Stochastic programs without duality gaps for objectives without a lower bound

# Stochastic programs without duality gaps for objectives without a lower bound

## Abstract

This paper studies parameterized stochastic optimization problems in finite discrete time that arise in many applications in operations research and mathematical finance. We prove the existence of solutions and the absence of a duality gap under conditions that relax the boundedness assumption made by Pennanen and Perkkiö in [Stochastic programs without duality gaps, Math. Program., 136(1):91–110,2012]. We apply the result to a utility maximization problem with an unbounded utility.

## 1 Introduction

Let be a complete probability space with a filtration of complete sub sigma-algebras of and consider the parametric dynamic stochastic optimization problem

 minimizeEf(x,u):=∫f(x(ω),u(ω),ω)dP(ω)over x∈N, ($P_{u}$)

where, for given integers and

 N={(xt)Tt=0|xt∈L0(Ω,Ft,P;Rnt)},

is the parameter and is an extended real-valued -measurable function, where . Here and in what follows, we define the expectation of a measurable function as unless the positive part is integrable2. The function is thus well-defined extended real-valued function on . We will assume throughout that the function is proper, lower semicontinuous and convex for every .

It was shown in [9] that, when applied to (($P_{u}$)), the conjugate duality framework of Rockafellar [17] allows for a unified treatment of many well-known duality frameworks in operations research and mathematical finance. In that context, the absence of a duality gap is equivalent to the closedness of the optimal value function

 φ(u)=infx∈NEf(x,u)

over an appropriate space of measurable functions . Pennanen and Perkkiö [14] gave a simple algebraic condition on the integrand that guarantees that the optimum in (($P_{u}$)) is always attained and is closed; see Section 2 below. The condition provides a far reaching generalization of the classical no-arbitrage condition in financial mathematics but it also assumes that is bounded from below. In the financial context, the lower bound excludes, e.g., portfolio optimization problems with utility functions that are not bounded from above. That such a bound is superfluous is suggested e.g. by the results of Rásonyi and Stettner [15] where the existence of solutions to portfolio optimization in the classical perfectly liquid market was obtained for more general utility functions.

This article relaxes the boundedness assumption on . A key result for this is Lemma 2 which, in turn, is based on local martingale techniques that are well-known in mathematical finance. We also prove an expression for the recession function of the optimal value function in terms of . This is of interest when, e.g., one studies robust no arbitrage and robust no scalable arbitrage properties and the related dominating markets; see [6, 13].

In Section 3 we apply the main results to an optimal investment problem in liquid markets. Under the assumptions that prices are bounded from below and that the initial prices are bounded, we recover an existence result by Rásonyi and Stettner [15, Theorem 6.2] who assumed a well-known asymptotic elasticity condition of the utility function. Moreover, here we obtain the closedness of the associated value function defined on an appropriate space of future liabilites that is important in valuation of contingent claims; see [12]. During the recent years, market models with proportional transaction costs or other nonlinear illiquidity effects have received an increasing interest [2, 5, 13]. Extensions of our results to such models will be analyzed in a forthcoming article.

## 2 Closedness of the value function

In this section we will first establish the closedness of the value function associated with (($P_{u}$)) in the space of measurable functions with respect to the convergence in measure. Using this, we will then establish the closedness on locally convex subspaces of . Recall that a function is closed if it is lower semicontinuous and either proper or a constant. A function is proper if it never takes the value and it is finite at some point.

An extended real-valued function on , for a complete probability space , is a normal integrand if is jointly measurable and is lower semicontinuous for all ; see [19, Corollary 14.34]. Thus, we may say that is an -measurable proper convex normal integrand on .

In all of the results, the statements concerning the recession functions are new. For a normal integrand, is defined -wise as the recession function of , i.e.

 f∞(x,u,ω)=supα>0f(¯x+αx,¯u+αu,ω)−f(¯x,¯u)α,

which is independent of the choice . By [16, Theorem 8.5], the supremum equals the limit as .

###### Theorem 1.

Assume that there exists such that

 f(x,u,ω)≥m(ω)

for all and that

 L={x∈N∣f∞(x,0)≤0}

is a linear space. Then

 φ(u)=infx∈NEf(x,u)

is closed and proper in and the infimum is attained for every . Moreover,

 φ∞(u)=infx∈NEf∞(x,u).
###### Proof.

Theorem 2 of [14] gives closedness of with respect to a locally convex topological space but the main argument of the proof establishes closedness with respect to the convergence in measure provided that has an integrable lower bound.

Let and be such that . Such exists by [14, Theorem 2]. We have that

 φ∞(u) =supα>0φ(¯u+αu)−φ(¯u)α=supα>0infx∈NEfα(x,u),

where

 fα(x,u)=f(¯x+αx,¯u+αu)−f(¯x,¯u)α.

We have

 φ∞(u)≤infx∈Nsupα>0Efα(x,u)≤infx∈NE[supα>0fα(x,u)]=infx∈XEf∞(x,u).

To prove the converse, let . For every positive integer , there is an with . The functions are non-decreasing in , so and we may proceed as in the proof of [14, Theorem 2] (where we may assume that for every by [14, Lemma 2]) to obtain a sequence of convex combinations such that almost surely. By Fatou’s Lemma,

 Ef∞(¯x,u)≤a,

which completes the proof. ∎

When extending the theorem above to objectives that do not have a uniform lower bound, a key role is played by the following lemma, where

 N⊥={v∈L1(Ω,F,Rn)|E(x⋅v)=0 ∀x∈N∞}

is the annihilator of .

###### Lemma 2.

Let and . If , then .

###### Proof.

Any can be represented as where and is the martingale defined as and . For , we have for all , so , where is a local martingale defined by

 mit=t∑t′=0xi⋅Δξit′.

Thus for . Since , we have that is a martingale ([4, Theorem 2]) and thus . ∎

The following additivity property of the extended real-valued expectation will often be used without a mention.

###### Lemma 3.

Let and be extended real-valued measurable functions. If either or , then

 E(ϕ1+ϕ2)=Eϕ1+Eϕ2.

The main contribution of this paper is contained in the following result which relaxes the lower bound of with respect to .

###### Theorem 4.

Assume that there exist and such that

 f(x,u,ω)≥x⋅v(ω)+λ[x⋅v(ω)]++β(ω)

for all and that

 L={x∈N∣f∞(x,0)≤0}

is a linear space. Then

 φ(u)=infx∈NEf(x,u)

is closed and proper in and the infimum is attained for every . Moreover,

 φ∞(u)=infx∈NEf∞(x,u).
###### Proof.

Note first that . Indeed, the lower bound on implies , so if belongs either to or , then and thus, , by Lemma 2. Applying Theorem 1 to the normal integrand we see that

 φv(u)=infx∈NEfv(x,u)

is closed and proper, the infimum is attained for every and that

 φ∞v(u)=infx∈NEf∞v(x,u).

If either or , the lower bound implies so that and , by Lemma 2. In both cases, Lemma 3 gives

 Efv(x,u)=Ef(x,u)−E(x⋅v)=Ef(x,u)

so that . Similarly, so that . ∎

Theorem 4 is not valid if, in the lower bound for , we allow to be zero.

###### Example 1.

Let and let be such that there exist with and nonzero independent of with . We define

 f(x,u,ω)=∥x−α(ω)∥+x⋅v(ω),

where so that and . The reader may verify that

 infx∈NEf(x,u)=0

and the only nontrivial candidate for is , but .

The rest of this section is concerned with closedness of the value function on locally convex subspaces of . We will assume from now on that the parameter belongs to a decomposable space which is in separating duality with another decomposable space under the bilinear form

 ⟨u,y⟩=E(u⋅y).

Recall that is decomposable if

 \mathbbm1Au+\mathbbm1Ω∖Au′∈U

whenever , and ; see e.g. [18].

The families of closed convex sets coincide in the weak and in the Mackey topologies [21, p. 132], so we may say that is closed in whenever it is so with respect to either of the topologies. The closed convex function

 φ∗(y)=supu∈U{⟨u,y⟩−φ(u)}

is called the conjugate of . When is closed, it has the dual representation

 φ(u)=supy∈Y{⟨u,y⟩−φ∗(y)};

see [17, Theorem 5]. Dual representations are behind many fundamental results in mathematical finance as in the classical formula where superhedging prices of contingent claims in liquid markets are given in terms of martingale measures; for this and some recent developments in more general market models, see [10] and [11].

Using traditional topological arguments on decomposable spaces (see e.g. Rockafellar [18] or Ioffe [3]), we may relax in Theorem 5 the lower bound on with respect to as well. The following theorem generalizes [14, Theorem 2] by relaxing the uniform lower boundedness of with respect to .

###### Theorem 5.

Assume that there exist , and a convex normal integrand on such that is -continuous, is finite on ,

 f(x,u,ω)≥x⋅v(ω)+λ[x⋅v(ω)]+−g(u,ω)

for all and such that

 L={x∈N∣f∞(x,0)≤0}

is a linear space. Then

 φ(u)=infx∈NEf(x,u)

is closed and proper in and the infimum is attained for every . Moreover,

 φ∞(u)=infx∈NEf∞(x,u).
###### Proof.

Applying Theorem 4 to the normal integrand , we get that

 φg(u)=infx∈NEfg(x,u)

is proper and closed on , that the infimum is attained for every and that

 φ∞g(u)=infx∈NEf∞g(x,u).

Closedness in implies closedness in the relative topology of which, by [14, Lemma 6], implies that is closed in . By Lemma 3, for all and thus

 φg=φ+Eg

on . The -continuity of implies the -closedness of .

As to the recession function, [16, Theorem 9.3], so

 φ∞g(u)=infx∈NEf∞(x,u)+Eg∞(u).

Similarly, closedness of and imply

 φ∞g=φ∞+(Eg)∞,

where, by the monotone convergence theorem, . ∎

The following lemma gives a sufficient condition for the first hypothesis in Theorem 5. The characterization involves the conjugate normal integrand of defined -wise as follows

 f∗(v,y,ω)=sup{x⋅v+u⋅y−f(x,u,ω)}.

By [19, Theorem 14.50], is indeed a normal integrand.

###### Lemma 6.

Assume that there exists and such that the function

 γ(v)=infy∈YEf∗(v,y)

is finite at and . Then

 f(x,u,ω)≥x⋅v(ω)+λ[x⋅v(ω)]+−g(u,ω)

for all , where for some and .

###### Proof.

The assumption means that there exist and such that

 f∗(λiv,yi)≤βi

where and . Equivalently,

 f(x,u,ω)≥λix⋅v(ω)+u⋅yi(ω)−βi(ω)i=1,2

for all , which implies

 f(x,u,ω)≥maxi{λix⋅v(ω)}+miniu⋅yi(ω)−miniβi(ω)

for all . ∎

## 3 Application to mathematical finance

We will consider optimal investment on a financial market with a finite set of assets from the point of view of an agent who has a financial liability described by a payment to be made at the terminal time. As is usual, we express the terminal wealth as a stochastic integral with respect to the adapted price process so that the problem can be written as

 minimizeEV(u−T−1∑t=0xt⋅Δst+1)overx∈N0, (ALM)

where , and is a nondecreasing nonconstant convex function with . The objective of the agent is thus to find a trading strategy that hedges against the liability as well as possible as measured by expected “disutility” at terminal time. The change of signs transforms the problem into a usual maximization of expected terminal utility. There is vast literature on ((ALM)) and on its extensions to more sophisticated models of optimal investment; see the references in [12].

The following theorem gives conditions for the closedness in of the value function of ((ALM)) and thus, the validity of the dual representation

 φ(u)=supy∈Y{⟨u,y⟩−φ∗(y)}.

Recall that satisfies the no-arbitrage condition if

 {T−1∑t=0xt⋅Δst+1∣∣ ∣∣T−1∑t=0xt⋅Δst+1≥0, x∈N}={0}. (NA)
###### Theorem 7.

Assume that we have ((NA)) and that there exists a martingale measure of such that for two different . If , then the value function of ((ALM)) is closed in and ((ALM)) has a solution for all .

###### Proof.

We fit ((ALM)) in the general model (($P_{u}$)) with3

 f(x,u,ω)=V(u−T−1∑t=0xt⋅Δst(ω))+δ0(xT).

We have

 f∞(x,0,ω)=V∞(−T−1∑t=0xt⋅Δst+1)+δ0(xT).

Since is nonconstant, we get for and for , so the linearity condition in Theorem 5 reduces to ((NA)). In order to apply Lemma 6, we calculate

 f∗(v,y,ω) =supx∈Rn,u∈R{x⋅v+uy−V(u−T−1∑t=0xt⋅Δst+1(ω))−δ0(xT)} =V∗(y)+supx∈Rn{T−1∑t=0xt⋅(yΔst+1(ω)+vt)} =V∗(y)+T−1∑t=0δ0(yΔst+1(ω)+vt).

We choose and so that

 Ef∗(λv,λy) =EV∗(λdQdP).

Here is finite for two different by assumption so that, by Lemma 6, we may apply Theorem 5. ∎

###### Remark 1.

The proof of Theorem 7 also gives a formula for the recession function of the value function associated with ((ALM)). That is, since the value function is closed, its recession function is closed as well which, by the recession formula in Theorem 5, is equivalent with the fact that the set

 C={u∈U|∃x∈N0: T−1∑t=0xt⋅Δst+1≥u}

of claims that can be superhedged without a cost is closed in . However, this result follows already from [14, Theorem 2].

###### Remark 2.

Under the assumptions of Theorem 7, if one is merely interested in the existence of solutions of ((ALM)), we point out the following. We may define

 f(x,u,ω)=V(c(ω)−T−1∑t=0xt⋅Δst+1(ω))+δ0(xT)

so that is independent of and the Fenchel inequality implies that

 f(x,u,ω)≥x⋅v(ω)+c(ω)λdQdP(ω)−V∗(λdQdP(ω))

for all , where is defined by . Thus, by Theorem 4, ((ALM)) has a solution for whenever is -integrable.

###### Remark 3.

In general, the price process does not admit a martingale measure for which the integrability condition in Theorem 7 is satisfied. For example, consider a one-period market model with a trivial -algebra , , , and a disutility function specified by

 V′(u)=∞∑n=1[\mathbbm1(−n−1,−n](u)(1+1n2)+\mathbbm1(n,n+1](u)(3−1n2)].

This is Example 7.3 in [15] where it has been shown that the optimal value of ((ALM)) for is finite but optimal solutions do not exist. The interested reader may verify that the unique martingale measure is given by and that whenever so that for every .

The “two--condition” in the theorem is close in spirit to [1, Assumption 4.2] since it implies a similar -condition for all sufficiently small [1, Corollary 4.4]. In the cited article the authors work in a continuous time setting, here two different suffice.

The following corollary says that the two--condition is implied by a simpler integrability condition for utility functions that satisfy either of the well-known asymptotic elasticity conditions

 V∗(λy) ≤CV∗t(y,ω)∀y∈[0,¯y] for % some C<+∞, ¯y>0, λ∈(0,1), (1) V∗(λy) ≤CV∗t(y,ω)∀y≥¯y for some% C<+∞, ¯y>0, λ>1. (2)

These conditions together with their equivalent formulations were introduced in [8] and [20], respectively.

###### Corollary 8.

Assume that we have ((NA)), satisfies (1) or (2), and that there exists a martingale measure of such that for some . If , then the value function of ((ALM)) is closed in and ((ALM)) has a solution for all .

###### Proof.

We use the notation from the proof of Theorem 7. For simplicity, we assume that . If we have (1), then

 EV∗(λ′dQdP) =E\mathbbm1{dQdP≤¯y}V∗(λ′dQdP)+E\mathbbm1{dQdP>¯y}V∗(λ′dQdP) ≤E\mathbbm1{dQdP≤¯y}V∗(λ′dQdP)+E\mathbbm1{dQdP>¯y}max{V∗(dQdP),V∗(λ′¯y)} ≤E\mathbbm1{dQdP≤¯y}CV∗(dQdP)+E\mathbbm1{dQdP>¯y}max{V∗(dQdP),CV∗(¯y)},

so is finite for some . Similary, if we have (2), then is finite for some . ∎

The next theorem gives conditions for the existence of a martingale measure used in Corollary 8. The third condition in the theorem means that the optimal value of ((ALM)) is finite for some positive initial endowment. We say that the price process is bounded from below if there is a constant such that -almost surely for all and . In the proof, much like in that of [8, Proposition 3.2], we approximate the disutility function by disutility functions that are bounded from below.

###### Theorem 9.

Assume that we have ((NA)), satisfies (1), the optimal value of ((ALM)) is finite for some , where is a constant, the price process is bounded from below, and that . Then there exists a martingale measure of with for some .

###### Proof.

For positive integers , we define functions on by

 ~φn(u)=infx∈N∞EVn(βu−T−1∑t=0xt⋅Δst+1),

where and . For , we get

 ~φ∗n(y) =supx∈N∞supu∈UE[uy−Vn(βu−T−1∑t=0xt⋅Δst+1)] =supx∈N∞E[y/βT−1∑t=0xt⋅Δst+1]+EV∗n(y/β) =δQ(y/β)+EV∗n(y/β),

where is the set of positive multiples of martingale measure densities of . Here the second equality follows from the interchange rule [19, Theorem 14.60], since, for every , there exists such that .

We have that , where

 φ(u) =infx∈NEV(βu−T−1∑t=0xt⋅Δst+1), φn(u) =infx∈NEVn(βu−T−1∑t=0xt⋅Δst+1).

We get as in the proof of Theorem 7 that are closed.4 Therefore and, by the above expression for , there exist such that

 limnE[cyn/β−V∗n(yn/β)+n−1]≥limnφn(c/β)≥φ(c/β)>−∞.

Since are nonnegative and is negative, are bounded in . Thus Komlós’ theorem (see e.g. [7]) implies the existence of a sequence of convex combinations such that converge to some -almost surely. By Fatou’s lemma and convexity,

 E[c¯y/β−V∗(¯y/β)]≥limsupnE[c~yn/β−V∗n(~yn/β)]≥φ(c/β)>−∞. (3)

Since satisfies (1), we get, as in the proof of Corollary 8, that is finite. Since increase to , there is, for every , an such that

 EV∗n(2−ν¯y/β)≥EV∗(2−ν¯y/β)−2−ν (4)

for all . Each of the functions is bounded, so, by a diagonalization argument, we may assume that

 EWν(2−ν~yn/β)≤EWν(2−ν¯y/β)+2−ν

for all . Since and since we have (4), we get that

 EWν(2−ν~yn/β)≤2−ν+1 (5)

for all . We define

 ~y=∞∑ν=02−ν~yn(ν).

Firstly, since increase to , we see from (3) that are bounded in and thus , where the first inequality follows from the lower semicontinuity and Fatou’s lemma. Secondly, it is not difficult to verify that are decreasing functions so that, by (5), . Thus

 EV∗(~y/β)=EV∗n(0)(~y/β)+∞∑ν=0EWν(~y/β)<∞.

Since the price process is bounded from below, we get from the monotone convergence theorem that

 E[~yβ\mathbbm1Ast+1]=∞∑ν=12−νE[~yνβ\mathbbm1Ast+1]=∞∑ν=12−νE[~yνβ\mathbbm1Ast]=E[~yβ\mathbbm1Ast]

for every and . Finally, by monotone convergence again, , so is a positive multiple of a martingale measure density.

Under the mild assumptions that the price process is bounded from below and that , we recover the following result on the existence of solutions by Rásonyi and Stettner in [15, Theorem 2.7].

###### Corollary 10.

Assume that we have ((NA)), there exists and such that

 V(λu)≥λγV(u)∀u≤~u, λ≥1,

the functions and

 Vt(u) =ess infxt∈L0(Ft)E[Vt+1(u+xt⋅Δst+1)∣Ft]

on are well-defined and proper, and that for all constants . If the price process is bounded from below and , then ((ALM)) has a solution for every .

###### Proof.

By [8, Corollary 6.1], (1) and the given growth condition for are equivalent. Thus, by Theorem 9, Corollary 8 and Remark 2, it suffices to show that ((ALM)) is finite for , where is a constant.

Let be such that . The proofs of Proposition 4.2 and Proposition 4.4 in [15] show that, outside an evanescent set,5 each

 ~Vt(u,ω)=E[Vt+1(