-
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
-
4 Journal of Applied Mathematics
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)
-
Journal of Applied Mathematics 5
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)
-
6 Journal of Applied Mathematics
(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
𝐻𝑢𝐼{𝑋𝑢(𝜔)∈[0,𝑐]}
𝑑𝑢 | F𝑡]
= 𝐸[𝐸[∫
∞
𝑆𝑐
𝐻𝑢𝐼{𝑋𝑢(𝜔)∈[0,𝑐]}
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
𝐻𝑢𝐼{𝑋𝑢(𝜔)∈[0,𝑐]}
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
𝑘𝑋𝑠] 𝑑𝑠 | F
𝑡]
≥ 𝐸[1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
𝑡]
− 𝐸[1
𝑘∫
𝑢+𝑘
𝑢
𝑋𝑠𝑑𝑠 | F
𝑡] ,
(44)
and, by the subadditive property of the sublinear expectation𝐸,
we derive that
𝐸[1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
𝑡] − 𝐸[
1
𝑘∫
𝑢+𝑘
𝑢
𝑋𝑠𝑑𝑠 | F
𝑡]
≥ 𝐸[1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
𝑡] −
1
𝑘∫
𝑢+𝑘
𝑢
𝐸 [𝑋𝑠| F
𝑡] 𝑑𝑠
≥ 𝐸[1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
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
𝑘𝑋𝑠] 𝑑𝑠 | F
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
𝑘𝑋𝑠] 𝑑𝑠 | F
𝑠0
] < 0, (47)
which is contradiction; we prove that (𝑋𝑡(𝜔) − 𝑝
𝑘𝑋𝑡(𝜔))/𝑘 is
a positive, measurable, and well-adapted process.
-
Journal of Applied Mathematics 7
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
𝑘𝑋𝑠] 𝑑𝑠 | F
𝑡]
= lim𝑘↓0
𝐸[1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
𝑡]
= 𝐸[lim𝑘↓0
1
𝑘∫
𝑡+𝑘
𝑡
𝑋𝑠𝑑𝑠 | F
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)
-
8 Journal of Applied Mathematics
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)
-
Journal of Applied Mathematics 9
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.
-
10 Journal of Applied Mathematics
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)
-
Journal of Applied Mathematics 11
We have
𝐸𝐺[∫
𝜏
𝑡
𝐿𝑢𝜀,𝑅,𝑛
(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
𝜀,𝑅,𝑛(𝑠, 𝑆
𝑡,𝑥
𝑠) − 𝐿𝑢
𝜀,𝑅(𝑠, 𝑆
𝑡,𝑥
𝑠)𝑑𝑠]
= ess sup𝑃∈𝑄
𝐸𝑃[∫
𝜏
𝑡
𝐿𝑢
𝜀,𝑅,𝑛(𝑠, 𝑆
𝑡,𝑥
𝑠)
− 𝐿𝑢𝜀,𝑅(𝑠, 𝑆
𝑡,𝑥
𝑠)𝑑𝑠] .
(105)
Since 𝜏 ≤ 𝜏𝑅
ess sup𝑃∈𝑄
𝐸𝑃[∫
𝜏
𝑡
𝐿𝑢
𝜀,𝑅,𝑛(𝑠, 𝑆
𝑡,𝑥
𝑠) − 𝐿𝑢
𝜀,𝑅(𝑠, 𝑆
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
𝐿𝑢𝜀,𝑅,𝑛
(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
𝐿𝑢 (𝑠, 𝑆𝑡,𝑥
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
𝐿𝑢 (𝑠, 𝑆𝑡,𝑥
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.
-
12 Journal of Applied Mathematics
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
𝜀,𝛿(𝜁) − 𝑓
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
𝜀,𝛿(𝜁) − 𝑓
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.
References
[1] H. P. McKean Jr., “Appendix: a free boundary problem forthe
heat equation arising from a problem in mathematicaleconomics,”
Industrial Management Review, vol. 6, pp. 32–39,1965.
[2] A. Bensoussan, “On the theory of option pricing,” Acta
Appli-candae Mathematicae, vol. 2, no. 2, pp. 139–158, 1984.
[3] F. Black and M. Scholes, “The pricing of options and
corporateliabilities,” Journal of Political Economy, vol. 81, no.
3, pp. 637–659, 1973.
[4] I. Karatzas, “On the pricing of American options,”
AppliedMathematics & Optimization, vol. 17, no. 1, pp. 37–60,
1988.
[5] N. EL Karoui and I. Karatzas, “Integration of the
optimalrisk in a stopping problem with absorption,” in Séminaire
deProbabilités XXIII, vol. 1372 of Lecture Notes inMathematics,
pp.405–420, Springer, Berlin, Germany, 1989.
[6] N. EL Karoui and I. Karatzas, “A new approach to the
Skorohodproblem and its applications,” Stochastics and Stochastic
Reports,vol. 34, pp. 57–82, 1991.
[7] J. L. Doob, Stochastic Processes, John Wiley & Sons, New
York,NY, USA, 1953.
[8] P. A. Meyer, “A decomposition theorem for
supermartingales,”Illinois Journal of Mathematics, vol. 6, pp.
193–205, 1962.
[9] P.-A. Meyer, “Decomposition of supermartingales: the
unique-ness theorem,” Illinois Journal of Mathematics, vol. 7, pp.
1–17,1963.
-
Journal of Applied Mathematics 13
[10] D. O. Kramkov, “Optional decomposition of
supermartingalesand hedging contingent claims in incomplete
security markets,”Probability Theory and Related Fields, vol. 105,
no. 4, pp. 459–479, 1996.
[11] R. Frey, “Superreplication in stochastic volatility models
andoptimal stopping,” Finance and Stochastics, vol. 4, no. 2, pp.
161–187, 2000.
[12] F. Knight, Risk, Uncertainty, and Profit, Houghton
MifflinHarcourt, Boston, Mass, USA, 1921.
[13] S. G. Peng, “Survey on normal distributions, central limit
theo-rem, Brownianmotion and the related stochastic calculus
undersublinear expectations,” Science in China, Series
A:Mathematics,vol. 52, no. 7, pp. 1391–1411, 2009.
[14] S. G. Peng, “Nonlinear expectations and stochastic
calculusunderuncertainty,” http://arxiv.org/abs/1002.4546.
[15] L. G. Epstein and S. Ji, “Ambiguous volatility and asset
pricingin continuous time,” Review of Financial Studies, vol. 26,
no. 7,pp. 1740–1786, 2013.
[16] J. Vorbrink, “Financial markets with volatility
uncertainty,”Journal of Mathematical Economics, vol. 53, pp. 64–78,
2014.
[17] W. Chen, “G-consistent price system and bid-ask pricingfor
European contingent claims underKnightian
uncertainty,”http://arxiv.org/abs/1308.6256.
[18] A. Friedman, Variational Principles and Free-Boundary
Prob-lems, Pure and Applied Mathematics, John Wiley & Sons,
NewYork, NY, USA, 1982.
[19] X. P. Li and S. G. Peng, “Stopping times and related
Itô’scalculus with G-Brownian motion,” Stochastic Processes
andTheir Applications, vol. 121, no. 7, pp. 1492–1508, 2011.
[20] M. Hu and S. Peng, “Extended conditional G-expectations
andrelated stoppingtimes,” http://arxiv.org/abs/1309.3829.
[21] Y. Z. Song, “Some properties on G-evaluation and its
applica-tions to G-martingale decomposition,” Science China
Mathe-matics, vol. 54, no. 2, pp. 287–300, 2011.
[22] W. Chen, “Time consistent G-expectationand bid-ask
dynamicpricing mechanisms for contingent claims under
uncertainty,”http://arxiv.org/abs/1111.4298v1.
[23] M. Hu, S. Ji, S. Peng, and Y. Song, “Comparison
theorem,Feynman-Kacformula and Girsanov transformation for
BSDEsdriven by G-Brownian motion,”
http://arxiv.org/abs/1212.5403v1.
[24] P.-L. Lions, “Optimal control of diffusion processes and
HJBequations part II: viscosity and uniqueness,”Communications
inPartial Differential Equations, vol. 8, no. 11, pp. 1229–1276,
1983.
[25] N. V. Krylov,ControlledDiffusion Processes, Springer,
NewYork,NY, USA, 1980.
[26] M. Nisio, Stochastic Control Theory, vol. 9 of ISI Lectures
Notes,Tata Institute, MacMillan India, 1981.
[27] M. Di Francesco, A. Pascucci, and S. Polidoro, “The
obstacleproblem for a class of hypoelliptic ultraparabolic
equations,”Proceedings of the Royal Society A: Mathematical,
Physical andEngineering Sciences, vol. 464, no. 2089, pp. 155–176,
2008.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of