#### Abstract

The target of this paper is to establish the bid-ask pricing framework for the American contingent claims against risky assets with G-asset price systems on the financial market under Knightian uncertainty. First, we prove G-Dooby-Meyer decomposition for G-supermartingale. Furthermore, we consider bid-ask pricing American contingent claims under Knightian uncertainty, by using G-Dooby-Meyer decomposition; we construct dynamic superhedge strategies for the optimal stopping problem and prove that the value functions of the optimal stopping problems are the bid and ask prices of the American contingent claims under Knightian uncertainty. Finally, we consider a free boundary problem, prove the strong solution existence of the free boundary problem, and derive 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 the pricing of the American option is by McKean [1]. There the problem of pricing the American option is transformed into a Stefan or free boundary problem. Solving the latter, McKean writes the American option price explicitly up to knowing a certain function, the optimal stopping boundary.

Bensoussan [2] presents a rigorous treatment for American contingent claims that can be exercised at any time before or at maturity. He adapts the Black and Scholes [3] methodology of duplicating the cash flow from such a claim to this situation by skillfully managing a self-financing portfolio that contains only the basic instruments of the market, that is, the stocks and the bond, and that entails no arbitrage opportunities before exercise. Bensoussan shows that the pricing of such claims is indeed possible and characterized the exercise time by means of an appropriate optimal stopping problem. In the study of the latter, Bensoussan employs the so-called “penalization method,” which forces rather stringent boundedness and regularity conditions on the payoff from the contingent claim.

From the theory of optimal stopping, it is well known that the value process of the optimal stopping problem can be characterized as the smallest supermartingale majorant to the stopping reward. Based on the Doob-Meyer decomposition for the supermartingale, a “martingale” treatment of the optimal stopping problem is used for handling pricing of the American option by Karatzas [4] and El Karoui and Karatzas [5, 6].

The Doob decomposition theorem was proved by and is named for Doob [7]. The analogous theorem in the continuous time case is the Doob-Meyer decomposition theorem proved by Meyer in [8, 9]. For the pricing American option problem in incomplete market, Kramkov [10] constructs the optional decomposition of supermartingale with respect to a family of equivalent local martingale measures. He calls such a representation optional because, in contrast to the Doob-Meyer decomposition, it generally exists only with an adapted (optional) process C. He applies this decomposition to the problem of hedging European and American style contingent claims in the setting of incomplete security markets. Using the optional decomposition, Frey [11] considers construction of superreplication strategies via optimal stopping which is similar to the optimal stopping problem that arises in the pricing of American-type derivatives on a family of probability space with equivalent local martingale measures.

For the realistic financial market, the asset price in the future is uncertain, the probability distribution of the asset price in the future is unknown, which is called Knightian uncertainty [12]. The probability distribution of the nature state in the future is unknown; investors have uncertain subjective belief, which makes their consumption and portfolio choice decisions uncertain and leads the uncertain asset price in the future. Pricing contingent claims against such assets under Knightian uncertainty is an open problem. Peng in [13, 14] constructs G frame work which is an analysis tool for nonlinear system and is applied in pricing European contingent claims under volatility uncertainty [15, 16].

The target of this paper is to establish the bid-ask pricing framework for the American contingent claims against risky assets with G-asset price systems (see [17]) on the financial market under Knightian uncertainty. Firstly, on sublinear expectation space, by using potential theory and sublinear expectation theory we construct G-Doob-Meyer decomposition for G-supermartingale, that is, a right continuous G-supermartingale could be decomposed as a G-martingale and a right continuous increasing process and the decomposition is unique. Second, we define bid and ask prices of the American contingent claim against the assets with G-asset price systems and apply the G-Doob-Meyer decomposition to prove that the bid and ask prices of American contingent claims under Knightian uncertainty could be described by the optimal stopping problems. Finally, we present a free boundary problem, and by using the penalization technique (see [18]) we derive that if there exists strong supersolution to the free boundary problem, then the strong solution to the free boundary problem exists. And by using truncation and regularization technique, we prove that the strong solution to the free boundary problem is the value function of the optimal stopping problem which is corresponding with pricing problem of the American contingent claim under Knightian 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 decomposition, we construct dynamic superhedge strategies for the optimal stopping problem and prove that the solution of the optimal stopping problem is the bid and ask prices of the American contingent claims under Knightian uncertainty. In Section 5, we consider a free boundary problem, prove the strong solution existence of the free boundary problem, and derive that the solution of the optimal stopping problem is equivalent to the strong solution to the free boundary problem.

#### 2. Preliminaries

Let be a given set and let be a linear space of real valued functions defined on containing constants. The space is also called the space of random variables.

*Definition 1. *A sublinear expectation is a functional satisfying(i)monotonicity: (ii)constant preserving: (iii)subadditivity: for each , (iv)positive homogeneity: The triple is called a sublinear expectation space.

In this section, we mainly consider the following type of sublinear expectation spaces : if then for , where denotes the linear space of functions satisfying

For each fixed , we take as our null space and denote as the quotient space. We set and extend to its completion under . Under the sublinear expectation can be continuously extended to the Banach space . Without loss generality, we denote the Banach space as . For the G-frame work, we refer to [13, 14].

In this paper we assume that , , , and are positive constants such that and .

*Definition 2. *Let and be two random variables in a sublinear expectation space ; and are called identically distributed, denoted by if

*Definition 3. *In a sublinear expectation space , a random variable is said to be independent of another random variable , if

*Definition 4 (G-normal distribution). *A random variable on a sublinear expectation space is called G-normal distributed if where is an independent copy of .

We denote by the collection of all symmetric matrices. Let be G-normal distributed random vectors on ; we define the following sublinear function:

*Remark 5. *For a random variable on the sublinear space , there are four typical parameters to character : where and describe the uncertainty of the mean and the variance of , respectively.

It is easy to check that if is G-normal distributed, then and we denote the G-normal distribution as . If is maximally distributed, then and we denote the maximal distribution (see [14]) as .

Let as Borel field subsets of . We are given a family of Borel subfields of , such that

*Definition 6. *We call a -dimensional stochastic process on a sublinear expectation space , if, for each , is a -dimensional random vector in .

*Definition 7. *Let and be -dimensional stochastic processes defined on a sublinear expectation space , for each ; is called the finite dimensional distribution of . and are said to be identically distributed, that is, , if where .

*Definition 8. *A process on the sublinear expectation space is called a G-Brownian motion if the following properties are satisfied:(i);(ii)For each , the increment is G-normal distributed by and is independent of , for each and .

From now on, the stochastic processes we will consider in the rest of this paper are all in the sublinear space .

#### 3. G-Doob-Meyer Decomposition for G-Supermartingale

*Definition 9. *A G-supermartingale (resp., G-submartingale) is a real valued process , well adapted to the family, such that If equality holds in (ii), the process is a G-martingale.

We will consider right continuous G-supermartingales; then if is right continuous G-supermartingale, (ii) in (16) holds with replaced by .

*Definition 10. *Let be an event in ; one defines capacity of as where is indicator function of event .

*Definition 11. *Process and are adapted to the filtration . One calls equivalent to , if and only if

For a right continuous G-supermartingale with is right continuous function of ; we can find a right continuous G-supermartingale equivalent to by defining Without loss generality, we denote .

*Definition 12. *For a positive constant , one defines stop time in as a positive, random variable such that .

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

Let be a right continuous G-supermartingale, denote as the last element of the process , and then the process is a G-supermartingale.

*Definition 13. *A right continuous increasing process is a well adapted stochastic process such that(i) a.s,(ii)for almost every , the function is positive, increasing, and right continuous. Let ; one will say that the right continuous increasing process is integrable if .

*Definition 14. *An increasing process is called natural if for every bounded, right continuous G-martingale we have

Lemma 15. *If is an increasing process and is bounded, right continuous G-martingale, then **In particular, condition (20) in Definition 14 is equivalent to *

*Proof. *For a partition of , with , we define Since is G-martingale and we finish the proof of the Lemma.

*Definition 16. *A positive right continuous G-supermartingale with is called a potential.

*Definition 17. *For , a process is said to be uniformly integrable on if

*Definition 18. *Let , and let be a right continuous process; we will say that it belongs to the class (GD) on this interval, if all the random variables are uniformly integrable and is stop time bounded by . If belongs to the class (GD) on every interval , , it will be said to belong locally to the class (GD).

If is an integrable right continuous, increasing process, then process is a negative G-supermartingale, and is a potential of the class (GD), which we will call the potential generated by .

Proposition 19. *(1) Any right continuous G-martingale belongs locally to class (GD).**(2) Any right continuous G-supermartingale , which is bounded from above, belongs locally to class (GD).**(3) Any right continuous supermartingale , which belongs locally to class (GD) and is uniformly integrable, belongs to class (GD).*

*Proof. *(1) If and is a stop time, , then G-martingale process has . Hence As , we have as ; then as , from which we prove (1).

(2) If and is a stop time, , then G-supermartingale process has . Suppose that is negative; then we complete the proof of (2) by using similar argument in proof (1).

(3) is uniformly integrable; we set The first part on the right-hand of the above equation is a G-martingale and equivalent to a right continuous process, and from (1) we know that it belongs to class (GD). We denote the second part in the above equation as ; it is a potential, that is, a positive right continuous G-supermartingale, and a.s. Next we will prove that belongs to class (GD). Since both and are stop times Consider that and locally belongs to (GD); that is, , which prove that We complete the proof.

Lemma 20. *Let be a right continuous G-supermartingale and a sequence of decomposed right continuous G-supermartingale: **where is G-martingale and is right continuous increasing process. Suppose that, for each , converge to in the topology, and are uniformly integrable in . Then the decomposition problem is solvable for the G-supermartingale ; more precisely, there are a right continuous increasing process and a G-martingale , such that .*

*Proof. *We denote by the weak topology ; a sequence of integrable random variables converges to a random variable in the -topology, if and only if is integrable, and Since are uniformly integrable in , by the properties of the sublinear expectation there exists a -convergent subsequence converging in the -topology to the random variables , for all rational values of . To simplify the notations, we will use converging to in the -topology for all rational values of . An integrable random variable is -measurable if and only if it is orthogonal to all bounded random variables such that ; it follows that is -measurable. For , and rational, where denote any set.

As converge to in topology, which is in a stronger topology than , the converge to random variables for rational, and the process is G-martingale; then there is a right continuous G-martingale , defined for all values of , such that for each rational . We define ; is a right continuous increasing process or at least becomes so after a modification on a set of measure zero. We complete the proof.

Lemma 21. *Let be a potential and belong to class (GD). One considers the measurable, positive, and well-adapted processes with the property that the right continuous increasing processes **are integrable, and the potentials they generate are majorized by . Then, for each , the random variables of all such processes are uniformly integrable.*

*Proof. *It is sufficient to prove that the are uniformly integrable.

(1) First we assume that is bounded by some positive constant ; then , and the uniform integrability follows.

We have that By using the subadditive property of the sublinear expectation , we derive that (2) In order to prove the general case, it will be enough to prove that any such that is majorized by is equal to a sum , where (i) generates a potential bounded by , and (ii) is smaller than some number , independent of , such that as . Define Set as goes to infinity ; therefore , and class (GD) property implies that . is a stop time, and before time . Hence from which we prove (ii). We will prove (i); first we prove that is bounded by :where we set and use Inequality (40) holds for each , for every rational and for every in consideration of the right continuity, which complete the proof.

Lemma 22. *Let be a potential and belong to class (GD), is a positive number, define , and then is a G-supermartingale. Denote by a right continuous version of ; then is potential.**Use the same notations as in Lemma 21. Let be a positive number, and . The process verifies the assumptions of Lemma 21, and their potentials increase to as .*

*Proof. *If For , We have that and, by the subadditive property of the sublinear expectation , we derive that Hence, we derive that for any , such that If there exits such that , the right continuous of implies that there exists such that on the interval . Thus which is contradiction; we prove that is a positive, measurable, and well-adapted process.

Since is right continuous G-supermartingale we finish the proof.

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

Theorem 23. *A potential belongs to class (GD) if and only if it is generated by some integrable right continuous increasing process.*

Theorem 24 (G-Doob-Meyer’s decomposition). *(1) is a right continuous G-supermartingale if and only if it belongs to class (GD) on every finite interval. More precisely, is then equal to the difference of a G-martingal and a right continuous increasing process : **(2) If the right continuous increasing process is natural, the decomposition is unique.*

*Proof. *(1) The necessity is obvious. We will prove the sufficiency; we choose a positive number and define the is a right continuous G-supermartingale of the class (GD), and by Theorem 23 there exists the following decomposition where is a G-martingal and is a right continuous increasing process.

Let , as in Lemma 22 the expression of that depend only on the values of on intervals , with small enough. As , they do not vary anymore once has reached values greater than , as again Lemma 20; we finish the proof of the Theorem.

(2) Assume that admits both decompositions: where and are G-martingale and , are natural increasing process. We define Then is a G-martingale, and, for every bounded and right continuous G-martingale , from Lemma 15 we have where , is a sequence of partitions of with converging to zero as . Since and are both G-martingale, we have For an arbitrary bonded random variable , we can select to be a right continuous equivalent process of , and we obtain that . We set ; therefore .

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

Theorem 25. * is a right continuous G-supermartingale; there exists a right continuous increasing process and adapted process , such that **where is G-Brownian motion.*

#### 4. Superhedging Strategies and Optimal Stopping

##### 4.1. Financial Model and G-Asset Price System

We consider a financial market with a nonrisky asset (bond) and a risky asset (stock) continuously trading in market. The price of the bond is given by where is the short interest rate; we assume a constant nonnegative short interest rate. We assume the risk asset with the G-asset price system (see [17]) on sublinear expectation space under Knightian uncertainty, for given and where is the generalized G-Brownian motion. The uncertain volatility is described by the G-Brownian motion . The uncertain drift can be rewritten as where is the asset return rate [22]. Then the uncertain risk premium of the G-asset price system is uncertain and distributed by [22], where is the interest rate of the bond.

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

Theorem 26 (G-Girsanov theorem). *Assume that is generalized G-Brownian motion on , and is defined by (61); there exists G-expectation space such that is G-Brownian motion under the G-expectation , and *

By the G-Girsanov theorem, the G-asset price system (58) of the risky asset can be rewritten on as follows: then by G-Itô formula we have

##### 4.2. Construction of Superreplication Strategies via Optimal Stopping

We consider the following class of contingent claims.

*Definition 27. *One defines a class of contingent claims with the nonnegative payoff having the following form: for some function such that the process is bounded below and càdlàg.

We consider a contingent claim with payoff defined in Definition 27 written on the stockes with maturity . We give definitions of superhedging (resp., subhedging) strategy and ask (resp., bid) price of the claim .

*Definition 28. *(1) A self-financing superstrategy (resp. substrategy) is a vector process (resp., ), where is the wealth process, is the portfolio process, and is the cumulative consumption process, such that where is an increasing, right continuous process with . The superstrategy (resp., substrategy) is called feasible if the constraint of nonnegative wealth holds (2) A superhedging (resp. subhedging) strategy against the contingent claim is a feasible self-financing superstrategy (resp., substrategy ) such that (resp., ). We denote by (resp., ) the class of superhedging (resp., subhedging) strategies against , and if (resp., ) is nonempty, is called superhedgeable (resp., subhedgeable).

(3) The ask-price at time of the superhedgeable claim is defined as and bid-price at time of the subhedgeable claim is defined as

Under uncertainty, the market is incomplete and the superhedging (resp., subhedging) strategy of the claim is not unique. The definition of the ask-price implies that the ask-price is the minimum amount of risk for the buyer to superhedging the claim; then it is coherent measure of risk of all superstrategies against the claim for the buyer. The coherent risk measure of all superstrategies against the claim can be regarded as the sublinear expectation of the claim; we have the following representation of bid-ask price of the claim via optimal stopping (Theorem 31).

Let be a filtration on G-expectation space , and and be -stopping times such that a.s. We denote by the set of all finite -stopping times with .

For given and , we define the function as the value function of the following optimal-stopping problem:

Proposition 29. *Consider two stopping times on filtration . Let denote some adapted and RCLL-stochastic process, which is bounded below. Then we have for two points and *

*Proof. *By the consistent property of the conditional G-expectation, for , , and thus we have There exists a sequence as , such that notice that we prove the Proposition.

Proposition 30. *The process is a G-supermartingale in .*

*Proof. *By Proposition 29, for Since , we have Thus, we derive that We prove the Proposition.

Theorem 31. *Assume that the financial market under uncertainty consists of the bond which has the price process satisfying (57) and risky assets with the price processes as the G-asset price systems (58) and can trade freely; the contingent claim which is written on the risky assets with the maturity has the class of the payoff defined in Definition 27, and the function is defined in (71). Then there exists a superhedging (resp., subhedging) strategy for , such that the process defined by **is the ask (resp., bid) price process against .*

*Proof. *The value function for the optimal stop time is a G-supermartingale; it is easily to check that is G-supermartingale. By G-Doob-Meyer decomposition Theorem 24where is a G-martingale and is an increasing process with . By G-martingale representation theorem [14, 21] where , is a G-martingale, and is an increasing process with . From the above equation, we have hence is a superhedging strategy.

Assume that is a superhedging strategy against ; then Taking conditional G-expectation on the both sides of (84) and noticing that the process is an increasing process with , we derive which implies that from which we prove that is the ask price against the claim at time . Similarly we can prove that is the bid price against the claim at time .

#### 5. Free Boundary and Optimal Stopping Problems

For given , , and , the G-asset price system (58) of the risky asset can be rewritten as follows:

We define the following deterministic function: where From Theorem 31 the price of an American option with expiry date and payoff function is the value function of the optimal stopping problem: We define operator as follows: where is the sublinear function defined by (9). We consider the free boundary problem Denote for And, for any compact subset of , we denote as the space of functions .

*Definition 32. *A function is a strong solution of problem (92) if almost everywhere in and it attains the final datum pointwisely. A function is a strong supersolution of problem (92) if .

We will prove the following existence results.

Theorem 33. *If there exists a strong supersolution of problem (92) then there also exists a strong solution of (92) such that in . Moreover for any and consequently, by the embedding theorem we have for any .*

Theorem 34. *Let be a strong solution to the free boundary problem (92) such that **form some constants , with sufficiently small so that **holds. Then we have **that is, the solution of the free boundary problem is the value function of the optimal stopping problem. In particular such a solution is unique.*

##### 5.1. Proof of Theorem 34

We employ a truncation and regularization technique to exploit the weak interior regularity properties of ; for we set for , , and, for denoting by the first exit time of from , it is easy check that is finite. As a first step we prove the following result: for every and such that , it holds thatFor fixed, positive, and small enough , we consider a function on with compact support and such that on . Moreover we denote by a regularizing sequence obtained by convolution of with the usual mollifiers; then for any we have and By G-Itô formula we have which implies that We have and, by dominated convergence, We have by sublinear expectation representation theorem (see [14]) there exists a family of probability space , such that Since where , for some , is the transition density of the solution of where is Wiener process in probability space and is adapted process such that . By Hölder inequality, we have ()