | H s ) K ( s ) (cid:18) D s P s + 12 ▽ Y P s σ Z ( s ) (cid:19) d s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) . Z t K ( s ) E ( U ′ ( W τ ) | H s ) d P s = E ( U ′ ( W τ ) | H t ) P t K ( t ) − E ( U ′ ( W τ ) | H ) P K (0) − Z t P s d (cid:18) E ( U ′ ( W τ ) | H s ) K ( s ) (cid:19) − (cid:20) P, E ( U ′ ( W τ ) | H · ) K ( · ) (cid:21) t . Hence, taking (20) into account, we obtain [0 ,τ ) ( t ) (cid:18) E ( U ′ ( W τ ) ( V τ − P t ) | H t ) K ( t ) (cid:19) + E ( U ′ ( W τ ) | H t ∧ τ ) P t ∧ τ K ( t ∧ τ ) − E ( U ′ ( W τ ) | H ) P K (0) − Z t ∧ τ P s d (cid:18) E ( U ′ ( W τ ) | H s ) K ( s ) (cid:19) − (cid:20) P, E ( U ′ ( W τ ) | H · ) K ( · ) (cid:21) t ∧ τ − Z t ∧ τ E ( U ′ ( W τ ) | H s ) K ( s ) (cid:18) D s P s + 12 ▽ Y P s σ Z ( s ) (cid:19) d s + E (cid:18) Z τ E ( U ′ ( W τ ) | H s ) K ( s ) (cid:18) D s P s + 12 ▽ Y P s σ Z ( s ) (cid:19) d s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) − E (cid:18) Z τ K ( s ) E ( U ′ ( W τ ) | H s ) d P s (cid:12)(cid:12)(cid:12)(cid:12) H t (cid:19) = 0 . Then by the uniqueness of the canonical decomposition in the previous equation (notice that the jump of [0 ,τ ) ( t ) is killed in the case that τ is predictive), we have E ( U ′ ( W τ ) ( V τ − P t ) | H t ) dd t (cid:18) K ( t ) (cid:19) − E ( U ′ ( W τ ) | H t ) K ( t ) (cid:18) D t P t + 12 ▽ Y P t σ Z ( t ) (cid:19) − K ( t ) dd t [ P, E ( U ′ ( W τ ) | H · )] t + dd t (cid:20) E ( U ′ ( W τ ) ( V τ − P · ) | H · ) , K ( · ) (cid:21) t . Moreover, we have the following specific conditions in the risk-neutral and risk-averse (exponential) cases.
Proposition 7
In the risk-neutral case, under the assumptions of Proposition 6, if ( P, X ) is an equilibrium,then D t P t + 12 ▽ Y P t σ Z ( t ) = 0 holds (see (14) and (19) . Also, if V t = P t , a.s. P ⊗ Leb, we have that L P t = 0 . (22) Proof.
As a consequence of Proposition 6 we have that, in the risk neutral case, for the functionals above, V t − P t ) dd t (cid:18) K ( t ) (cid:19) − K ( t ) (cid:18) D t P t + 12 ▽ Y P t d[ Z ] t d t (cid:19) By the competitiveness of prices E ( V t |F t ) = P t , so by taking conditional expectations w.r.t F t we obtainthat D t P t + 12 ▽ Y P t d[ Z ] t d t = 0 and if V t = P t , a.s. P ⊗ Leb, dd t (cid:18) K ( t ) (cid:19) = 0 . (23)Now by Proposition 5 we obtain (22).Consider the risk-averse case when U ( x ) = γe γx with γ < . If the noise traders total demand Z is Gaussianwe can apply the following relationship between vertical and Fréchet or Malliavin derivatives (see Theorem6.1 in [12]): E (cid:0) D Zt U ( W τ ) (cid:12)(cid:12) H t (cid:1) = ∇ Z E ( U ( W τ ) | H t ) . E (cid:16) U ′ ( W τ ) ( V τ − P t ) + D Zt U ( W τ ) (cid:12)(cid:12)(cid:12) H t (cid:17) = 0 , we have that ∇ Z E ( U ( W τ ) | H t ) = − E (cid:16) U ′ ( W τ ) ( V τ − P t ) (cid:12)(cid:12)(cid:12) H t (cid:17) . Since U ′ ( x ) = γU ( x ) , we obtain ∇ Z E ( U ′ ( W τ ) | H t ) = − γ E (cid:16) U ′ ( W τ ) ( V τ − P t ) (cid:12)(cid:12)(cid:12) H t (cid:17) d [ P, E ( U ′ ( W τ ) | H · )]d t = ∇ Y P t ∇ Z E (cid:16) U ′ ( W τ ) (cid:12)(cid:12)(cid:12) H t (cid:17) σ Z ( t ) = K ( t, t ) E (cid:16) U ′ ( W τ ) ( V τ − P t ) (cid:12)(cid:12)(cid:12) H t (cid:17) σ Z ( t ) . Then (21) becomes E ( U ′ ( W τ ) ( V τ − P t ) | H t ) (cid:18) dd t (cid:18) K (cid:19) + γK ( t ) σ Z ( t ) (cid:19) − E ( U ′ ( W τ ) | H t ) K ( t ) (cid:18) D t P t + 12 ▽ P t σ Z ( t ) (cid:19) = 0 . Furthermore, if V t ≡ V we have that ( V − P t ) (cid:18) dd t (cid:18) K (cid:19) + γK ( t ) σ Z ( t ) (cid:19) − K ( t ) (cid:18) D t P t + 12 ▽ Y P t σ Z ( t ) (cid:19) = 0 . Taking the conditional expectations w.r.t F t , by the competitiveness of prices E ( V |F t ) = P t , we obtain that D t P t + 12 ▽ Y P t σ Z ( t ) = 0 . Provided that V = P t , a.s. P ⊗ Leb , we have that dd t (cid:18) K ( t ) (cid:19) + γK ( t ) σ Z ( t ) = 0 . (24)Then we have the following proposition. Proposition 8
Consider the risk-averse case with utility function is U ( x ) = γe γx , γ < . Let V t ≡ V andassume that (15) holds. Also assume that Z is Gaussian. If ( P, X ) is an equilibrium, we have (19) : D t P t + 12 ▽ Y P t σ Z ( t ) = 0 and, if V = P t , a.s. P ⊗ Leb , we have that L P t = γσ Z ( ∇ Y P t ) . Proof.
By (24) K ( t ) dd t K ( t ) = γK ( t ) σ Z ( t ) , now by Proposition 5 and Proposition 3 L P t = γK ( t ) K ( t ) σ Z ( t ) = γ ( ∇ Y P t ) σ Z ( t ) . emark 6 Finally, according to [14] and for the above functionals if the horizon τ is random and inde-pendent of the rest of processes involved, in an equilibrium situation we have dd t P ( τ > t ) K ( t ) = 0 , then dd t K ( t ) = K ( t ) dd t P ( τ > t ) , and by Proposition 5 L P t = K ( t ) d K ( t )d t = K ( t ) K ( t ) dd t P ( τ > t )= ▽ Y P t dd t P ( τ > t ) . Consider the following class of functionals P t = H ( t, ξ t ) , t ≥ , ξ t := Z t λ ( s, P s )d Y s , where λ ∈ C , is a strictly positive function and H ∈ C , with H ( t, · ) strictly increasing for every t ≥ . Then, by using the Itô’s formula and omitting the arguments in the functions, we have d P t = ∂ Hλ d Y t + (cid:18) ∂ H + 12 ∂ Hλ σ Z (cid:19) d t. Furthermore, we have that D t P t = ∂ H + ∂ H D t ξ t = ∂ H − ∂ H ▽ Y ξ t σ Z = ∂ H − ∂ H ▽ Y λσ Z = ∂ H − ∂ H∂ λ ▽ Y P t σ Z and ▽ Y P t = ∂ λ ▽ Y P t ∂ H + λ∂ H ▽ Y ξ t = ∂ λ ▽ Y P t ∂ H + λ ∂ H. Consequently, D t P t + 12 ▽ Y P t σ Z = ∂ H + 12 ∂ Hλ σ Z . Then, under the condition D t P t + 12 ▽ Y P t σ Z = 0 , we have that ∂ H + 12 ∂ Hλ σ Z = 0 , and by Proposition 4, K ( s, t ) = λ ( s, P s ) η s ∂ H ( t, ξ t ) η t , η t := E (cid:18)Z t ∂ H∂ λ d Y s (cid:19) . Therefore K ( s, t ) = K ( s ) K ( t ) , with K ( s ) = λ ( s, P s ) η s , K ( t ) = ∂ H ( t, ξ t ) η t . By using the Itô formula we obtain that d (cid:18) K ( s ) (cid:19) = η s d (cid:18) λ (cid:19) + 1 λ d η s + d (cid:20) η, λ (cid:21) s = − η s ∂ λλ d s − η s ∂ λλ ∂ Hλ d Y s − η s λ ∂ λ − ∂ λ ) λλ ( ∂ H ) λ σ Z d s + 1 λ η s ∂ λ∂ H d Y s − ( ∂ λ∂ H ) λ σ Z η s d s = − η s ∂ λλ d s − ∂ λ ( ∂ H ) σ Z η s d s = − η s (cid:18) ∂ λ ( ∂ H ) σ Z + ∂ λλ (cid:19) d s, Then, we have that L P t = K ( t ) dd t K ( t ) = K ( t ) K ( t ) η t (cid:18) ∂ λ ( ∂ H ) σ Z + ∂ λλ (cid:19) = ∂ Hλ (cid:18) ∂ λ ( ∂ H ) σ Z + ∂ λλ (cid:19) = ∂ H (cid:18) ∂ λ + 12 σ Z ( λ∂ H ) ∂ λ (cid:19) Hence, we will have an equilibrium price rule, in the risk-neutral case, if ∂ H + 12 ∂ Hλ σ Z = 0 ,∂ λ + 12 σ Z ( λ∂ H ) ∂ λ = 0 . and in the non risk-neutral case, for the exponential risk aversion, if ∂ H + 12 ∂ Hλ σ Z = 0 , and L P t = ∂ H (cid:18) ∂ λ + 12 σ Z ( λ∂ H ) ∂ λ (cid:19) = γ ( ∇ Y P t ) σ Z = γK ( t ) K ( t ) σ Z = γ ( ∂ Hλ ) σ Z that is ∂ λλ + 12 σ Z ( ∂ H ) ∂ λ = γ∂ Hσ Z . (25)We can identify some particular cases. For the risk-neutral case • λ ( t, x ) = λ > , and H ( t, x ) harmonic with H ( , x ) strictly increasing . Notice that in this case it issufficient to require that H ( t, x ) is C , since (10) is trivially satisfied.18 If we take H ( t, x ) = x and λ ( t, x ) = λ > , we have P t = P + λY t , that corresponds to the Bachelier model for Z Gaussian. • If H ( t, x ) = x and λ ( t, x ) = λx, we have P t = P e λY t − λ t that is the Black-Scholes model. For the non risk-neutral model • Note that H ( t, x ) harmonic and λ constant cannot be an equilibrium. Therefore equilibrium pricescannot be a function of the spot aggregate demand. • If we take H ( t, x ) = x and λ ( t, x ) = Cx (1 − x ) , with C > , we have that (25) becomes ∂ xx λ = γ that is γ = − C. This model will give prices in (0 , and if Y is a Brownian motion B we have that d P t = CP t (1 − P t )d B t and this is the well-known Kimura model in population genetics, see Kimura(1964). It is apparent that depending on the behaviour of the fundamental value and the aggregate demand of thenoise traders we can have an equilibrium with one or another equilibrium pricing rule. If the aggregatedemand Z of the noise traders is a Brownian motion with variance σ Z , Y = X + Z will be also an F -Brownian motion with variance σ Z , because of Theorem 2 and the Lévy Theorem. Consequently, we willhave an equilibrium if the pricing rule P t ( Y · t ) is such that P T ( Y · T ) = V T . Note also that the strategy X willbe just obtained as the canonical decomposition of the F -Brownian motion Y under the filtration H .Consider the case where Z is a Brownian motion with variance σ and V t ≡ V . In such a situation we have anecessary and sufficient condition for and equilibrium for both, the risk-neutral case and the risk-adverse caseunder the exponential utility. Also in both cases the equilibrium pricing rules give prices that are continuousdiffusions: d P t = λ ( t, P t )d Y t where d Y t = σ d W t and W is a standard Brownian motion. In the risk-neutral case λ ( t, x ) satisfies ∂ t λ + 12 λ σ ∂ xx λ = 0 , and in the risk-adverse case ∂ t λ + 12 λ σ ∂ xx λ = γλ σ . In any case the additional necessary and sufficient condition to have an equilibrium model is to find a strategysuch that P T = V and at the same time Y is certainly a Brownian motion with variance σ . We have to find α t ( V ) , with α t ( x ) F t -measurable, such that the equation d Y t = α t ( V )d t + d Z t , ≤ t ≤ T, V = P T and independent of Z, has a strong solution. In order to do so, we can look for certain α ( t, x, Y · t ) , where x is a fixed value of P T and try to find a strong solution of d Y t = α t ( x )d t + d Z t , ≤ t ≤ T, later we can insert V instead of y , but we need Y to be a Brownian motion with variance σ . Sufficientconditions to have a strong solution are given, e.g., in Theorem 4.6, Lipster and Shiryaev (2001). Then α t ( x ) has to be the drift in the canonical decomposition of Y when Y T and Z · t are known at time t. Thefollowing propositions are useful to find α, here we assume that F t = ¯ σ ( W · t , t ≥ , ¯ σ denotes the σ -fieldcorresponding to the usual augmentation of the natural filtration. Proposition 9
Assume that for any bounded and measurable function f there exists a B [0 , T ]) ⊗ F T -measurable process ξ such that f ( P T ) = E ( f ( P T ))+ Z T E ( f ( P T ) ξ t |F t ) d W t , with R T | ξ t | d t < ∞ . Then W · − R · α t ( P T )d t is an ( F t ∨ σ ( P T )) -Brownian motion with α t ( P T ) = E ( ξ t |F t ∨ σ ( P T )) Proof.
Let f be a measurable and bounded function and A ∈ F s , with s ≤ t. Then E (( W t − W s ) A f ( P T )) = E (cid:18) A Z ts E ( f ( P T ) ξ u |F u ) d u (cid:19) = E (cid:18) A f ( P T ) Z ts E ( ξ u |F u ∨ σ ( P T )) d u (cid:19) . Proposition 10
Suppose that d P P T |F t ( x |F t ) = L T ( x ; W · t ) µ (d x ) is a regular version of the conditional probability of P T given F t , µ being a reference measure and such that i ) L T ( x ; W · t ) > for all ( x, ω ) µ ⊗ P -a.s. ,ii ) ∇ W Z R f ( x ) L T ( x ; W · t ) µ (d x ) = Z R f ( x ) ∇ W L T ( x ; W · t ) µ (d x ) . Then W · − R · α t ( P T )d t is an ( F t ∨ σ ( P T )) -Brownian motion with α t ( x ) = ∇ W log L T ( x ; W · t ) , provided that log L T ( x ; W · t ) ∈ C ( W ) . Proof.
Let f be a measurable and bounded function ∇ W E ( f ( P T ) |F t ) = ∇ W Z R f ( x ) L T ( x ; W · t ) µ (d x )= Z R f ( x ) ∇ W L T ( x ; W · t ) µ (d x )= Z R f ( x ) ∇ W log L T ( x ; W · t ) L T ( x ; W · t ) µ (d x )= E ( f ( P T ) ∇ W log L T ( P T ; W · t ) |F t ) . xample 3 Assume that P t = P + σW t . Then P T |F t ∼ N (cid:0) P + σW t , σ ( T − t ) (cid:1) , that is d P P T |F t ( x |F t ) = 1 p πσ ( T − t ) exp (cid:26) − σ ( T − t ) ( x − P − σW t ) (cid:27) d x, then α t ( x ) = σ ( x − P − σW t ) σ ( T − t ) , that is α t ( P T ) = W T − W t T − t . Example 4
Assume that P t = P + R t G ( u, P u )d W u . G ∈ C , , E (cid:16)R T G ( t, P t )d t (cid:17) < ∞ . Let p s,t ( x, y ) thetransition density corresponding to the Markov process P . Then according to the previous proposition α t ( y ) = ∇ W log p t,T ( P t , y )= ∂ log p t,T ( P t , y ) ∇ W P t = ∂ log p t,T ( P t , y ) G ( t, P t ) . For instance, we can consider the simple case where P t = P + R t σP u d W u , then, for s ≤ t , P t = P s exp (cid:8) σ ( W t − W s ) − σ ( t − s ) (cid:9) , and p s,t ( x ; y ) = 1 p πσ ( T − t ) exp ( − σ ( T − t ) (cid:18) log y − log x − σ ( t − s ) (cid:19) ) y , consequently α t ( y ) = ∂ log p t,T ( P t , y ) G ( t, P t ) = σ log y − log P t − σ ( t − s ) σ ( T − t ) , that is α t ( P T ) = W T − W t T − t . Notice that in thsi case we obtain the same result as in the previous example. This is not surprising since inboth cases to know P T is the same as to know W T since P T is an increasing function of W T . Obviously thiswill not be the case for a general diffusion. The simplest case where this does not happen is the equilibriumprice model P t = P + Z t λ ( s )d W s , with ∂ t λ = γλ σ . arising in the non-risk-neutral model. Now P T | P t ∼ N (cid:16) P , R t λ ( s )d s (cid:17) , then α t ( x ) = ∇ W log L T ( x ; W · t )= λ ( t ) ( x − P t ) R Tt λ ( s )d s . If we consider the Kimura model, with risk-aversion parameter γ < , P t = P − γ Z t P t (1 − P t )d W t , hen the transition density is given by (see Kimura(1964)) p t,T ( P t , x ) = 1 p πγ ( T − t ) p P t (1 − P t ) (cid:16)p x (1 − x ) (cid:17) exp − γ T − t ) − (cid:16) log x (1 − P t )(1 − x ) P t (cid:17) γ ( T − t ) , and we have that α t ( y ) = ∇ W log L T ( x ; W · t )= 12 (1 − P t ) + log x (1 − P t )(1 − x ) P t γ ( T − t ) . Example 5
We can consider the case where the privilege information is the time, say τ , where a Brownianmotion reaches for the first time a level a. Then, assume that P T = h ( T ∧ τ ) for a measurable and bounded function h . Now we have that P T = P + Z T ∇ W E ( h ( T ∧ τ ) |F t ) d W t , then since f τ ( u |F t ) = W t − a p π ( u − t ) exp ( − ( W t − a ) u − t ) ) { τ>t } , we obtain that α t ( u ) = ∇ W log f τ ( u |F t ) = (cid:18) W t − a − W t − au − t (cid:19) { τ>t } . Acknowledgement : This paper is devoted to the memory of our beloved colleague José Fajardo Barbachán,who passed away before we could complete together this work.
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