# The Quasi-Sure Limit of Convex Combinations of Nonnegative Measurable Functions

Authors: Yulian Fan
Journal: Journal of Function Spaces
 (2019)

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## Abstract

This study shows that, for a sequence of nonnegative valued measurable functions, a sequence of convex combinations converges to a nonnegative function in the quasi-sure sense. This can be used to prove some existence results in multiprobabilities models, and an example application in finance is discussed herein.


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## Body

## 1. Introduction

Stochastic models are widely used in practical applications and most are built on a probability space. However, in practice, situations with many possible probabilities occur. A few examples include the uncertain drift or uncertain volatility cases in economic models. These multiprobabilities are called ambiguity, model uncertainty, or Knight uncertainty. Reference [1] emphasized the significant distinction between risk and ambiguity and [2] showed that the distinction between risk and ambiguity is behaviorally significant. To study economic problems while considering ambiguity, mathematical models must be built on a multiprobability space.Reference [3] first developed the theory of nonlinear g-expectation which nontrivially generalizes the classical linear expectation from a probability space to a space with a set of uniformly absolute continuous probabilities. Considering this theory, [4] studied the stochastic differential recursive utility with drift ambiguity. However, many economic and financial problems involve significant volatility uncertainty, which is characterized by a family of nondominated probability measures. Motivated by volatility uncertainty in statistics, risk measures, and super-hedging in finance, [5] introduced a nonlinear expectation, called the G-expectation, which can be regarded as the upper expectation of a specific family of nondominated probability measures. Subsequently, [6, 7] introduced a new type of “G-Brownian motion” and presented the related calculus of Itô’s type. Reference [8] developed a representation of the G-expectation and G-Brownian motion. Reference [9] studied the martingale representation theorem for the G-expectation. Reference [10] determined the properties of hitting times for the G-martingales. References [11, 12] studied the G-BSDEs in G-expectation space.This paper provides a convergence result in the multiprobability space. In such spaces, under the upper expectation ofE[·] defined in (4), the corresponding Fatou’s lemma and dominated convergence theorem no longer hold. Therefore, the convergence result reported in this paper will be useful for future studies.This paper is organized as follows. In Section2, we prove that, for a sequence of nonnegative measurable functions, there is a sequence of convex combinations which converges to a nonnegative function in the quasi-sure sense. In Section 3, we use the results of Section 2 to prove an existence result in a multiprobabilities model.


## 2. Main Results

LetΩ be a complete separable metric space, B(Ω) the Borel σ-algebra of Ω, and M the collection of all probability measures on (Ω,B(Ω)). Let L0(Ω) be the space of all B(Ω)-measurable real functions.Consider a given subsetP⊆M.Denote(1)cA≔supP∈P⁡PA,A∈BΩ.Then,c(·) is a Choquet capacity; see [13–15]. A set A is called polar if c(A)=0, and we say a property holds “quasi-surely”(q.s.) if it holds outside a polar set. Let Xn,X∈L0(Ω), n∈N. The sequence Xn is said to converge in capacity c to X, denoted by Xn⟶cX, n→∞, if (2)limn→∞⁡cXn-X≥ϵ=0,∀ϵ>0.The sequence Xn is said to mutually converge in capacity c, if (3)limm,n→∞⁡cXm-Xn≥ϵ=0,∀ϵ>0.The upper expectationE[·] of P is defined as follows (see [16] ): for each X∈L0(Ω) such that EP[X] exists for each P∈P,(4)EX≔supP∈P⁡EPX.Lemma 1.
LetXn,X∈L0(Ω), n∈N. Then,
(i)Xn→X,n→∞, q.s., if (5)c⋂n=1∞⋃ν=1∞Xn+ν-X≥ϵ=0,∀ϵ>0.
(ii)Xn is Cauchy q.s., if (6)c⋂n=1∞⋃ν=1∞Xn+ν-Xn≥ϵ=0,∀ϵ>0.Proof.
(i) We choose{ϵk}, such that ϵk>0 and ϵk→0, k→∞. (7)cXn→Xc=c⋃k=1∞⋂n=1∞⋃ν=1∞Xn+ν-X≥ϵk≤∑k=1∞c⋂n=1∞⋃ν=1∞Xn+ν-X≥ϵk=0.Thus, we obtain Xn→X q.s.
(ii) Similar to proof (i), we choose{ϵk}, such that ϵk>0 and ϵk→0, k→∞. (8)cXn-Xm→0c=c⋃k=1∞⋂n=1∞⋃ν=1∞Xn+ν-Xn≥ϵk≤∑k=1∞c⋂n=1∞⋃ν=1∞Xn+ν-Xn≥ϵk=0.Therefore, Xn is Cauchy q.s.Lemma 2.
LetXn,X∈L0(Ω), n∈N, {ϵn} be a positive number sequence, and ϵn→0.
(i) If(9)∑n=1∞cXn-X≥ϵn<∞,then Xn→X q.s.
(ii) If(10)∑n=1∞cXn+ν-Xn≥ϵn<∞,then Xn is Cauchy q.s.Proof.
(i) For anyϵ>0, there exists n0 such that, for all n≥n0, ϵn<ϵ. Then, we have (11)c⋂n=1∞⋃ν=1∞Xn+ν-X≥ϵ≤∑n=k∞cXn-X≥ϵ≤∑n=k∞cXn-X≥ϵn.Letting k→∞ on the right side of the above inequality yields (12)c⋂n=1∞⋃ν=1∞Xn+ν-X≥ϵ=0,and by Lemma 1, we have Xn→X q.s.
(ii) Similar to proof (i).Lemma 3.
LetXn∈L0(Ω), n∈N.
(i) IfXn⟶cX, then there exists a subsequence {Xnk} of {Xn} such that Xnk→X q.s.
(ii) IfXn,n∈N, mutually converges in capacity c, then there exist a subsequence {Xnk} of {Xn} and X∈L0(Ω) such that Xnk→X q.s.Proof.
(i) SinceXn⟶cX, for each ϵk=1/2k, k∈N, there exists nk∈N, such that (13)cXn-X≥12k<12k,n≥nk.We can choose nk↑∞,k→∞. Then, we have (14)∑k=1∞cXnk-X≥12k<∞.By Lemma 2, Xnk→X,k→∞, q.s.
(ii) Similar to proof (i), there exists a subsequence{Xnk}, which is Cauchy q.s. Then, there exists X∈L0(Ω) such that Xnk→X q.s.
IfL0(Ω) is the space of all B(Ω)-measurable functions in R∪{+∞}, the sequence Xn mutually converges in capacity c, if (15)limm,n→∞⁡cXm-Xn≥ϵandmin⁡Xn,Xm≤ϵ-1=0,∀ϵ>0.The results of the above lemmas still hold.Letconv(Xn,Xn+1,…) denote the convex combination of Xn,Xn+1,…. Using a similar argument as in Lemma A1.1 of [17], we obtain the following.Theorem 4.
Let(Xn)n≥1 be a sequence of [0,∞[ valued measurable functions. There exists a sequence gn∈conv(Xn,Xn+1,…) such that (gn)n≥1 converges to a [0,∞] valued function g q.s.Proof.
Letu:R+∪{0}∪{+∞}→[0,1] be defined as u(x)=e-x. Define sn as(16)sn=inf⁡ug∣g∈convXn,Xn+1,…and choose gn∈conv(Xn,Xn+1,…) so that (17)ugn≤sn+ϵn,where 0≤ϵn→0.
It is clear thatsn is a bounded increasing sequence, so there exists s0 such that limn→∞u(gn)=s0.
On the compact (metrisable) space[0,∞], (xn)n≥1 is Cauchy if and only if for each α>0 there is n0 so that for all n,m≥n0 we have |xn-xm|≤α or min⁡(xn,xm)≥α-1. From the properties of u(x), we have that for α>0 there is β>0 so that |x-y|>α and min⁡(x,y)≤α-1 imply 1/2u(x)+1/2u(y)>u(x+y/2)+β.
For a givenα>0, we can take β as above and, with the convexity of u, we obtain (18)β1gn-gm>αandmin⁡gn,gm≤α-1+ugn+gm2≤12ugn+12ugm.Then, (19)-β1gn-gm>αandmin⁡gn,gm≤α-1≥ugn+gm2-12ugn-12ugm.Without loss of generality, we can set m≥n. By (16), u(gn+gm/2)≥sn. Taking the expectation about each P∈P and letting n→∞, we obtain (20)-βlimn→∞⁡Pgn-gm>αandmin⁡gn,gm≤α-1≥limn→∞⁡EPugn+gm2-12ugn-12ugm=EPlimn→∞⁡ugn+gm2-12ugn-12ugm≥EPlimn→∞⁡sn-12sn-12sm-ϵn-ϵm=0.Thus, we have (21)limn→∞⁡Pgn-gm>αandmin⁡gn,gm≤α-1=0.
Therefore,(22)limn→∞⁡cgn-gm>αandmin⁡gn,gm≤α-1≥supP∈P⁡limn→∞⁡Pgn-gm>αandmin⁡gn,gm≤α-1=0.That is, gn mutually converges in capacity c, and, by Lemma 3, there exist a subsequence (gnk)k≥1 and a [0,∞] valued function g such that gnk→g, q.s.


## 3. Application of Theorem4

We set, forp>0, (23)Lp≔X∈L0Ω:EXp<∞,(24)N≔X∈L0Ω:X=0,q.s.,and denote Lp=Lp/N.Consider the following optimization problem in finance:(25)infXT∈Lξ⁡Elξ-XT,undertheconstraintEXT≤x,where XT is the terminal wealth of the hedging portfolio at terminal time T, Lξ={X:0≤X≤ξ}, ξ is the nonnegative contingent claim which the investor attempts to hedge, E[|ξ|2]<∞, l is the loss function which is an increasing convex function defined on [0,∞), and x is the constraint regarding the initial wealth. The corresponding hedging problem in the single probability model was introduced in [18] and is referred to as efficient hedging.We use the result of the Theorem4 to prove the existence of the solution of problem (25).Theorem 5.
There is a solutionX~T∈Lξ to problem (25).Proof.
LetLξx consist of elements of Lξ that satisfy E[XT]≤x and let (XTn) be a minimizing sequence for (25) in Lξx. By Theorem 4, there exists a sequence X~Tn belonging to conv{XTn,XTn+1,…} such that X~Tn→X~T, q.s. Since XTn∈Lξx, i.e., 0≤XTn≤ξ and E[XTn]≤x, we obtain 0≤X~Tn,X~T≤ξ, E[X~Tn]≤x, and (26)EX~T=supP∈P⁡EPlimn→∞⁡X~Tn=supP∈P⁡limn→∞⁡EPX~Tn≤supP∈P⁡limn→∞⁡EX~Tn=limn→∞⁡EX~Tn≤x,where the second equality sign is a result of the dominated convergence theorem under probability P. Therefore, X~T∈Lξx.
Similarly, sinceX~Tn→X~T, q.s., we have (27)Elξ-X~T=supP∈P⁡EPlξ-X~T=supP∈P⁡EPlimn→∞⁡lξ-X~Tn=supP∈P⁡limn→∞⁡EPlξ-X~Tn≤supP∈P⁡limn→∞⁡Elξ-X~Tn=limn→∞⁡Elξ-X~Tn
By the convexity ofE[·] and function l and X~Tn belonging to conv{XTn,XTn+1,…}, we can conclude that E[l(ξ-X~Tn)]] is not larger than the corresponding convex combination of E[l(ξ-XTm)]], m≥n. And because (XTn) is a minimizing sequence for (25) in Lξx, (28)limn→∞⁡Elξ-XTn=infXT∈Lξx⁡Elξ-XT.Then we have (29)Elξ-X~T≤infXT∈Lξx⁡Elξ-XT.
So(30)Elξ-X~T=infXT∈Lξx⁡Elξ-XT.



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