Research Article G-Doob-Meyer Decomposition and Its ... · e Doob decomposition theorem was proved by and is named for Doob [ ]. e analogous theorem in the contin-uous time case is

Research ArticleG-Doob-Meyer Decomposition and Its Applications inBid-Ask Pricing for Derivatives under Knightian Uncertainty

Wei Chen

School of Economics, Shandong University, Jinan 250100, China

Correspondence should be addressed to Wei Chen; [email protected]

Received 20 April 2015; Revised 23 July 2015; Accepted 9 August 2015

Academic Editor: Jafar Biazar

Copyright © 2015 Wei Chen. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The target of this paper is to establish the bid-ask pricing framework for the American contingent claims against risky assets withG-asset price systems on the financial market under Knightian uncertainty. First, we prove G-Dooby-Meyer decomposition forG-supermartingale. Furthermore, we consider bid-ask pricing American contingent claims under Knightian uncertainty, by usingG-Dooby-Meyer decomposition; we construct dynamic superhedge strategies for the optimal stopping problem and prove that thevalue functions of the optimal stopping problems are the bid and ask prices of the American contingent claims under Knightianuncertainty. Finally, we consider a free boundary problem, prove the strong solution existence of the free boundary problem, andderive that the value function of the optimal stopping problem is equivalent to the strong solution to the free boundary problem.

1. Introduction

The earliest and one of the most penetrating analyses on thepricing of the American option is by McKean [1]. There theproblem of pricing the American option is transformed intoa Stefan or free boundary problem. Solving the latter,McKeanwrites the American option price explicitly up to knowing acertain function, the optimal stopping boundary.

Bensoussan [2] presents a rigorous treatment for Amer-ican contingent claims that can be exercised at any timebefore or at maturity. He adapts the Black and Scholes[3] methodology of duplicating the cash flow from such aclaim to this situation by skillfully managing a self-financingportfolio that contains only the basic instruments of themarket, that is, the stocks and the bond, and that entailsno arbitrage opportunities before exercise. Bensoussan showsthat the pricing of such claims is indeed possible and charac-terized the exercise time by means of an appropriate optimalstopping problem. In the study of the latter, Bensoussanemploys the so-called “penalization method,” which forcesrather stringent boundedness and regularity conditions onthe payoff from the contingent claim.

From the theory of optimal stopping, it is well knownthat the value process of the optimal stopping problem canbe characterized as the smallest supermartingale majorant to

the stopping reward. Based on the Doob-Meyer decomposi-tion for the supermartingale, a “martingale” treatment of theoptimal stopping problem is used for handling pricing of theAmerican option by Karatzas [4] and El Karoui and Karatzas[5, 6].

The Doob decomposition theorem was proved by and isnamed for Doob [7]. The analogous theorem in the contin-uous time case is the Doob-Meyer decomposition theoremproved by Meyer in [8, 9]. For the pricing American optionproblem in incomplete market, Kramkov [10] constructs theoptional decomposition of supermartingale with respect toa family of equivalent local martingale measures. He callssuch a representation optional because, in contrast to theDoob-Meyer decomposition, it generally exists only with anadapted (optional) process C. He applies this decomposi-tion to the problem of hedging European and Americanstyle contingent claims in the setting of incomplete securitymarkets. Using the optional decomposition, Frey [11] con-siders construction of superreplication strategies via optimalstopping which is similar to the optimal stopping problemthat arises in the pricing of American-type derivatives on afamily of probability space with equivalent local martingalemeasures.

For the realistic financial market, the asset price in thefuture is uncertain, the probability distribution of the asset

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015, Article ID 910809, 13 pageshttp://dx.doi.org/10.1155/2015/910809

2 Journal of Applied Mathematics

price in the future is unknown, which is called Knightianuncertainty [12]. The probability distribution of the naturestate in the future is unknown; investors have uncertainsubjective belief, which makes their consumption and port-folio choice decisions uncertain and leads the uncertainasset price in the future. Pricing contingent claims againstsuch assets under Knightian uncertainty is an open problem.Peng in [13, 14] constructs G frame work which is ananalysis tool for nonlinear system and is applied in pricingEuropean contingent claims under volatility uncertainty[15, 16].

The target of this paper is to establish the bid-ask pricingframework for the American contingent claims against riskyassets with G-asset price systems (see [17]) on the financialmarket under Knightian uncertainty. Firstly, on sublinearexpectation space, by using potential theory and sublinearexpectation theory we construct G-Doob-Meyer decompo-sition for G-supermartingale, that is, a right continuous G-supermartingale could be decomposed as aG-martingale anda right continuous increasing process and the decompositionis unique. Second, we define bid and ask prices of theAmerican contingent claim against the assets with G-assetprice systems and apply the G-Doob-Meyer decompositionto prove that the bid and ask prices of American contingentclaims under Knightian uncertainty could be described bythe optimal stopping problems. Finally, we present a freeboundary problem, and by using the penalization technique(see [18]) we derive that if there exists strong supersolutionto the free boundary problem, then the strong solution to thefree boundary problem exists. And by using truncation andregularization technique, we prove that the strong solutionto the free boundary problem is the value function ofthe optimal stopping problem which is corresponding withpricing problem of the American contingent claim underKnightian uncertainty.

The rest of this paper is organized as follows. In Section 2,we give preliminaries for the sublinear expectation theory.In Section 3 we prove G-Doob-Meyer decomposition for G-supermartingale. In Section 4, using G-Doob-Meyer decom-position, we construct dynamic superhedge strategies for theoptimal stopping problem and prove that the solution of theoptimal stopping problem is the bid and ask prices of theAmerican contingent claims under Knightian uncertainty.In Section 5, we consider a free boundary problem, provethe strong solution existence of the free boundary problem,and derive that the solution of the optimal stopping problemis equivalent to the strong solution to the free boundaryproblem.

2. Preliminaries

LetΩ be a given set and letH be a linear space of real valuedfunctions defined onΩ containing constants. The spaceH isalso called the space of random variables.

Definition 1. A sublinear expectation 𝐸 is a functional 𝐸 :H → 𝑅 satisfying

(i) monotonicity:

𝐸 [𝑋] ≥ 𝐸 [𝑌] if 𝑋 ≥ 𝑌, (1)

(ii) constant preserving:

𝐸 [𝑐] = 𝑐 for 𝑐 ∈ 𝑅, (2)

(iii) subadditivity: for each𝑋,𝑌 ∈ H,

𝐸 [𝑋 + 𝑌] ≤ 𝐸 [𝑋] + 𝐸 [𝑌] , (3)

(iv) positive homogeneity:

𝐸 [𝜆𝑋] = 𝜆𝐸 [𝑋] for 𝜆 ≥ 0. (4)

The triple (Ω,H, 𝐸) is called a sublinear expectation space.

In this section, we mainly consider the following type ofsublinear expectation spaces (Ω,H, 𝐸): if 𝑋

1, 𝑋

2, . . . , 𝑋

𝑛∈

H then 𝜑(𝑋1, 𝑋

2, . . . , 𝑋

𝑛) ∈ H for 𝜑 ∈ 𝐶

𝑏,Lip(𝑅𝑛), where

𝐶𝑏,Lip(𝑅

𝑛) denotes the linear space of functions 𝜙 satisfying

𝜙 (𝑥) − 𝜙 (𝑦) ≤ 𝐶 (1 + |𝑥|

𝑚+𝑦

𝑚

)𝑥 − 𝑦

for 𝑥, 𝑦 ∈ 𝑅, some 𝐶 > 0, 𝑚 ∈ 𝑁 is depending on 𝜙.(5)

For each fixed 𝑝 ≥ 1, we takeH𝑝0= {𝑋 ∈ H, 𝐸[|𝑋|𝑝] =

0} as our null space and denote H/H𝑝0as the quotient

space. We set ‖𝑋‖𝑝:= (𝐸[|𝑋|

𝑝])1/𝑝 and extend H/H𝑝

0to

its completion Ĥ𝑝under ‖ ⋅ ‖

𝑝. Under ‖ ⋅ ‖

𝑝the sublinear

expectation 𝐸 can be continuously extended to the Banachspace (Ĥ

𝑝, ‖ ⋅ ‖

𝑝). Without loss generality, we denote the

Banach space (Ĥ𝑝, ‖ ⋅ ‖

𝑝) as 𝐿𝑝

𝐺(Ω,H, 𝐸). For the G-frame

work, we refer to [13, 14].In this paper we assume that 𝜇, 𝜇, 𝜎, and 𝜎 are positive

constants such that 𝜇 ≤ 𝜇 and 𝜎 ≤ 𝜎.

Definition 2. Let 𝑋1and 𝑋

2be two random variables in a

sublinear expectation space (Ω,H, 𝐸); 𝑋1and 𝑋

2are called

identically distributed, denoted by𝑋1

𝑑

= 𝑋2if

𝐸 [𝜙 (𝑋1)] = 𝐸 [𝜙 (𝑋

2)] ∀𝜙 ∈ 𝐶

𝑏,Lip (𝑅𝑛) . (6)

Definition 3. In a sublinear expectation space (Ω,H, 𝐸), arandom variable 𝑌 is said to be independent of anotherrandom variable𝑋, if

𝐸 [𝜙 (𝑋, 𝑌)] = 𝐸 [𝐸 [𝜙 (𝑥, 𝑌)]𝑥=𝑋

] . (7)

Definition 4 (G-normal distribution). A random variable 𝑋on a sublinear expectation space (Ω,H, 𝐸) is calledG-normaldistributed if

𝑎𝑋 + 𝑏𝑋 = √𝑎2 + 𝑏2𝑋 for 𝑎, 𝑏 ≥ 0, (8)

where𝑋 is an independent copy of𝑋.

Journal of Applied Mathematics 3

We denote by 𝑆(𝑑) the collection of all 𝑑 × 𝑑 sym-metric matrices. Let 𝑋 be G-normal distributed randomvectors on (Ω,H, 𝐸); we define the following sublinearfunction:

𝐺 (𝐴) :=1

2𝐸 [⟨𝐴𝑋,𝑋⟩] , 𝐴 ∈ 𝑆 (𝑑) . (9)

Remark 5. For a random variable 𝑋 on the sublinear space(Ω,H, 𝐸), there are four typical parameters to character𝑋:

𝜇𝑋= 𝐸𝑋,

𝜇𝑋= −𝐸 [−𝑋] ,

𝜎2

𝑋= 𝐸𝑋

2,

𝜎2

𝑋= −𝐸 [−𝑋

2] ,

(10)

where [𝜇𝑋, 𝜇

𝑋] and [𝜎2

𝑋, 𝜎

2

𝑋] describe the uncertainty of the

mean and the variance of𝑋, respectively.It is easy to check that if𝑋 is G-normal distributed, then

𝜇𝑋= 𝐸𝑋 = 𝜇

𝑋= −𝐸 [−𝑋] = 0, (11)

and we denote the G-normal distribution as𝑁({0}, [𝜎2, 𝜎2]).If𝑋 is maximally distributed, then

𝜎2

𝑋= 𝐸𝑋

2= 𝜎

2

𝑋= −𝐸 [−𝑋

2] = 0, (12)

and we denote the maximal distribution (see [14]) as𝑁([𝜇, 𝜇], {0}).

Let F as Borel field subsets of Ω. We are given a family{F

𝑡}𝑡∈𝑅+

of Borel subfields ofF, such that

F𝑠⊂ F

𝑡, 𝑠 < 𝑡. (13)

Definition 6. We call (𝑋𝑡)𝑡∈𝑅

a 𝑑-dimensional stochastic pro-cess on a sublinear expectation space (Ω,H, 𝐸,F, {F}

𝑡∈𝑅+

),if, for each 𝑡 ∈ 𝑅, 𝑋

𝑡is a 𝑑-dimensional random vector in

H.

Definition 7. Let (𝑋𝑡)𝑡∈𝑅

and (𝑌𝑡)𝑡∈𝑅

be 𝑑-dimensionalstochastic processes defined on a sublinear expectation space(Ω,H, 𝐸,F, {F}

𝑡∈𝑅+

), for each 𝑡 = (𝑡1, 𝑡2, . . . , 𝑡

𝑛) ∈ T;

𝐹𝑋

𝑡[𝜑] := 𝐸 [𝜑 (𝑋

𝑡)] , ∀𝜑 ∈ 𝐶

𝑙,Lip (𝑅𝑛×𝑑

) (14)

is called the finite dimensional distribution of𝑋𝑡.𝑋 and𝑌 are

said to be identically distributed, that is,𝑋 𝑑= 𝑌, if

𝐹𝑋

𝑡[𝜑] = 𝐹

𝑌

𝑡[𝜑] , ∀𝑡 ∈ T, ∀𝜑 ∈ 𝐶

𝑙,Lip (𝑅𝑛×𝑑

) , (15)

whereT := {𝑡 = (𝑡1, 𝑡2, . . . , 𝑡

𝑛) : ∀𝑛 ∈ 𝑁, 𝑡

𝑖∈ 𝑅, 𝑡

𝑖̸= 𝑡𝑗, 0 ≤

𝑖, 𝑗 ≤ 𝑛, 𝑖 ̸= 𝑗}.

Definition 8. A process (𝐵𝑡)𝑡≥0

on the sublinear expectationspace (Ω,H, 𝐸,F, {F}

𝑡∈𝑅+

) is called a G-Brownianmotion ifthe following properties are satisfied:

(i) 𝐵0(𝜔) = 0;

(ii) For each 𝑡, 𝑠 > 0, the increment 𝐵𝑡+𝑠

− 𝐵𝑡is G-normal

distributed by𝑁({0}, [𝑠𝜎2, 𝑠𝜎2]) and is independent of(𝐵

𝑡1

, 𝐵𝑡2

, . . . , 𝐵𝑡𝑛

), for each 𝑛 ∈ 𝑁 and 𝑡1, 𝑡2, . . . , 𝑡

𝑛∈

(0, 𝑡].

From now on, the stochastic processes we will considerin the rest of this paper are all in the sublinear space(Ω,H, 𝐸,F, {F}

𝑡∈𝑅+

).

3. G-Doob-Meyer Decomposition forG-Supermartingale

Definition 9. A G-supermartingale (resp., G-submartingale)is a real valued process {𝑋

𝑡}, well adapted to the F

𝑡family,

such that

(i) 𝐸 [𝑋𝑡] < ∞ ∀𝑡 ∈ 𝑅+,

(ii) 𝐸 [𝑋𝑡+𝑠

| F𝑠] ≤ (resp.≥)𝑋

𝑠

∀𝑡 ∈ 𝑅+, ∀𝑠 ∈ 𝑅

+.

(16)

If equality holds in (ii), the process is a G-martingale.

We will consider right continuous G-supermartingales;then if {𝑋

𝑡} is right continuousG-supermartingale, (ii) in (16)

holds withF𝑡replaced byF

𝑡+.

Definition 10. Let 𝐴 be an event inF𝑡+; one defines capacity

of 𝐴 as

𝑐 (𝐴) = 𝐸 [𝐼𝐴] , (17)

where 𝐼𝐴is indicator function of event 𝐴.

Definition 11. Process 𝑋𝑡and 𝑌

𝑡are adapted to the filtration

F𝑡. One calls 𝑌

𝑡equivalent to𝑋

𝑡, if and only if

𝑐 (𝑌𝑡̸= 𝑋

𝑡) = 0. (18)

For a right continuous G-supermartingale {𝑋𝑡} with

𝐸[𝑋𝑡] is right continuous function of 𝑡; we can find a right

continuous G-supermartingale {𝑌𝑡} equivalent to {𝑋

𝑡} by

defining


𝑌𝑡(𝜔) :=

{

{

{

𝑋𝑡+(𝜔) = lim

𝑠↓𝑡

𝑋𝑠(𝜔) , for any 𝜔 ∈ Ω such that the limit exits

0, otherwise.(19)

Without loss generality, we denoteF𝑡= F

𝑡+.

Definition 12. For a positive constant𝑇, one defines stop time𝜏 in [0, 𝑇] as a positive, random variable 𝜏(𝜔) such that {𝜏 ≤𝑇} ∈ F

𝑇.

In [19, 20], authors discuss the definition of stop time andits related theory in G frame work.

Let {𝑋𝑡} be a right continuousG-supermartingale, denote

𝑋∞as the last element of the process𝑋

𝑡, and then the process

{𝑋𝑡}0≤𝑡≤∞

is a G-supermartingale.

Definition 13. A right continuous increasing process is a welladapted stochastic process {𝐴

𝑡} such that

(i) 𝐴0= 0 a.s,

(ii) for almost every 𝜔, the function 𝑡 → 𝐴𝑡(𝜔) is posi-

tive, increasing, and right continuous. Let 𝐴∞(𝜔) :=

lim𝑡→∞

𝐴𝑡(𝜔); one will say that the right continuous

increasing process is integrable if 𝐸[𝐴∞] < ∞.

Definition 14. An increasing process 𝐴 is called natural if forevery bounded, right continuousG-martingale {𝑀

𝑡}0≤𝑡𝑛}] ≤ 𝐸 [

𝑋𝑎 𝐼{|𝑋𝑇|>𝑛}

] . (26)


As 𝑛 ⋅ 𝑐(|𝑋𝑇| > 𝑛) ≤ 𝐸[|𝑋

𝑇|] ≤ 𝐸[|𝑋

𝑎|], we have

𝑐(|𝑋𝑇| > 𝑛) → 0 as 𝑛 → ∞; then 𝐸[|𝑋

𝑎|𝐼{|𝑋𝑇|>𝑛}

] ≤

(𝐸[|𝑋𝑑|2])1/2(𝑐(𝐼

{|𝑋𝑇|>𝑛}

))1/2

→ 0 as 𝑛 → ∞, from whichwe prove (1).

(2) If 𝑎 < ∞ and 𝑇 is a stop time, 𝑇 ≤ 𝑎, then G-super-martingale process {𝑋

𝑡} has 𝑋

𝑇≥ 𝐸[𝑋

𝑎| F

𝑇]. Suppose that

{𝑋𝑡} is negative; then

𝐸 [−𝑋𝑇𝐼{𝑋𝑇𝑛}] + 𝐸 [𝑌

𝑇𝐼{𝑇>𝑎}

]

≤ 𝐸 [𝑌𝑇𝐼{𝑇≤𝑎, 𝑌

𝑇>𝑛}] + 𝐸 [𝑌

𝑎] .

(29)

Consider that lim𝑎→∞

𝐸[𝑌𝑎] = 0 and {𝑌

𝑡} locally belongs to

(GD); that is, lim𝑛→∞

𝐸[𝑌𝑇𝐼{𝑇≤𝑎, 𝑌

𝑇>𝑛}] = 0, which prove that

lim𝑛→∞

𝐸 [𝑌𝑇𝐼{𝑌𝑇>𝑛}] = 0. (30)

We complete the proof.

Lemma 20. Let {𝑋𝑡} be a right continuous G-supermartingale

and {𝑋𝑛𝑡} a sequence of decomposed right continuous G-

supermartingale:

𝑋𝑛

𝑡= 𝑀

𝑛

𝑡− 𝐴

𝑛

𝑡, (31)

where {𝑀𝑛𝑡} is G-martingale and {𝐴𝑛

𝑡} is right continuous

increasing process. Suppose that, for each 𝑡, 𝑋𝑛𝑡converge to

𝑋𝑡in the 𝐿1

𝐺(Ω) topology, and 𝐴𝑛

𝑡are uniformly integrable

in 𝑛. Then the decomposition problem is solvable for theG-supermartingale {𝑋

𝑡}; more precisely, there are a right

continuous increasing process {𝐴𝑡} and a G-martingale {𝑀

𝑡},

such that 𝑋𝑡= 𝑀

𝑡− 𝐴

𝑡.

Proof. We denote by𝑤 the weak topology𝑤(𝐿1𝐺(Ω), 𝐿

∞

𝐺(Ω));

a sequence of integrable random variables 𝑓𝑛converges to

a random variable 𝑓 in the 𝑤-topology, if and only if 𝑓 isintegrable, and

lim𝑛→∞

𝐸 [𝑓𝑛𝑔] = 𝐸 [𝑓𝑔] , ∀𝑔 ∈ 𝐿

∞

𝐺(Ω) . (32)

Since 𝐴𝑛𝑡are uniformly integrable in 𝑛, by the properties of

the sublinear expectation 𝐸[⋅] there exists a 𝑤-convergent

subsequence𝐴𝑛𝑘𝑡converging in the𝑤-topology to the random

variables 𝐴𝑡, for all rational values of 𝑡. To simplify the

notations, we will use𝐴𝑛𝑡converging to𝐴

𝑡in the𝑤-topology

for all rational values of 𝑡. An integrable random variable 𝑓 isF

𝑡-measurable if and only if it is orthogonal to all bounded

random variables 𝑔 such that 𝐸[𝑔 | F𝑡] = 0; it follows that

𝐴

𝑡isF

𝑡-measurable. For 𝑠 < 𝑡, 𝑠 and 𝑡 rational,

𝐸 [(𝐴𝑛

𝑡− 𝐴

𝑛

𝑠) 𝐼

𝐵] ≥ 0, (33)

where 𝐵 denote anyF set.As 𝑋𝑛

𝑡converge to 𝑋

𝑡in 𝐿1

𝐺(Ω) topology, which is in

a stronger topology than 𝑤, the 𝑀𝑛𝑡converge to random

variables 𝑀𝑡for 𝑡 rational, and the process {𝑀

𝑡} is G-

martingale; then there is a right continuous G-martingale{𝑀

𝑡}, defined for all values of 𝑡, such that 𝑐(𝑀

𝑡̸= 𝑀

𝑡) = 0

for each rational 𝑡. We define 𝐴𝑡= 𝑋

𝑡+ 𝑀

𝑡; {𝐴

𝑡} is a right

continuous increasing process or at least becomes so aftera modification on a set of measure zero. We complete theproof.

Lemma 21. Let {𝑋𝑡} be a potential and belong to class (GD).

One considers the measurable, positive, and well-adaptedprocesses𝐻 = {𝐻

𝑡} with the property that the right continuous

increasing processes

𝐴 (𝐻) = {𝐴𝑡(𝐻, 𝜔)} = {∫

𝑡

0

𝐻𝑠(𝜔) 𝑑𝑠} (34)

are integrable, and the potentials 𝑌(𝐻) = {𝑌𝑡(𝐻, 𝜔)} they

generate are majorized by 𝑋𝑡. Then, for each 𝑡, the random

variables 𝐴𝑡(𝐻) of all such processes 𝐴(𝐻) are uniformly

integrable.

Proof. It is sufficient to prove that the 𝐴∞(𝐻) are uniformly

integrable.(1) First we assume that 𝑋

𝑡is bounded by some positive

constant 𝐶; then 𝐸[𝐴2∞(𝐻)] ≤ 2𝐶

2, and the uniformintegrability follows.

We have that

𝐴2

∞(𝐻, 𝜔)

= 2∫

∞

0

[𝐴∞(𝐻, 𝜔) − 𝐴

𝑢(𝐻, 𝜔)] 𝑑𝐴

𝑢(𝐻, 𝜔)

= 2∫

∞

0

[𝐴∞(𝐻, 𝜔) − 𝐴

𝑢(𝐻, 𝜔)]𝐻

𝑢(𝜔) 𝑑𝑢.

(35)

By using the subadditive property of the sublinear expecta-tion 𝐸, we derive that

𝐸 [𝐴2

∞(𝐻, 𝜔)] = 𝐸 [𝐸 [𝐴

2

∞(𝐻, 𝜔) | F

𝑡]]

≤ 2𝐸 [∫

∞

0

𝐻𝑢𝐸 [𝐴

∞(𝐻, 𝜔) − 𝐴

𝑢(𝐻, 𝜔) | F

𝑢] 𝑑𝑢]

= 2𝐸 [∫

∞

0

𝐻𝑢𝑌𝑢(𝐻) 𝑑𝑢] ≤ 2𝐶𝐸[∫

∞

0

𝐻𝑢𝑑𝑢]

= 2𝐶𝐸 [𝑌0(𝐻)] ≤ 2𝐶

2.

(36)


(2) In order to prove the general case, it will be enoughto prove that any 𝐻 such that 𝑌(𝐻) is majorized by {𝑋

𝑡}

is equal to a sum 𝐻𝑐 + 𝐻𝑐, where (i) 𝐴(𝐻𝑐) generates a

potential bounded by 𝑐, and (ii) 𝐸[𝐴∞[𝐻

𝑐]] is smaller than

some number 𝜀𝑐, independent of 𝐻, such that 𝜀

𝑐→ 0 as

𝑐 → 0. Define

𝐻𝑐

𝑡(𝜔) = 𝐻

𝑡(𝜔) 𝐼

{𝑋𝑡(𝜔)∈[0,𝑐]}

,

𝐻𝑐𝑡= 𝐻

𝑡− 𝐻

𝑐

𝑡.

(37)

Set

𝑇𝑐(𝜔) = inf {𝑡 : such that 𝑋

𝑡(𝜔) ≥ 𝑐} , (38)

as 𝑐 goes to infinity lim𝑐→∞

𝑇𝑐(𝜔) = ∞; therefore 𝑋

𝑇𝑐 → 0,

and class (GD) property implies that 𝐸[𝑋𝑇𝑐] → 0. 𝑇𝑐 is a

stop time, and 𝐼{𝑋𝑡(𝜔)∈[0,𝑐]}

= 1 before time 𝑇𝑐. Hence

𝐸 [𝐴∞(𝐻

𝑐)] = 𝐸 [∫

∞

0

𝐻𝑢(1 − 𝐼

{𝑋𝑢(𝜔)∈[0,𝑐]}

)] 𝑑𝑢

≤ 𝐸[∫

∞

0

𝐻𝑢𝑑𝑢]

= 𝐸 [𝐴∞(𝐻) − 𝐴

𝑇𝑐 (𝐻)]

= 𝐸 [𝐸 [𝐴∞(𝐻) − 𝐴

𝑇𝑐 (𝐻) | F

𝑡]]

= 𝐸 [𝑌𝑇𝑐 (𝐻)] ≤ 𝐸 [𝑋

𝑇𝑐 (𝐻)] ≤ 𝜀

𝑐,

for large enough 𝑐,

(39)

fromwhich we prove (ii).We will prove (i); first we prove that𝑌(𝐻

𝑐) is bounded by 𝑐:

𝑌𝑡(𝐻

𝑐) = 𝐸 [𝐴

∞(𝐻

𝑐) − 𝐴

𝑡(𝐻

𝑐) | F

𝑡]

= 𝐸 [∫

∞

𝑡

𝐻𝑢𝐼{𝑋𝑢(𝜔)∈[0,𝑐]}

𝑑𝑢 | F𝑡]

≤ 𝐸[∫

∞

𝑆𝑐


𝑑𝑢 | F𝑡]

= 𝐸[𝐸[∫

∞

𝑆𝑐


𝑑𝑢 | F𝑆𝑐] | F

𝑡]

= 𝐸 [𝑌𝑆𝑐 | F

𝑡] ≤ 𝑐,

(40)

where we set

𝑆𝑐(𝜔) = inf {𝑡 : such that 𝑋

𝑡(𝜔) ≤ 𝑐} (41)

and use

∫

𝑆𝑐(𝜔)

𝑡


𝑑𝑢 = 0. (42)

Inequality (40) holds for each 𝑡, for every rational 𝑡 andfor every 𝑡 in consideration of the right continuity, whichcomplete the proof.

Lemma 22. Let {𝑋𝑡} be a potential and belong to class (GD),

𝑘 is a positive number, define 𝑌𝑡= 𝐸[𝑋

𝑡+𝑘| F

𝑡], and then {𝑌

𝑡}

is a G-supermartingale. Denote by {𝑝𝑘𝑋𝑡} a right continuous

version of {𝑌𝑡}; then {𝑝

𝑘𝑋𝑡} is potential.

Use the same notations as in Lemma 21. Let 𝑘 be a positivenumber, and 𝐻

𝑡,𝑘(𝜔) = (𝑋

𝑡(𝜔) − 𝑝

𝑘𝑋𝑡(𝜔))/𝑘. The process

𝐻𝑘= {𝐻

𝑡,𝑘} verifies the assumptions of Lemma 21, and their

potentials increase to {𝑋𝑡} as 𝑘 → 0.

Proof. If 𝑡 < 𝑢

𝐸 [1

𝑘(∫

𝑢

0

[𝑋𝑠− 𝑝

𝑘𝑋𝑠] 𝑑𝑠 − ∫

𝑡

0

[𝑋𝑠− 𝑝

𝑘𝑋𝑠] 𝑑𝑠) |

F𝑡] = 𝐸[

1

𝑘∫

𝑢

𝑡

[𝑋𝑠− 𝑝

𝑘𝑋𝑠] 𝑑𝑠 | F

𝑡] .

(43)

For 𝑠 ≥ 𝑡, 𝐸[𝑝𝑘𝑋𝑠| F

𝑡] = 𝐸[𝐸[𝑋

𝑠+𝑘| F

𝑠] | F

𝑡] = 𝐸[𝑋

𝑠+𝑘|

F𝑡].We have that

𝐸[1

𝑘∫

𝑢

𝑡

[𝑋𝑠− 𝑝


𝑡]

≥ 𝐸[1

𝑘∫

𝑡+𝑘

𝑡

𝑋𝑠𝑑𝑠 | F

𝑡]

− 𝐸[1

𝑘∫

𝑢+𝑘

𝑢


𝑡] ,

(44)

and, by the subadditive property of the sublinear expectation𝐸, we derive that

𝐸[1

𝑘∫

𝑡+𝑘

𝑡


𝑡] − 𝐸[

1

𝑘∫

𝑢+𝑘

𝑢


𝑡]

≥ 𝐸[1

𝑘∫

𝑡+𝑘

𝑡


𝑡] −

1

𝑘∫

𝑢+𝑘

𝑢

𝐸 [𝑋𝑠| F

𝑡] 𝑑𝑠

≥ 𝐸[1

𝑘∫

𝑡+𝑘

𝑡


𝑡] − 𝑋

𝑡

≥ −𝐸[1

𝑘∫

𝑡+𝑘

𝑡

(𝑋𝑡− 𝑋

𝑠) 𝑑𝑠 | F

𝑡]

≥ −1

𝑘∫

𝑡+𝑘

𝑡

𝐸 [𝑋𝑡− 𝑋

𝑠| F

𝑡] 𝑑𝑠 ≥ 0.

(45)

Hence, we derive that for any 𝑢, 𝑡 such that 𝑢 > 𝑡

𝐸 [1

𝑘∫

𝑢

𝑡

[𝑋𝑠− 𝑝


𝑡] ≥ 0. (46)

If there exits 𝑠0≥ 0 such that (1/𝑘)[𝑋

𝑠0

−𝑝𝑘𝑋𝑠0

] < 0, the rightcontinuous of {𝑋

𝑡} implies that there exists 𝛿 > 0 such that

(1/𝑘)[𝑋𝑠− 𝑝

𝑘𝑋𝑠] < 0 on the interval [𝑠

0, 𝑠0+ 𝛿]. Thus

𝐸[1

𝑘∫

𝑠0+𝛿

𝑠0

[𝑋𝑠− 𝑝


𝑠0

] < 0, (47)

which is contradiction; we prove that (𝑋𝑡(𝜔) − 𝑝

𝑘𝑋𝑡(𝜔))/𝑘 is

a positive, measurable, and well-adapted process.


Since {𝑋𝑡} is right continuous G-supermartingale

lim𝑠↓𝑡

𝑋𝑠= 𝑋

𝑡,

lim𝑘↓0

𝑌𝑡(𝐻

𝑘) = lim

𝑘↓0

𝐸[1

𝑘∫

∞

𝑡

[𝑋𝑠− 𝑝


𝑡]

= lim𝑘↓0

𝐸[1

𝑘∫

𝑡+𝑘

𝑡


𝑡]

= 𝐸[lim𝑘↓0

1

𝑘∫

𝑡+𝑘

𝑡


𝑡] = 𝑋

𝑡;

(48)

we finish the proof.

From Lemmas 20, 21, and 22 we can prove the followingtheorem.

Theorem23. Apotential {𝑋𝑡} belongs to class (GD) if and only

if it is generated by some integrable right continuous increasingprocess.

Theorem 24 (G-Doob-Meyer’s decomposition). (1) {𝑋𝑡} is a

right continuous G-supermartingale if and only if it belongsto class (GD) on every finite interval. More precisely, {𝑋

𝑡} is

then equal to the difference of a G-martingal 𝑀𝑡and a right

continuous increasing process 𝐴𝑡:

𝑋𝑡= 𝑀

𝑡− 𝐴

𝑡. (49)

(2) If the right continuous increasing process 𝐴 is natural,the decomposition is unique.

Proof. (1) The necessity is obvious. We will prove the suffi-ciency; we choose a positive number 𝑎 and define

𝑋

𝑡(𝜔) := 𝑋

𝑡(𝜔) , 𝑡 ∈ [0, 𝑎] ,

𝑋

𝑡(𝜔) := 𝑋

𝑎(𝜔) , 𝑡 > 𝑎;

(50)

the {𝑋𝑡} is a right continuous G-supermartingale of the

class (GD), and by Theorem 23 there exists the followingdecomposition

𝑋

𝑡= 𝑀

𝑡− 𝐴

𝑡, (51)

where {𝑀𝑡} is a G-martingal and {𝐴

𝑡} is a right continuous

increasing process.Let 𝑎 → ∞, as in Lemma 22 the expression of 𝑌

𝑡(𝐻

𝑘)

that𝐴𝑡depend only on the values of {𝑋

𝑡} on intervals [0, 𝑡+𝜀],

with 𝜀 small enough. As 𝑎 → ∞, they do not vary anymoreonce 𝑎 has reached values greater than 𝑡, as again Lemma 20;we finish the proof of the Theorem.

(2) Assume that𝑋 admits both decompositions:

𝑋𝑡= 𝑀

𝑡− 𝐴

𝑡= 𝑀

𝑡− 𝐴

𝑡, (52)

where𝑀𝑡and𝑀

𝑡are G-martingale and 𝐴

𝑡, 𝐴

𝑡are natural

increasing process. We define

{𝐶𝑡:= 𝐴

𝑡− 𝐴

𝑡=𝑀

𝑡−𝑀

𝑡} . (53)

Then {𝐶𝑡} is a G-martingale, and, for every bounded and right

continuous G-martingale {𝜉𝑡}, from Lemma 15 we have

𝐸 [𝜉𝑡(𝐴

𝑡− 𝐴

𝑡)] = 𝐸 [∫

(0,𝑡]

𝜉𝑠−𝑑𝐶

𝑠]

= lim𝑛→∞

𝑚𝑛

∑

𝑘=1

𝜉𝑡𝑛

𝑗−1

[𝐶𝑡(𝑛)

𝑗

− 𝐶𝑡(𝑛)

𝑗−1

] ,

(54)

where Π𝑛= {𝑡

(𝑛)

0, . . . , 𝑡

(𝑛)

𝑚𝑛

}, 𝑛 ≥ 1 is a sequence of partitionsof [0, 𝑡]with max

1≤𝑗≤𝑚𝑛

(𝑡(𝑛)

𝑗−𝑡

(𝑛)

𝑗−1) converging to zero as 𝑛 →

∞. Since 𝜉 and 𝐶 are both G-martingale, we have

𝐸 [𝜉𝑡(𝑛)

𝑗−1

(𝐶𝑡(𝑛)

𝑗

− 𝐶𝑡(𝑛)

𝑗−1

)] = 0,

and thus 𝐸 [𝜉𝑡𝑗−1

(𝐴

𝑡− 𝐴

𝑡)] = 0.

(55)

For an arbitrary bonded random variable 𝜉, we can select {𝜉𝑡}

to be a right continuous equivalent process of {𝐸[𝜉 | F𝑡]},

and we obtain that 𝐸[𝜉(𝐴𝑡− 𝐴

𝑡)] = 0. We set 𝜉 = 𝐼

𝐴

𝑡̸=𝐴

𝑡

;therefore 𝑐(𝐴

𝑡̸= 𝐴

𝑡) = 0.

By Theorem 24 and G-martingale decomposition the-orem in [14, 21], we have the following G-Doob-Meyertheorem.

Theorem 25. {𝑋𝑡} is a right continuous G-supermartingale;

there exists a right continuous increasing process 𝐴𝑡and

adapted process 𝜂𝑡, such that

𝑋𝑡= ∫

𝑡

0

𝜂𝑠𝑑𝐵

𝑠− 𝐴

𝑡, (56)

where 𝐵𝑡is G-Brownian motion.

4. Superhedging Strategies andOptimal Stopping

4.1. Financial Model and G-Asset Price System. We considera financial market with a nonrisky asset (bond) and a riskyasset (stock) continuously trading in market. The price 𝑃(𝑡)of the bond is given by

𝑑𝑃 (𝑡) = 𝑟𝑃 (𝑡) 𝑑𝑡 𝑃 (0) = 1, (57)

where 𝑟 is the short interest rate; we assume a constantnonnegative short interest rate. We assume the risk asset withthe G-asset price system ((𝑆

𝑢)𝑢≥𝑡, 𝐸) (see [17]) on sublinear

expectation space (Ω,H, 𝐸,F, (F𝑡))underKnightian uncer-

tainty, for given 𝑡 ∈ [0, 𝑇] and 𝑥 ∈ 𝑅

𝑑𝑆𝑡,𝑥

𝑢= 𝑆

𝑡,𝑥

𝑢𝑑𝐵

𝑡= 𝑆

𝑡,𝑥

𝑢(𝑑𝑏

𝑡+ 𝑑𝐵

𝑡) ,

𝑆𝑡,𝑥

𝑡= 𝑥,

(58)

where 𝐵𝑡is the generalized G-Brownian motion. The uncer-

tain volatility is described by the G-Brownian motion 𝐵𝑡. The

uncertain drift 𝑏𝑡can be rewritten as

𝑏𝑡= ∫

𝑡

0

𝜇𝑢𝑑𝑢, (59)


where 𝜇𝑡is the asset return rate [22]. Then the uncertain risk

premium of the G-asset price system

𝜃𝑡= 𝜇

𝑡− 𝑟, (60)

is uncertain and distributed by𝑁([𝜇−𝑟, 𝜇−𝑟], {0}) [22], where𝑟 is the interest rate of the bond.

Define

𝐵𝑡:= 𝐵

𝑡− 𝑟𝑡 = 𝑏

𝑡+ 𝐵

𝑡− 𝑟𝑡; (61)

we have the following G-Girsanov theorem (presented in [17,23]).

Theorem 26 (G-Girsanov theorem). Assume that (𝐵𝑡)𝑡≥0

is generalized G-Brownian motion on (Ω,H, 𝐸,F𝑡), and

𝐵𝑡is defined by (61); there exists G-expectation space

(Ω,H, 𝐸𝐺,F𝑡) such that 𝐵

𝑡is G-Brownian motion under the

G-expectation 𝐸𝐺, and

𝐸 [𝐵2

𝑡] = 𝐸

𝐺[𝐵

2

𝑡] ,

−𝐸 [−𝐵2

𝑡] = −𝐸

𝐺[−𝐵

2

𝑡] .

(62)

By the G-Girsanov theorem, the G-asset price system(58) of the risky asset can be rewritten on (Ω,H, 𝐸𝐺,F

𝑡) as

follows:

𝑑𝑆𝑡,𝑥

𝑢= 𝑆

𝑡,𝑥

𝑢(𝑟𝑑𝑡 + 𝑑𝐵

𝑡) ,

𝑆𝑡,𝑥

𝑡= 𝑥;

(63)

then by G-Itô formula we have

𝑆𝑡,𝑥

𝑢= 𝑥 exp (𝑟 (𝑢 − 𝑡) + 𝐵

𝑢−𝑡−1

2(⟨𝐵

𝑢⟩ − ⟨𝐵

𝑡⟩)) ,

𝑢 > 𝑡.

(64)

4.2. Construction of Superreplication Strategies via OptimalStopping. We consider the following class of contingentclaims.

Definition 27. One defines a class of contingent claims withthe nonnegative payoff 𝜉 ∈ 𝐿2

𝐺(Ω

𝑇) having the following

form:

𝜉 = 𝑓 (𝑆𝑡,𝑥

𝑇) (65)

for some function 𝑓 : Ω → 𝑅 such that the process

𝑓𝑢:= 𝑓 (𝑆

𝑡,𝑥

𝑢) (66)

is bounded below and càdlàg.

We consider a contingent claim 𝜉 with payoff defined inDefinition 27 written on the stockes 𝑆

𝑡with maturity 𝑇. We

give definitions of superhedging (resp., subhedging) strategyand ask (resp., bid) price of the claim 𝜉.

Definition 28. (1) A self-financing superstrategy (resp. sub-strategy) is a vector process (𝑌, 𝜋, 𝐶) (resp., (−𝑌, 𝜋, 𝐶)), where𝑌 is the wealth process, 𝜋 is the portfolio process, and𝐶 is thecumulative consumption process, such that

𝑑𝑌𝑡= 𝑟𝑌

𝑡𝑑𝑡 + 𝜋

𝑡𝑑𝐵

𝑡− 𝑑𝐶

𝑡,

(resp. − 𝑑𝑌𝑡= −𝑟𝑌

𝑡𝑑𝑡 + 𝜋

𝑡𝑑𝐵

𝑡− 𝑑𝐶

𝑡) ,

(67)

where 𝐶 is an increasing, right continuous process with 𝐶0=

0.The superstrategy (resp., substrategy) is called feasible if theconstraint of nonnegative wealth holds

𝑌𝑡≥ 0, 𝑡 ∈ [0, 𝑇] . (68)

(2) A superhedging (resp. subhedging) strategy againstthe contingent claim 𝜉 is a feasible self-financing superstrat-egy (𝑌, 𝜋, 𝐶) (resp., substrategy (−𝑌, 𝜋, 𝐶)) such that 𝑌

𝑇= 𝜉

(resp., −𝑌𝑇= −𝜉). We denote by H(𝜉) (resp., H(−𝜉)) the

class of superhedging (resp., subhedging) strategies against𝜉, and if H(𝜉) (resp., H(−𝜉)) is nonempty, 𝜉 is calledsuperhedgeable (resp., subhedgeable).

(3) The ask-price 𝑋(𝑡) at time 𝑡 of the superhedgeableclaim 𝜉 is defined as

𝑋 (𝑡) = inf {𝑥 ≥ 0 : ∃ (𝑌𝑡, 𝜋

𝑡, 𝐶

𝑡)

∈H (𝜉) such that 𝑌𝑡= 𝑥} ,

(69)

and bid-price 𝑋(𝑡) at time 𝑡 of the subhedgeable claim 𝜉 isdefined as

𝑋(𝑡) = sup {𝑥 ≥ 0 : ∃ (−𝑌

𝑡, 𝜋

𝑡, 𝐶

𝑡)

∈H(−𝜉) such that − 𝑌

𝑡= −𝑥} .

(70)

Under uncertainty, the market is incomplete and thesuperhedging (resp., subhedging) strategy of the claim is notunique. The definition of the ask-price 𝑋(𝑡) implies that theask-price 𝑋(𝑡) is the minimum amount of risk for the buyerto superhedging the claim; then it is coherent measure ofrisk of all superstrategies against the claim for the buyer. Thecoherent risk measure of all superstrategies against the claimcan be regarded as the sublinear expectation of the claim; wehave the following representation of bid-ask price of the claimvia optimal stopping (Theorem 31).

Let (G𝑡) be a filtration on G-expectation space (Ω,H,

𝐸𝐺,F, (F

𝑡)𝑡≥0), and 𝜏

1and 𝜏

2be (G

𝑡)-stopping times such

that 𝜏1≤ 𝜏

2a.s. We denote by G

𝜏1,𝜏2

the set of all finite (G𝑡)-

stopping times 𝜏 with 𝜏1≤ 𝜏 ≤ 𝜏

2.

For given 𝑡 ∈ [0, 𝑇] and 𝑥 ∈ 𝑅+, we define the function

𝑉𝐴𝑚

: [0, 𝑇] × Ω → 𝑅 as the value function of the followingoptimal-stopping problem:

𝑉𝐴𝑚

(𝑡, 𝑆𝑡) := sup

]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓]]

= sup]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆])] .

(71)


Proposition 29. Consider two stopping times 𝜏 ≤ 𝜏 onfiltration F. Let (𝑓

𝑡)𝑡≥0

denote some adapted and RCLL-stochastic process, which is bounded below. Then we have fortwo points 𝑠, 𝑡 ∈ [0, 𝜏] and 𝑠 < 𝑡

ess sup𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑠[𝑓𝜏]} = 𝐸

𝐺

𝑠[ess sup

𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑡[𝑓𝜏]}] . (72)

Proof. By the consistent property of the conditional G-expectation, for 𝜏 ∈ F

𝜏,𝜏, 𝑠, 𝑡 ∈ [0, 𝜏], and 𝑠 < 𝑡

𝐸𝐺

𝑠[𝑓𝜏] = 𝐸

𝐺

𝑠[𝐸

𝐺

𝑡[𝑓𝜏]] ≤ 𝐸

𝐺

𝑠[ess sup

𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑡[𝑓𝜏]}] ; (73)

thus we have

ess sup𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑠[𝑓𝜏]} ≤ 𝐸

𝐺

𝑠[ess sup

𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑡[𝑓𝜏]}] . (74)

There exists a sequence {𝜏𝑛} → 𝜏

∗∈ [𝜏, 𝜏] as 𝑛 → ∞, such

that

lim𝑛→∞

𝐸𝐺

𝑡[𝑓

𝜏𝑛

] = 𝐸𝐺

𝑡[𝑓𝜏∗] = ess sup

𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑡[𝑓𝜏]} ; (75)

notice that

𝐸𝐺

𝑠[ess sup

𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑡[𝑓𝜏]}] = 𝐸

𝐺

𝑠[𝐸

𝐺

𝑡[𝑓𝜏∗]] = 𝐸

𝐺

𝑠[𝑓𝜏∗]

≤ ess sup𝜏∈F𝜏,𝜏

{𝐸𝐺

𝑠[𝑓𝜏]} ;

(76)

we prove the Proposition.

Proposition 30. The process 𝑉𝐴𝑚(𝑡, 𝑆𝑡)0≤𝑡≤𝑇

is a G-supermartingale in (Ω,H, 𝐸𝐺,F,F

𝑡).

Proof. By Proposition 29, for 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇

𝐸𝐺

𝑠[ sup]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆])]] = sup

]∈F𝑡,𝑇

𝐸𝐺

𝑠[𝑓 (𝑆])] . (77)

SinceF𝑡,𝑇

⊆ F𝑠,𝑇, we have

sup]∈F𝑡,𝑇

𝐸𝐺

𝑠[𝑓 (𝑆])] ≤ sup

]∈F𝑠,𝑇

𝐸𝐺

𝑠[𝑓 (𝑆])] . (78)

Thus, we derive that

𝐸𝐺

𝑠[ sup]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆])]] ≤ sup

]∈F𝑠,𝑇

𝐸𝐺

𝑠[𝑓 (𝑆])] . (79)

We prove the Proposition.

Theorem 31. Assume that the financial market under uncer-tainty consists of the bond which has the price process satisfying(57) and risky assets with the price processes as the G-asset pricesystems (58) and can trade freely; the contingent claim 𝜉 which

is written on the risky assets with the maturity 𝑇 > 0 has theclass of the payoff defined in Definition 27, and the function𝑉𝐴𝑚(𝑡, 𝑆

𝑡) is defined in (71). Then there exists a superhedging

(resp., subhedging) strategy for 𝜉, such that the process 𝑉 =(𝑉

𝑡)0≤𝑡≤𝑇

defined by

𝑉𝑡:= 𝑒

−𝑟(𝑇−𝑡)𝑉𝐴𝑚

(𝑡, 𝑆𝑡) ,

(𝑟𝑒𝑠𝑝. − 𝑒−𝑟(𝑇−𝑡)ess sup

]∈F𝑡,𝑇

𝐸𝐺

𝑡[−𝑓]])

(80)

is the ask (resp., bid) price process against 𝜉.

Proof. Thevalue function for the optimal stop time𝑉𝐴𝑚(𝑡, 𝑆𝑡)

is a G-supermartingale; it is easily to check that 𝑒−𝑟𝑡𝑉𝑡

is G-supermartingale. By G-Doob-Meyer decompositionTheorem 24

𝑒−𝑟𝑡𝑉𝑡= 𝑀

𝑡− 𝐶

𝑡, (81)

where 𝑀𝑡is a G-martingale and 𝐶

𝑡is an increasing process

with𝐶0= 0. By G-martingale representation theorem [14, 21]

𝑀𝑡= 𝐸

𝐺[𝑀

𝑇] + ∫

𝑡

0

𝜂𝑠𝑑𝐵

𝑡− 𝐾

𝑡, (82)

where 𝜂𝑠∈ 𝐻

1

𝐺(0, 𝑇), −𝐾

𝑡is a G-martingale, and 𝐾

𝑡is an

increasing process with 𝐾0= 0. From the above equation,

we have

𝑒−𝑟𝑡𝑉𝑡= 𝐸

𝐺[𝑀

𝑇] + ∫

𝑡

0

𝜂𝑠𝑑𝐵

𝑡− (𝐾

𝑡+ 𝐶

𝑡) ; (83)

hence (𝑉𝑡, 𝑒𝑟𝑡𝜂𝑡, ∫

𝑡

0𝑒𝑟𝑠𝑑(𝐶

𝑠+𝐾

𝑠)𝑑𝑠) is a superhedging strategy.

Assume that (𝑌𝑡, 𝜋

𝑡, 𝐶

𝑡) is a superhedging strategy against

𝜉; then

𝑒−𝑟𝑡𝑌𝑡= 𝑒

−𝑟𝑇𝜉 − ∫

𝑇

𝑡

𝜋𝑡𝑑𝐵

𝑡+ 𝐶

𝑡. (84)

Taking conditional G-expectation on the both sides of (84)and noticing that the process 𝐶

𝑡is an increasing process with

𝐶0= 0, we derive

𝑒−𝑟𝑡𝑌𝑡≥ 𝐸

𝐺

𝑡[𝑒−𝑟𝑇

𝜉] , (85)

which implies that

𝑌𝑡≥ 𝐸

𝐺

𝑡[𝑒−𝑟(𝑇−𝑡)

𝜉] ≥ 𝐸𝐺

𝑡[𝑒

−𝑟(𝑇−𝑡)ess sup]∈F𝑇,𝑇

[𝑓]]]

≥ 𝑒−𝑟(𝑇−𝑡)ess sup

]∈F𝑇,𝑇

𝐸𝐺

𝑡[𝑓]] ≥ 𝑒

−𝑟(𝑇−𝑡)ess sup]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓]]

= 𝑉𝑡

(86)

from which we prove that 𝑉𝑡= 𝑒

−𝑟(𝑇−𝑡)𝑉𝐴𝑚(𝑡, 𝑆

𝑡) is the ask

price against the claim 𝜉 at time 𝑡. Similarly we can provethat −𝑒−𝑟(𝑇−𝑡) ess sup]∈F

𝑡,𝑇

𝐸𝐺

𝑡[−𝑓]] is the bid price against the

claim 𝜉 at time 𝑡.


5. Free Boundary and OptimalStopping Problems

For given 𝑡 ∈ [0, 𝑇], 𝑥 ∈ 𝑅𝑑, and 𝑑 = 1, the G-asset pricesystem (58) of the risky asset can be rewritten as follows:

𝑑𝑆𝑡,𝑥

𝑢= 𝑆

𝑡,𝑥

𝑢(𝑟𝑑𝑡 + 𝑑𝐵

𝑡) ,

𝑆𝑡,𝑥

𝑡= 𝑥.

(87)

We define the following deterministic function:

𝑢𝑎(𝑡, 𝑥) := 𝑒

−𝑟(𝑇−𝑡)𝑉𝐴𝑚

(𝑡, 𝑆𝑡,𝑥

𝑡) , (88)

where

𝑉𝐴𝑚

(𝑡, 𝑆𝑡,𝑥

𝑡) = ess sup

]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆

𝑡,𝑥

] )] . (89)

From Theorem 31 the price of an American option withexpiry date 𝑇 and payoff function 𝑓 is the value function ofthe optimal stopping problem:

𝑢𝑎(𝑡, 𝑥) := 𝑒

−𝑟(𝑇−𝑡)ess sup]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆])] . (90)

We define operator 𝐿 as follows:

𝐿𝑢 = 𝐺 (𝐷2𝑢) + 𝑟𝐷𝑢 + 𝜕

𝑡𝑢, (91)

where 𝐺(⋅) is the sublinear function defined by (9). Weconsider the free boundary problem

L𝑢 := max {𝐿𝑢 − 𝑟𝑢, 𝑓 − 𝑢} = 0, in [0, 𝑇] × 𝑅,

𝑢 (𝑇, ⋅) = 𝑓 (𝑇, ⋅) , in 𝑅.(92)

Denote

S𝑇:= [0, 𝑇] × 𝑅, (93)

for 𝑝 ≥ 1

S𝑝(S

𝑇) := {𝑢 ∈ 𝐿

𝑝(S

𝑇) : 𝐷

2𝑢,𝐷𝑢, 𝜕

𝑡𝑢 ∈ 𝐿

𝑝(S

𝑇)} . (94)

And, for any compact subset 𝐷 of S𝑇, we denote S𝑝loc (𝐷) as

the space of functions 𝑢 ∈ S𝑝(𝐷).

Definition 32. A function 𝑢 ∈ S1loc(S𝑇) ∩ 𝐶(𝑅 × [0, 𝑇]) is astrong solution of problem (92) ifL𝑢 = 0 almost everywherein S

𝑇and it attains the final datum pointwisely. A function

𝑢 ∈ S1loc(S𝑇) ∩ 𝐶(𝑅 × [0, 𝑇]) is a strong supersolution ofproblem (92) ifL𝑢 ≤ 0.

We will prove the following existence results.

Theorem33. If there exists a strong supersolution 𝑢 of problem(92) then there also exists a strong solution 𝑢 of (92) suchthat 𝑢 ≤ 𝑢 in S

𝑇. Moreover 𝑢 ∈ S𝑝

𝑙𝑜𝑐(S

𝑇) for any 𝑝 ≥ 1

and consequently, by the embedding theorem we have 𝑢 ∈𝐶1,𝛼

𝐵,𝑙𝑜𝑐(S

𝑇) for any 𝛼 ∈ [0, 1].

Theorem 34. Let 𝑢 be a strong solution to the free boundaryproblem (92) such that

|𝑢 (𝑡, 𝑥)| ≤ 𝐶𝑒𝜆|𝑥|2

, (𝑡, 𝑥) ∈ S𝑇

(95)

form some constants 𝐶, 𝜆 with 𝜆 sufficiently small so that

𝐸𝐺[exp(𝜆 sup

𝑡≤𝑢≤𝑇

𝑆𝑡,𝑥

𝑢

2

)] < ∞ (96)

holds. Then we have

𝑢 (𝑡, 𝑥) = 𝑒−𝑟(𝑇−𝑡)ess sup

]∈F𝑡,𝑇

𝐸𝐺

𝑡[𝑓 (𝑆])] ; (97)

that is, the solution of the free boundary problem is the valuefunction of the optimal stopping problem. In particular such asolution is unique.

5.1. Proof of Theorem 34. We employ a truncation and reg-ularization technique to exploit the weak interior regularityproperties of 𝑢; for 𝑅 > 0 we set for 𝑅 > 0, 𝐵

𝑅:= {𝑥 ∈ 𝑅

𝑑|

|𝑥| < 𝑅}, and, for 𝑥 ∈ 𝐵𝑅denoting by 𝜏

𝑅the first exit time of

𝑆𝑡,𝑥

𝑢from𝐵

𝑅, it is easy check that𝐸𝐺[𝜏

𝑅] is finite. As a first step

we prove the following result: for every (𝑡, 𝑥) ∈ [0, 𝑇]×𝐵𝑅and

𝜏 ∈ F𝑡,𝑇

such that 𝜏 ∈ [𝑡, 𝜏𝑅], it holds that

𝑢 (𝑡, 𝑥) = 𝐸𝐺[𝑢 (𝜏, 𝑆

𝑡,𝑥

𝜏)] − 𝐸

𝐺[∫

𝜏

𝑡

𝐿𝑢 (𝑠, 𝑆𝑡,𝑥

𝑠) 𝑑𝑠] . (98)

For fixed, positive, and small enough 𝜀, we consider a function𝑢𝜀,𝑅 on 𝑅𝑑+1 = 𝑅2 with compact support and such that𝑢𝜀,𝑅

= 𝑢 on [𝑡, 𝑇 − 𝜀] × 𝐵𝑅. Moreover we denote by (𝑢𝜀,𝑅,𝑛)

𝑛∈𝑁

a regularizing sequence obtained by convolution of 𝑢𝜀,𝑅 withthe usual mollifiers; then for any 𝑝 ≥ 1 we have 𝑢𝜀,𝑅,𝑛 ∈S𝑝(𝑅2) and

lim𝑛→∞

𝐿𝑢

𝜀,𝑅,𝑛− 𝐿𝑢

𝜀,𝑅𝐿𝑝([𝑡,𝑇−𝜀]×𝐵𝑅)= 0. (99)

By G-Itô formula we have

𝑢𝜀,𝑅,𝑛

(𝜏, 𝑆𝑡,𝑥

𝜏) = 𝑢

𝜀,𝑅,𝑛(𝑡, 𝑥) +

1

2∫

𝜏

𝑡

𝐷2𝑢𝜀,𝑅,𝑛

𝑑 ⟨𝐵⟩𝑠

+ ∫

𝜏

𝑡

𝑟𝐷𝑢𝜀,𝑅,𝑛

𝑑𝑠 + ∫

𝜏

𝑡

𝜕𝑠𝑢𝜀,𝑅,𝑛

𝑑𝑠

+ ∫

𝜏

𝑡

𝐷𝑢𝜀,𝑅,𝑛

𝑑𝐵𝑠,

(100)

which implies that

𝐸𝐺[𝑢

𝜀,𝑅,𝑛(𝜏, 𝑆

𝑡,𝑥

𝜏)] = 𝑢

𝜀,𝑅,𝑛(𝑡, 𝑥) + ∫

𝜏

𝑡

𝐿𝑢𝜀,𝑅,𝑛

𝑑𝑠. (101)

We have

lim𝑛→∞

𝑢𝜀,𝑅,𝑛

(𝑡, 𝑥) = 𝑢𝜀,𝑅(𝑡, 𝑥) (102)

and, by dominated convergence,

lim𝑛→∞

𝐸𝐺[𝑢

𝜀,𝑅,𝑛(𝜏, 𝑆

𝑡,𝑥

𝜏)] = 𝐸

𝐺[𝑢

𝜀,𝑅(𝜏, 𝑆

𝑡,𝑥

𝜏)] . (103)


We have

𝐸𝐺[∫

𝜏

𝑡


(𝑠, 𝑆𝑡,𝑥

𝑠) 𝑑𝑠]

− 𝐸𝐺[∫

𝜏

𝑡

𝐿𝑢𝜀,𝑅(𝑠, 𝑆

𝑡,𝑥

𝑠) 𝑑𝑠]

≤ 𝐸𝐺[∫

𝜏

𝑡

𝐿𝑢

𝜀,𝑅,𝑛(𝑠, 𝑆

𝑡,𝑥

𝑠) − 𝐿𝑢

𝜀,𝑅(𝑠, 𝑆

𝑡,𝑥

𝑠)𝑑𝑠] ;

(104)

by sublinear expectation representation theorem (see [14])there exists a family of probability space 𝑄, such that

𝐸𝐺[∫

𝜏

𝑡

𝐿𝑢


𝑡,𝑥

𝑠) − 𝐿𝑢


𝑡,𝑥

𝑠)𝑑𝑠]

= ess sup𝑃∈𝑄

𝐸𝑃[∫

𝜏

𝑡

𝐿𝑢


𝑡,𝑥

𝑠)

− 𝐿𝑢𝜀,𝑅(𝑠, 𝑆

𝑡,𝑥

𝑠)𝑑𝑠] .

(105)

Since 𝜏 ≤ 𝜏𝑅

ess sup𝑃∈𝑄

𝐸𝑃[∫

𝜏

𝑡

𝐿𝑢


𝑡,𝑥

𝑠) − 𝐿𝑢


𝑡,𝑥

𝑠)𝑑𝑠]

≤ ess sup𝑃∈𝑄

𝐸𝑃[∫

𝑇−𝜀

𝑡

𝐿𝑢

𝜀,𝑅,𝑛(𝑠, 𝑦) − 𝐿𝑢

𝜀,𝑅(𝑠, 𝑦)

⋅𝐼|𝑆𝑡,𝑥

𝑠|≤𝐵𝑅

𝑑𝑠] ≤ ess sup𝑃∈𝑄

∫

𝑇−𝜀

𝑡

∫𝐵𝑅

𝐿𝑢

𝜀,𝑅,𝑛(𝑠, 𝑦)

− 𝐿𝑢𝜀,𝑅(𝑠, 𝑦)

Γ𝑃(𝑡, 𝑥; 𝑠, 𝑦) 𝑑𝑦 𝑑𝑠,

(106)

where Γ𝑃(𝑡, 𝑥; ⋅, ⋅) ∈ 𝐿

𝑞([𝑡, 𝑇] × 𝐵

𝑅), for some 𝑞 > 1, is the

transition density of the solution of

𝑑𝑋𝑡,𝑥

𝑠= 𝑋

𝑡,𝑥

𝑠(𝑟𝑑𝑠 + 𝜎

𝑠,𝑃𝑑𝑊

𝑠,𝑃) , (107)

where 𝑊𝑠,𝑃

is Wiener process in probability space (Ω𝑡, 𝑃,

F𝑃,F𝑃𝑡) and 𝜎

𝑠,𝑃is adapted process such that 𝜎

𝑠,𝑃∈ [𝜎, 𝜎].

By Hölder inequality, we have (1/𝑝 + 1/𝑞 = 1)

∫

𝑇−𝜀

𝑡

∫𝐵𝑅

𝐿𝑢

𝜀,𝑅,𝑛(𝑠, 𝑦) − 𝐿𝑢

𝜀,𝑅(𝑠, 𝑦)

⋅ Γ𝑃(𝑡, 𝑥; 𝑠, 𝑦) 𝑑𝑦 𝑑𝑠 ≤

𝐿𝑢

𝜀,𝑅,𝑛(𝑠, 𝑦)

− 𝐿𝑢𝜀,𝑅(𝑠, 𝑦)

𝐿𝑞([𝑡,𝑇]×𝐵𝑅)

Γ𝑃 (𝑡, 𝑥; 𝑠, 𝑦)𝐿𝑝([𝑡,𝑇]×𝐵

𝑅),

(108)

and then we obtain that

lim𝑛→∞

𝐸𝐺[∫

𝜏

𝑡


(𝑠, 𝑆𝑡,𝑥

𝑠)]

= 𝐸𝐺[∫

𝜏

𝑡

𝐿𝑢𝜀,𝑅(𝑠, 𝑆

𝑡,𝑥

𝑠)] .

(109)

This concludes the proof of (98), since 𝑢𝜀,𝑅 = 𝑢 on [𝑡, 𝑇 − 𝜀] ×𝐵𝑅and 𝜀 > 0 is arbitrary.

Since 𝐿𝑢 ≤ 0, we have for any 𝜏 ∈ F𝑡,𝑇

𝐸𝐺∫

𝜏

𝑡


𝑠) 𝑑𝑠 ≤ 0; (110)

we infer from (98) that

𝑢 (𝑡, 𝑥) ≥ 𝐸𝐺[𝑢 (𝜏 ∧ 𝜏

𝑅, 𝑆𝑡,𝑥

𝜏∧𝜏𝑅

)] . (111)

Next we pass to the limit as 𝑅 → +∞: we have

lim𝑅→+∞

𝜏 ∧ 𝜏𝑅= 𝜏, (112)

and by the growth assumption (95)

𝑢 (𝜏 ∧ 𝜏

𝑅, 𝑆𝑡,𝑥

𝜏∧𝜏𝑅

)≤ 𝐶 exp(𝜆 sup

𝑡≤𝑠≤𝑇

𝑆𝑡,𝑥

𝑠

2

) . (113)

As 𝑅 → +∞

𝑢 (𝑡, 𝑥) ≥ 𝐸𝐺[𝑢 (𝜏, 𝑆

𝑡,𝑥

𝜏)] ≥ 𝐸

𝐺[𝑓 (𝜏, 𝑆

𝑡,𝑥

𝜏)] . (114)

This shows that

𝑢 (𝑡, 𝑥) ≥ sup𝜏∈F𝑡,𝑇

𝐸𝐺[𝑓 (𝜏, 𝑆

𝑡,𝑥

𝜏)] . (115)

We conclude the proof by putting

𝜏0= inf {𝑠 ∈ [𝑡, 𝑇] | 𝑢 (𝑠, 𝑆𝑡,𝑥

𝑠) = 𝑓 (𝑠, 𝑆

𝑡,𝑥

𝑠)} . (116)

Since 𝐿𝑢 = 0 a.e., where 𝑢 > 𝜙, it holds

𝐸𝐺[∫

𝜏0∧𝜏𝑅

𝑡


𝑠) 𝑑𝑠] = 0 (117)

and from (98) we derive that

𝑢 (𝑡, 𝑥) = 𝐸𝐺[𝑢 (𝜏

0∧ 𝜏

𝑅, 𝑆𝑡,𝑥

𝜏0∧𝜏𝑅

)] . (118)

Repeating the previous argument to pass to the limit in 𝑅, weobtain

𝑢 (𝑡, 𝑥) = 𝐸𝐺[𝑢 (𝜏

0, 𝑆𝑡,𝑥

𝜏0

)] = 𝐸𝐺[𝑓 (𝜏

0, 𝑆𝑡,𝑥

𝜏0

)] . (119)

Therefore, we finish the proof.

5.2. Free Boundary Problem. Here we consider the freeboundary problem on a bounded cylinder. We denote thebounded cylinders as the form [0, 𝑇] × 𝐻

𝑛, where (𝐻

𝑛) is an

increasing covering of𝑅𝑑 (𝑑 = 1).Wewill prove the existenceof a strong solution to problem

max {𝐿𝑢, 𝑓 − 𝑢} = 0, in 𝐻(𝑇) := [0, 𝑇] × 𝐻,

𝑢|𝜕𝑃𝐻(𝑇)

= 𝑓,

(120)

where𝐻 is a bounded domain of 𝑅𝑑 and

𝜕𝑃𝐻(𝑇) := 𝜕𝐻 (𝑇) \ ({𝑇} × 𝐻) (121)

is the parabolic boundary of𝐻(𝑇).We assume the following condition on the payoff func-

tion.


Assumption 35. The payoff function 𝜉 = 𝑓(𝑆𝑡,𝑥𝑢) has the

following assumption expressed by the sublinear function:

−𝐺 (−𝐷2𝑓) ≥ 𝑐 in 𝐻, (122)

where 𝐺(⋅) is the sublinear function defined by (9).

Theorem 36. One assumes assumption 5.1 holds. Problem(120) has a strong solution 𝑢 ∈ S1

𝑙𝑜𝑐(𝐻(𝑇)) ∩ 𝐶(𝐻(𝑇)).

Moreover 𝑢 ∈ S𝑝𝑙𝑜𝑐(𝐻(𝑇)) for any 𝑝 > 1.

Proof. The proof is based on a standard penalization tech-nique (see [18]). We consider a family (𝛽

𝜀)𝜀∈[0,1]

of smoothfunctions such that, for any 𝜀, function 𝛽

𝜀is increasing

bounded on 𝑅 and has bounded first order derivative, suchthat

𝛽𝜀(𝑠) ≤ 𝜀, 𝑠 > 0,

lim𝜀→0

𝛽𝜀(s) = −∞, 𝑠 < 0.

(123)

We denote by 𝑓𝛿 the regularization of 𝑓 and consider thefollowing penalized and regularized problem and denote thesolution as 𝑢

𝜀,𝛿

𝐿𝑢 = 𝛽𝜀(𝑢 − 𝑓

𝛿) , in 𝐻(𝑇) ,

𝑢|𝜕𝑃𝐻(𝑇)

= 𝑓𝛿;

(124)

Lions [24], Krylov [25], and Nisio [26] prove that problem(124) has a unique viscosity solution 𝑢

(𝜀,𝛿)∈ 𝐶

2,𝛼(𝐻(𝑇)) ∩

𝐶(𝐻(𝑇)) with 𝛼 ∈ [0, 1].Next, we firstly prove the uniform boundedness of the

penalization term:

𝛽𝜀(𝑢

𝜀,𝛿− 𝑓

𝛿)≤ 𝑐, in 𝐻(𝑇) , (125)

with 𝑐 independent of 𝜀 and 𝛿.By construction 𝛽

𝜀≤ 𝜀, it suffices to prove the lower

bound in (125). By continuity, 𝛽𝜀(𝑢𝜀,𝛿−𝑓

𝛿) has a minimum 𝜁

in𝐻(𝑇) and we may suppose

𝛽𝜀(𝑢

𝜀,𝛿(𝜁) − 𝑓

𝛿(𝜁)) ≤ 0; (126)

otherwise we prove the lower bound. If 𝜁 ∈ 𝜕𝑃𝐻(𝑇) then

𝛽𝜀(𝑢


𝛿(𝜁)) = 𝛽

𝜀(0) = 0. (127)

On the other hand, if 𝜁 ∈ 𝐻(𝑇), then we recall that 𝛽𝜀is

increasing and consequently 𝑢(𝜀,𝛿)

− 𝑓𝛿 also has a (negative)

minimum in 𝜁. Thus, we have

𝐿𝑢𝜀,𝛿(𝜁) − 𝐿𝑓

𝛿(𝜁) ≥ 0 ≥ 𝑢


𝛿(𝜁) . (128)

By Assumption 35 on 𝑓, we have that 𝐿𝑓𝛿(𝜁) is boundeduniformly in 𝛿. Therefore, by (128), we deduce

𝛽𝜀(𝑢

(𝜀,𝛿)(𝜁) − 𝑓

𝛿(𝜁)) = 𝐿𝑢

(𝜀,𝛿)(𝜁) ≥ 𝐿𝑓

𝛿(𝜁) ≥ 𝑐, (129)

where 𝑐 is a constant independent of 𝜀, 𝛿 and this proves (125).Secondly, we use theS𝑝 interior estimate combined with

(125), to infer that, for every compact subset 𝐷 in 𝐻(𝑇) and𝑝 ≥ 1, the norm ‖𝑢

𝜀,𝛿‖S𝑝(𝐷) is bounded uniformly in 𝜀 and 𝛿.

It follows that (𝑢𝜀,𝛿) converges as 𝜀, 𝛿 → 0 weakly in S𝑝 on

compact subsets of𝐻(𝑇) to a function 𝑢. Moreover

lim sup𝜀,𝛿

𝛽𝜀(𝑢

𝜀,𝛿− 𝑓

𝛿) ≤ 0, (130)

so that 𝐿𝑢 ≤ 𝑓 a.e. in 𝐻(𝑇). On the other hand, 𝐿𝑢 = 𝑓 a.e.in set {𝑢 > 𝑓}.

Finally, it is straightforward to verify that 𝑢 ∈ 𝐶(𝐻(𝑇))and assumes the initial-boundary conditions, by using stan-dard arguments based on themaximumprinciple and barrierfunctions.

Proof ofTheorem 33. Theproof ofTheorem 33 about the exis-tence theorem for the free boundary problem on unboundeddomains is similar to [27] by using Theorem 36 about theexistence theorem for the free boundary problem on theregular bounded cylindrical domain.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

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Research Article G-Doob-Meyer Decomposition and Its ... · e Doob decomposition theorem was proved by and is named for Doob [ ]. e analogous theorem in the contin-uous time case is