A model for a large investor trading at market indifference prices. II: Continuous-time case

# A model for a large investor trading at market indifference prices. II: Continuous-time case

[ [    [ [ Technische Universität Berlin and Carnegie Mellon University Institut für Mathematik
Technische Universität Berlin
Straße des 17. Juni 136
10623 Berlin
Germany
Department of Mathematical Sciences
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, Pennsylvania 15213-3890
USA
\smonth9 \syear2011\smonth1 \syear2014
\smonth9 \syear2011\smonth1 \syear2014
\smonth9 \syear2011\smonth1 \syear2014
###### Abstract

We develop from basic economic principles a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices. In this model, the market makers compete with their quotes for the investor’s orders and trade among themselves to attain Pareto optimal allocations. We first consider the case of simple strategies and then, in analogy to the construction of stochastic integrals, investigate the transition to general continuous dynamics. As a result, we show that the model’s evolution can be described by a nonlinear stochastic differential equation for the market makers’ expected utilities.

[
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\doi

10.1214/14-AAP1059 \volume25 \issue5 2015 \firstpage2708 \lastpage2742 \docsubtyFLA \newproclaimAssumption[Theorem]Assumption\newproclaimDefinition[Theorem]Definition\newproclaimExample[Theorem]Example \newproclaimQuestion[Theorem]Question\newproclaimRemark[Theorem]Remark

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Continuous-time model for a large investor

{aug}

A]\fnmsPeter \snmBank\thanksrefT1label=e1]bank@math.tu-berlin.de and B]\fnmsDmitry \snmKramkov\corref\thanksrefT2label=e2]kramkov@cmu.edu \thankstextT1Supported in part by NSF under Grant DMS-05-05021. \thankstextT2The author also holds a part-time position at the University of Oxford. Supported in part by NSF under Grant DMS-05-05414 and by the Carnegie Mellon Portugal Program.

class=AMS] \kwd[Primary ]91G10 \kwd91G20 \kwd[; secondary ]52A41 \kwd60G60 Bertrand competition \kwdcontingent claims \kwdequilibrium \kwdindifference prices \kwdliquidity \kwdlarge investor \kwdPareto allocation \kwdprice impact \kwdsaddle functions \kwdnonlinear stochastic integral \kwdrandom field

## 1 Introduction

A typical financial model presumes that the prices of traded securities are not affected by an investor’s buy and sell orders. From a practical viewpoint, this assumption is justified as long as his trading volume remains small enough to be easily covered by market liquidity. An opposite situation occurs, for instance, when an economic agent has to sell a large block of shares over a short period of time; see, for example, Almgren and Chriss  and Schied and Schöneborn . This and other examples motivate the development of financial models for a “large” trader, where the dependence of market prices on his strategy, called a price impact or a demand pressure, is taken into account.

Hereafter, we assume that the interest rate is zero and, in particular, is not affected by the large investor. As usual in mathematical finance, we describe a (self-financing) strategy by a predictable process where is the number of stocks held just before time and is a finite time horizon. The role of a “model” is to define a predictable process representing the evolution of the cash balance for the strategy . We denote by the marginal price process of traded stocks, that is, is the price at which one can trade an infinitesimal quantity of stocks at time . Recall that in the standard model of a “small” agent the price does not depend on and

 Xt(Q)=∫t0QudSu−QtSt.

In mathematical finance, a common approach is to specify the price impact of trades exogenously, that is, to postulate it as one of the inputs. For example, Frey and Stremme , Platen and Schweizer , Papanicolaou and Sircar  and Bank and Baum  choose a stochastic field of reaction functions, which explicitly state the dependence of the marginal prices on the investor’s current holdings, Çetin, Jarrow and Protter in  start with a stochastic field of supply curves, which define the prices in terms of traded quantities (changes in holdings), and Cvitanić and Ma  make the drift and the volatility of the price process dependent on a trading strategy; we refer the reader to the recent survey  by Gökay, Roch and Soner for more details and additional references. Note that in all these models the processes and , of the cash balance and of the marginal stock price, only depend on the “past” of the strategy , in the sense that

 Xt(Q)=Xt(Qt),St(Q)=St(Qt), (1)

where denotes the process “stopped” at with .

The exogenous nature of the above models facilitates their calibration to market data; see, for example,  by Çetin, Jarrow and Protter. There are, however, some disadvantages. For example, the models in [13, 23, 22, 4, 8] and  do not satisfy the natural “closability” property for a large investor model:

 ∣∣Qn∣∣≤1n⟹XT(Qn)→0,n→∞, (2)

while in Cvitanić and Ma  the stock price is not affected by a jump in investor’s holdings: .

In our project, we seek to derive the dependence of prices on strategies endogenously by relying on the framework developed in financial economics. A starting point here is the postulate that, at any given moment, a price reflects a balance between demand and supply or, more formally, it is an output of an equilibrium. In addition to the references cited below, we refer the reader to the book  by O’Hara and the survey  by Amihud, Mendelson and Pedersen.

To be more specific, denote by the terminal price of the traded security, which we assume to be given exogenously, that is, for every strategy . Recall that in a small agent model the absence of arbitrage implies the existence of an equivalent probability measure such that

 St=EQ[ψ|Ft],0≤t≤T, (3)

where is the -field describing the information available at time . This result is often called the fundamental theorem of asset pricing; in full generality, it has been proved by Delbaen and Schachermayer in [11, 12]. The economic nature of this pricing measure does not matter in the standard, small agent, setup. However, it becomes important in an equilibrium-based construction of models for a large trader where it typically originates from a Pareto optimal allocation of wealth and is given by the expression (4) below.

We shall consider an economy formed by market participants, called hereafter the market makers, whose preferences for terminal wealth are defined by utility functions , , and an identical subjective probability measure . It is well known in financial economics that the Pareto optimality of the market makers’ wealth allocation yields the pricing measure  defined by

 dQdP=vmu′m(αm),m=1,…,M, (4)

where is a normalizing constant.

It is natural to expect that in the case when the strategy is not anymore negligible an expression similar to (3) should still hold true for the marginal price process:

 St(Q)=EQt(Q)[ψ|Ft(Q)],0≤t≤T. (5)

This indicates that the price impact at time described by the mapping may be attributed to two common aspects of market’s microstructure: {longlist}

. Models focusing on information aspects naturally occur in the presence of an insider, where , the information available to the market makers at time , is usually generated by the sum of and the cumulative demand process of “noise” traders; see Glosten and Milgrom , Kyle  and Back and Baruch , among others.

. In view of (4), this reflects how , the Pareto optimal allocation of the total wealth or “inventory” induced by , affects the valuation of marginal trades. Note that the random variable is measurable with respect to the terminal -field [not with respect to the current -field !].

In our study, we shall focus on the inventory aspect of price formation and disregard the informational component. We assume that the market makers share the same exogenously given filtration as the large trader and, in particular, their information flow is not affected by his strategy :

 Ft(Q)=Ft,0≤t≤T.

Note that this informational symmetry is postulated only regarding the externally given random outcome. As we shall discuss below, in inventory based models, the actual form of the map , or, equivalently, is implied by game-theoretical features of the interaction between the market makers and the investor. In particular, it depends on the knowledge the market makers possess at time about the subsequent evolution of the investor’s strategy, conditionally on the forthcoming random outcome on .

For example, the models in Grossman and Miller , Garleanu, Pedersen and Poteshman  and German  rely on a setup inspired by the Arrow–Debreu equilibrium. Their framework implicitly assumes that right from the start the market makers have full knowledge of the investor’s future strategy (of course, contingent on the unfolding random scenario). In this case, the resulting pricing measures and the Pareto allocations do not depend on time:

 Qt(Q)=Q(Q),αt(Q)=α(Q),0≤t≤T, (6)

and are determined by the budget equations:

 EQ(Q)[αm(0)]=EQ(Q)[αm(Q)],m=1,…,M,

and the clearing condition:

 M∑m=1αm(Q)=M∑m=1αm(0)+∫T0QtdSt(Q).

Here, and are defined in terms of by (4) and (5). The positive sign in the clearing condition is due to our convention to interpret as the number of stocks held by the market makers. It is instructive to note that in the case of exponential utilities, when with a risk-aversion , the stock price in these models depends only on the “future” of the strategy:

 St(Q)=St((Qs)t≤s≤T),0≤t≤T,

which is just the opposite of (1).

In our model, the interaction between the market makers and the investor takes place according to a Bertrand competition; a similar framework (but with a single market maker and only in a one-period setting) was used in Stoll . The key economic assumptions can be summarized as follows: {longlist}[2.]

After every trade, the market makers can redistribute new income to form a Pareto allocation.

As a result of a trade, the expected utilities of the market makers do not change. The first condition assumes that the market makers are able to find the most effective way to share among themselves the risk of the resulting total endowment, thus producing a Pareto optimal allocation. The second assumption is a consequence of a Bertrand competition which forces the market makers to quote the most aggressive prices without lowering their expected utilities; in the limit, these utilities are left unchanged.

Our framework implicitly assumes that at every time the market makers have no a priori knowledge about the subsequent trading strategy of the economic agent (even conditionally on the future random outcome). As a consequence, the marginal price process and the cash balance process are related to as in (1). Similarly, the dependence on of the pricing measures and of the Pareto optimal allocations is nonanticipative in the sense that

 Qt(Q)=Qt(Qt),αt(Q)=αt(Qt),0≤t≤T,

which is quite opposite to (6).

In , we studied the model in a static, one-step, setting. The current paper deals with the general continuous-time framework. Building on the single-period case in an inductive manner, we first define simple strategies, where the trades occur only at a finite number of times; see Theorem 2.2. The main challenge is then to show that this construction allows for a consistent passage to general predictable strategies. For instance, it is an issue to verify that the cash balance process is stable with respect to uniform perturbations of the strategy and, in particular, that the closability property (2) and its generalizations stated in Questions 2.2 and 2.2 hold.

These stability questions are addressed by deriving and analyzing a nonlinear stochastic differential equation for the market makers’ expected utilities; see (60) in Theorem 4.5. A key role is played by the fact, that together with the strategy , these utilities form a “sufficient statistics” in the model, that is, they uniquely determine the Pareto optimal allocation of wealth among the market makers. The corresponding functional dependencies are explicitly given as gradients of the stochastic field of aggregate utilities and its saddle conjugate; here we rely on our companion paper .

An outline of this paper is as follows. In Section 2, we define the model and study the case when the investor trades according to a simple strategy. In Section 3, we provide a conditional version of the well-known parameterization of Pareto optimal allocations and recall basic results from  concerning the stochastic field of aggregate utilities and its conjugate. With these tools at hand, we formally define the strategies with general continuous dynamics in Section 4. We conclude with Section 5 by showing that the construction of strategies in Section 4 is consistent with the original idea based on the approximation by simple strategies. In the last two sections, we restrict ourselves to a Brownian setting, due to convenience of references to Kunita .

## 2 Model

### 2.1 Market makers and the large investor

We consider a financial model where market makers quote prices for a finite number of stocks. Uncertainty and the flow of information are modeled by a filtered probability space satisfying the standard conditions of right-continuity and completeness; the initial -field is trivial, is a finite maturity and .

As usual, we identify random variables differing on a set of -measure zero; stands for the metric space of such equivalence classes with values in endowed with the topology of convergence in probability; , , denotes the Banach space of -integrable random variables. For a -field and a set denote and , , the respective subsets of and consisting of all -measurable random variables with values in .

The way the market makers serve the incoming orders crucially depends on their attitude toward risk, which we model in the classical framework of expected utility. Thus, we interpret the probability measure as a description of the common beliefs of our market makers (same for all) and denote by market maker ’s utility function for terminal wealth.

{Assumption}

Each , , is a strictly concave, strictly increasing, continuously differentiable, and bounded from above function on the real line satisfying

 limx→∞um(x)=0. (7)

The normalizing condition (7) is added only for notational convenience. Our main results will be derived under the following additional condition on the utility functions, which, in particular, implies their boundedness from above.

{Assumption}

Each utility function , , is twice continuously differentiable and its absolute risk aversion coefficient is bounded away from zero and infinity, that is, for some ,

 1c≤am(x)≜−u′′m(x)u′m(x)≤c,x∈R.

The prices quoted by the market makers are also influenced by their initial endowments , where is an -measurable random variable describing the terminal wealth of the th market maker (if the large investor, introduced later, will not trade at all). We assume that the initial allocation is Pareto optimal in the sense of:

{Definition}

Let be a -field contained in . A vector of -measurable random variables is called a Pareto optimal allocation given the information or just a -Pareto allocation if

 E[∣∣um(αm)∣∣|G]<∞,m=1,…,M, (8)

and there is no other allocation with the same total endowment,

 M∑m=1βm=M∑m=1αm, (9)

leaving all market makers not worse and at least one of them better off in the sense that

 E[um(βm)|G]≥E[um(αm)|G]for all m=1,…,M (10)

and

 P[E[um(βm)|G]>E[um(αm)|G]]>0for some m∈{1,…,M}. (11)

A Pareto optimal allocation given the trivial -field is simply called a Pareto allocation.

In other words, Pareto optimality is a stability requirement for an allocation of wealth which ensures that there are no mutually beneficial trades that can be struck between market makers.

Finally, we consider an economic agent or investor who is going to trade dynamically in the financial market formed by a bank account and stocks. We assume that the interest rate on the bank account is given exogenously and is not affected by the investor’s trades; for simplicity of notation, we set it to be zero. The stocks pay terminal dividends . Their prices are computed endogenously and depend on investor’s order flow.

As the result of trading with the investor, up to and including time , the total endowment of the market makers may change from to

 Σ(ξ,θ)≜Σ0+ξ+⟨θ,ψ⟩=Σ0+ξ+J∑j=1θjψj, (12)

where and are, respectively, the cash amount and the number of assets acquired by the market makers from the investor; they are -measurable random variables with values in and , respectively. Our model will assume that is allocated among the market makers in the form of an -Pareto allocation. For this to be possible, we have to impose:

{Assumption}

For every and , there is an allocation with total random endowment defined in (12) such that

 E[um(βm)]>−∞,m=1,…,M. (13)

See (36) for an equivalent reformulation of this assumption in terms of the aggregate utility function. For later use, we verify its conditional version.

###### Lemma 2.1

Under Assumptions 2.1 and 2.1, for every -field and random variables and there is an allocation with total endowment such that

 E[um(βm)|G]>−∞,m=1,…,M. (14)
{pf}

Clearly, it is sufficient to verify (14) on each of the -measurable sets

 An≜{ω∈Ω\dvtx∣∣ξ(ω)∣∣+∣∣θ(ω)∣∣≤n},n≥1,

which shows that without loss of generality we can assume and to be bounded when proving (14). Then can be written as a convex combination of finitely many points , with -measurable weights , . By Assumption 2.1, for each there is an allocation with the total endowment such that

 E[um(βmk)]>−∞,m=1,…,M.

Thus, the allocation

 β≜K∑k=1λkβk

has the total endowment and, by the concavity of the utility functions, satisfies (13), and hence, also (14).

### 2.2 Simple strategies

An investment strategy of the agent is described by a predictable -dimensional process , where is the cumulative number of the stocks sold by the investor through his transactions up to time . For a strategy to be self-financing we have to complement by a corresponding predictable process describing the cumulative amount of cash spent by the investor. Hereafter, we shall call such an a cash balance process.

{Remark}

Our description of a trading strategy follows the standard practice of mathematical finance except for the sign: positive values of or now mean short positions for the investor in stocks or cash, and hence total long positions for the market makers. This convention makes future notation more simple and intuitive.

To facilitate the understanding of the economic assumptions behind our model, we consider first the case of a simple strategy where trading occurs only at a finite number of times, that is,

 Qt=N∑n=1θn1(τn−1,τn](t),0≤t≤T, (15)

with stopping times and random variables , . It is natural to expect that, for such a strategy , the cash balance process has a similar form:

 Xt=N∑n=1ξn1(τn−1,τn](t),0≤t≤T, (16)

with , . In our model, these cash amounts will be determined by (forward) induction along with a sequence of conditionally Pareto optimal allocations such that each is an -Pareto allocation with the total endowment

 Σ(ξn,θn)=Σ0+ξn+⟨θn,ψ⟩.

Recall that at time , before any trade with the investor has taken place, the market makers have the initial Pareto allocation and the total endowment . After the first transaction of stocks and in cash, the total random endowment becomes . The central assumptions of our model, which will allow us to identify the cash amount uniquely, are that, as a result of the trade: {longlist}[2.]

The random endowment is redistributed between the market makers to form a new Pareto allocation .

The market makers’ expected utilities do not change:

Proceeding by induction, we arrive at the re-balance time with the economy characterized by an -Pareto allocation of the random endowment . We assume that after exchanging securities and in cash the market makers will hold an -Pareto allocation of satisfying the key condition of the preservation of expected utilities:

 E[um(αmn+1)|Fτn]=E[um(αmn)|Fτn],m=1,…,M. (17)

The fact that this inductive procedure indeed works is ensured by the following result, established in a single-period framework in , Theorem 2.6.

###### Theorem 2.2

Under Assumptions 2.1 and 2.1, every sequence of stock positions as in (15) yields a unique sequence of cash balances as in (16) and a unique sequence of allocations such that, for each , is an -Pareto allocation of preserving the market makers’ expected utilities in the sense of (17).

{pf}

The proof follows from Lemma 2.1 above, Lemma 2.3 below and a standard induction argument.

###### Lemma 2.3

Let Assumption 2.1 hold and consider a -field and random variables and . Suppose there is an allocation which has the total endowment and satisfies the integrability condition (14).

Then there are a unique and a unique -Pareto allocation with the total endowment such that

 E[um(αm)|G]=γm,m=1,…,M.
{pf}

The uniqueness of such and is a consequence of the definition of the -Pareto optimality and the strict concavity and monotonicity of the utility functions. Indeed, let and be another such pair. The allocation

 βm≜(˜αm+ξ−˜ξM)1{˜ξ<ξ}+αm1{˜ξ≥ξ},m=1,…,M,

has the same total endowment as . If the -measurable set is not empty, then because the utility functions are strictly increasing, dominates in the sense of Definition 2.1 and we get a contradiction with the -Pareto optimality of . Hence, and then, by symmetry, . In this case, the allocation has the same total endowment as and . If then, in view of the strict concavity of the utility functions, dominates both and , contradicting their -Pareto optimality.

To verify the existence, we shall use a conditional version of the argument from the proof of Theorem 2.6 in . To facilitate references, we assume hereafter that is integrable, that is, . This extra condition does not restrict any generality as, if necessary, we can replace the reference probability measure with the equivalent measure such that

 dQdP=const11+|γ|.

Note that because is -measurable this change of measure does not affect -Pareto optimality.

For , denote by the family of allocations with total endowments less than or equal to such that

 E[um(βm)|G]≥γm,m=1,…,M.

Since the utility functions are increasing and converge to as and because there is an allocation of satisfying (14), the set

 H≜{η∈L0(G,R)\dvtxB(η)≠∅}

is nonempty. For instance, it contains the random variable

 ˜η≜M∞∑n=1n(1An−1An−1),

where, for

 An≜{ω∈Ω\dvtxE[um(βm+n)|G](ω)≥γm(ω),m=1,…,M}.

Indeed, by construction, is -measurable and, as ,

 E[um(βm+˜η/M)|G]≥γm,m=1,…,M.

Hence, the allocation belongs to .

If , then the set is convex (even with respect to -measurable weights) by the concavity of the utility functions. Moreover, this set is bounded in :

 limz→∞supβ∈B(η)P[|β|≥z]=0.

Indeed, from the properties of utility functions in Assumption 2.1 we deduce that

 x−≜max(0,−x)≤−um(x)u′m(0),x∈R.

Hence, for ,

 E[(βm)−]≤1u′m(0)E[−um(βm)]≤1u′m(0)E[−γm]<∞,

implying that the set is bounded in . The boundedness of in then follows after we recall that

 M∑m=1βm≤Σ+η,β∈B(η).

Observe that if the random variables belong to , then so does their minimum . It follows that there is a decreasing sequence in such that its limit is less than or equal to every element of . Let , . As , the family of all possible convex combinations of is bounded in . By Lemma A1.1 in Delbaen and Schachermayer , we can then choose convex combinations of , , converging almost surely to a random variable . It is clear that

 M∑m=1αm≤Σ+ξ. (18)

Since the utility functions are bounded above and, by the convexity of , , an application of Fatou’s lemma yields

 E[um(αm)|G]≥limsupn→∞E[um(ζmn)|G]≥γm,m=1,…,M. (19)

It follows that . The minimality property of then immediately implies that in (18) and (19) we have, in fact, equalities and that is a -Pareto allocation.

In Section 4, we shall prove a more constructive version of Theorem 2.2, namely, Theorem 4.1, where the cash balances and the Pareto allocations will be given as explicit functions of their predecessors and of the new position .

The main goal of this paper is to extend the definition of the cash balance processes from simple to general predictable strategies . This task has a number of similarities with the construction of a stochastic integral with respect to a semi-martingale. In particular, we are interested in the following questions.

{Question}

For simple strategies that converge to another simple strategy in , that is, such that

 (Qn−Q)∗T≜sup0≤t≤T∣∣Qnt−Qt∣∣→0, (20)

do the corresponding cash balance processes converge in as well:

 (Xn−X)∗T→0?
{Question}

For every sequence of simple strategies converging in to a predictable process , does the sequence of their cash balance processes converge to a predictable process in ?

Naturally, when we have an affirmative answer to Question 2.2, the process should be called the cash balance process for the strategy . Note that a predictable process can be approximated by simple processes as in (20) if and only if it has LCRL (left-continuous with right limits) trajectories.

The construction of cash balance processes and processes of Pareto allocations for general strategies will be accomplished in Section 4, while the answers to Questions 2.2 and 2.2 will be given in Section 5. These results rely on the parameterization of Pareto allocations in Section 3.1 and the properties of sample paths of the stochastic field of aggregate utilities established in  and recalled in Section 3.2.

## 3 Random fields associated with Pareto allocations

Let us collect in this section some notation and results which will allow us to work efficiently with conditional Pareto allocations. We first recall some terminology. For a set a map is called a random field; is continuous, convex, etc., if its sample paths are continuous, convex, etc., for all . A random field is called a stochastic field if, for , , that is, the random variable is -measurable.

### 3.1 Parameterization of Pareto allocations

We begin by recalling the results and notation from  concerning the classical parameterization of Pareto allocations. As usual in the theory of such allocations, a key role is played by the aggregate utility function

 r(v,x)≜supx1+⋯+xM=xM∑m=1vmum(xm),v∈(0,∞)M,x∈R. (21)

We shall rely on the properties of this function stated in Section 3 of . In particular, is continuously differentiable and the upper bound in (21) is attained at the unique vector in determined by either

 vmu′m(ˆxm)=∂r∂x(v,x),m=1,…,M, (22)

or, equivalently,

 um(ˆxm)=∂r∂vm(v,x),m=1,…,M. (23)

Following , we denote by

 A≜(0,∞)M×R×RJ, (24)

the parameter set of Pareto allocations in our economy. An element will often be represented as . Here, is a Pareto weight and and stand for, respectively, a cash amount and a number of stocks owned collectively by the market makers.

According to Lemma 3.2 in , for , the random vector defined by

 vmu′m(πm(a))=∂r∂x(v,Σ(x,q)),m=1,…,M, (25)

forms a Pareto allocation and, conversely, for , every Pareto allocation of the total endowment is given by (25) for some . Moreover, if and only if for some constant and, therefore, (25) defines a one-to-one correspondence between the Pareto allocations with total endowment and the set

 SM≜{w∈(0,1)M\dvtxM∑m=1wm=1},

the interior of the simplex in . Following , we denote by

 π\dvtxA→L0(RM),

the random field of Pareto allocations given by (25). Clearly, the sample paths of this random field are continuous. From the equivalence of (22) and (23), we deduce that the Pareto allocation can be equivalently defined by

 um(πm(a))=∂r∂vm(v,Σ(x,q)),m=1,…,M. (26)

In Corollary 3.2 below, we provide the description of the conditional Pareto allocations in our economy, which is analogous to (25). The proof of this corollary relies on the following general and well-known fact, which is a conditional version of Theorem 3.1 in .

###### Theorem 3.1

Consider the family of market makers with utility functions satisfying Assumption 2.1. Let be a -field and . Then the following statements are equivalent: {longlist}[2.]

The allocation is -Pareto optimal.

Integrability condition (8) holds and there is a -measurable random variable with values in such that

 λmu′m(αm)=∂r∂x(λ,Σ),m=1,…,M, (27)

where and the function is defined in (21). Moreover, such a random variable is defined uniquely in .

{pf}

: It is enough to show that

 u′m(αm)u′1(α1)∈L0(G,(0,∞)),m=1,…,M. (28)

Indeed, in this case, define

 λm≜1/u′m(αm)∑Mk=11/u′k(αk),m=1,…,M,

and observe that, as are strictly decreasing functions, is the only allocation of such that

 λmu′m(αm)=λ1u′1(α1),m=1,…,M.

However, in view of (22), an allocation with such property is provided by (27).

Clearly, every obeying (27) also satisfies the equality above and, hence, is defined uniquely.

Suppose (28) fails to hold for some index , for example, for . Then we can find a random variable such that

 |ξ|≤1,(u′1(α1−1)+u′2(α2−1))|ξ|∈L1(R), (29)

and the set

 A≜{ω∈Ω\dvtxE[u′1(α1)ξ|G](ω)<0

has positive probability. For instance, we can take

 ξ≜ζ−˜E[ζ|G]1+u′1(α1−1)+u′2(α2−1),

where

 ζ≜u′2(α2)u′1(α1)+u′2(α2)

and is the expectation under the probability measure with the density

 d˜PdP=constu′1(α1)+u′2(α2)1+u′1(α1−1)+u′2(α2−1).

Indeed, in this case, (29) holds easily, while, as direct computations show

 A={ω∈Ω\dvtx˜E[(ζ−˜E[ζ|G])2|G](ω)>0}

and because is not -measurable.

From the continuity of the first derivatives of the utility functions, we deduce the existence of such that the set

 B≜{ω∈Ω\dvtxE[u′1(α1−εξ)ξ|G](ω)<0

also has positive probability. Denoting and observing that, by the concavity of utility functions,

 u1(α1) ≤ u1(α1−η)+u′1(α1−η)η, u2(α2) ≤ u2(α2+η)−u′2(α2+η)η,

we obtain that the allocation

 β1=α1−η,β2=α2+η,βm=αm,m=3,…,M,

satisfies (9), (10) and (11), thus contradicting the -Pareto optimality of .

: For every allocation with the same total endowment as , we have

 M∑m=1λmum(βm)≤r(λ,Σ)=M∑m=1λmum(αm), (30)

where the last equality is equivalent to (27) in view of (22). Granted integrability as in (8), this clearly implies the -Pareto optimality of .

From Theorem 3.1 and the definition of the random field in (25), we obtain the following corollary.

###### Corollary 3.2

Let Assumptions 2.1 and 2.1 hold and consider a -field and random variables and .

Then for every the random vector forms a -Pareto allocation. Conversely, every -Pareto allocation of the total endowment is given by for some .

{pf}

The only delicate point is to show that the allocation

 αm≜πm(λ,ξ,θ),m=1,…,M,

satisfies the integrability condition (8). Lemma 2.1 implies the existence of an allocation of satisfying (14). The result now follows from inequality (30) which holds true by the properties of .

### 3.2 Stochastic field of aggregate utilities and its conjugate

A key role in the construction of the general investment strategies will be played by the stochastic field of aggregate utilities and its saddle conjugate stochastic field given by

 Ft(a) ≜ E[r(v,Σ(x,q))|Ft],a=(v,x,q)∈A, (31) Gt(b) ≜ supv∈(0,∞)Minfx∈R[⟨v,u⟩+xy−Ft(v,x,q)], (32) \eqntextb=(u,y,q)∈B, (33)

where , the aggregate utility function is given by (21), the parameter set is defined in (24), and

 B≜(−∞,0)M×(0,∞)×RJ.

These stochastic fields are studied in . For the convenience of future references, we recall below some of their properties.

First, we need to introduce some notation. For a nonnegative integer and an open subset of denote by the Fréchet space of -times continuously differentiable maps with the topology generated by the semi-norms

 (34)

Here, is a compact subset of , is a multi-index of nonnegative integers, , and

 Dk≜∂|k|∂xk11⋯∂xkdd. (35)

In particular, for , is the identity operator and .

For a metric space we denote by the space of RCLL (right-continuous with left limits) maps of to .

Suppose now that Assumptions 2.1 and 2.1 hold. Note that in  instead of Assumption 2.1 we used the equivalent condition:

 E[r(v,Σ(x,q))]>−∞,(v,x,q)∈A; (36)

see Lemma 3.2 in  for the proof of equivalence. Theorem 4.1 and Corollary 4.3 in  describe in detail the properties of the sample paths of the stochastic fields and . In particular, these sample paths belong to and for every , , and with we have the invertibility relations

 w = ∂Gt∂u(∂Ft∂v(a,t),1,q)/(M∑m=1∂Gt∂um(∂Ft∂v(a),1,q)), (37) x = Gt(∂Ft∂v(a),1,q), (38)