Martingale Property and Capacity under G-Framework

The main purpose of this article is to study the symmetric martingale property and capacity defined by G-expectation introduced by Peng (cf. http://arxiv.org/PS_cache/math/pdf/0601/0601035v2.pdf ) in 2006. We show that the G-capacity can not be dynamic, and also demonstrate the relationship between symmetric G-martingale and the martingale under linear expectation. Based on these results and path-wise analysis, we obtain the martingale characterization theorem for G Brownian motion without Markovian assumption. This theorem covers the Levy's martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy's theorem.


INTRODUCTION
In 1969, Robert C. Merton introduced stochastic calculus to finance, see [20], and indeed to the broader field of economics, beginning an amazing decade of developments.The most famous pricing formula for the European call option was given by Black and Scholes in 1973, see [2].As these developments unfolded, Cox and Ross [6] notice that, without loss of generality, to price some derivative security as an option, one would get the correct result by assuming that all of the securities have the same expected rate of return.This is the "risk neutral" pricing method.This notion is first given by Harrison and Kreps [14], who formalize (under conditions) the near equivalence of the absence arbitrage with the existence of some new "risk neutral" probability measure under which all expected rates of return are indeed equal to the current risk free rate.To get this "risk-neutral" probability measure, Harrison and Kreps apply Girsanov's theorem to change the measure.
Black & Scholes's result and Cox's result have been widely used since it appeared.But their work deeply depends on certain assumptions, e.g., the interest rate and the volatility of the stock price remain constant and known.In fact, the interest rate and the volatility of the stock price are not always constant and known, which are called mean uncertainty and volatility uncertainty.As for the mean uncertainty, Girsanov's theorem or Peng's g-expectation is a powerful tool to solve the problem, see [3] and [11].How to deal with the volatility uncertainty is a big problem, see [1], [15], [19], [27].As in [9], the main difficulty is that we have to deal with a series of probability measures which are not absolutely continuous with respect to one single probability measure.It shows that this problem can not be solved in a given probability space.
In 2006, Peng made a change to the heat equation that Brownian motion satisfies, see [23], [24], and constructed the G-normal distribution via the modified heat equation.With this G-normal distribution, a nonlinear expectation is given which is called G-expectation and the related conditional expectation is constructed, which is a kind of dynamic coherent risk measure introduced by Delbaen in [7].Under this framework, the canonical process is a G-Brownian motion.The stochastic calculus of Itô's type with respect to the G-Brownian motion and the related Itô's formula are also derived.G-Brownian motion, different from Brownian motion in the classical case, is not defined on a given probability space.
It is interesting to get a pricing formula (Black-Scholes formula) in G-framework.To do this, it is important to give the Girsanov theorem in this framework.Its proof depends heavily on the martingale characterization of Brownian motion, due to Lèvy.This theorem enables us to recognize a Brownian motion just with one or two martingale properties of a process.
give the martingale characterization of G-Brownian motion when the corresponding G-heat equation is uniformly parabolic, see in section 5. G-expectation theory has received more attention since Peng's basic paper appeared, see [23], [24], [25].Soner et al [21] study the G-martingale problem under some condition, and investigate the representation theorem for all G-martingales including the non-symmetric martingale based on a class of backward stochastic differential equations.In the current work, we are focused on symmetric G-martingale, our method is totally different from the method in [21].As for the martingale characterization for G-Brownian motion, our method is different from [29], which is based on viscosity solution theory for nonlinear parabolic equation.But in this paper, the path wise analysis is important to underly our result, which is a different approach from [29].Meanwhile, the result in this paper is very useful for further application in finance, especially for option pricing problems with volatility uncertainty.By using the result in this paper, we can prove the Girsanov type theorem under G-framework and the pricing formula.
The rest of the paper is organized as follows.In section 2, we review the G-framework established in [23] and adapt it according to our objective.In section 3, we investigate some properties of G-expectation by using stochastic control, define the capacity via G-expectation, and show that it is not filtration consistent.In section 4, we investigate the relation between the symmetric Gmartingale and the martingale under each probability measure P v , when the corresponding G-heat equation is uniformly parabolic.In section 5, based on path wise analysis, we give the martingale characterization for G-Brownian motion.The last section is conclusion and discussion about this work and future work.
This space is used throughout the rest of this paper.For each T > 0, consider the following spaces of ąřrandom variables ąś where l i p(R m ) is the collection of bounded Lipschitz continuous functions on R m .
Obviously, it holds that . We further denote We set B t (ω) = ω t , and define where u(t, x) is the viscosity solution to the following G-heat equation For any where ϕ m is obtained via the backward deduction: . . .
Remark 2.2.As in [23], the G-expectation satisfies Besides (a) (d), the conditional G-expectation still satisfies the properties: Remark 2.4.By theory of stochastic control as in [30], we know that, for any fixed T > 0, ) , E is the linear expectation under weiner measure, and G-expectation becomes linear expectation when σ 0 = 1, {B t } t≥0 is Brownian motion under Weiner measure.
Here Λ = {v, v is progressively measurable and quadratic integrable s.t., Lemma 2.5.For any X ∈ L 0 i p ( ), if E G [|X |] = 0, then for any ω ∈ Ω, we have X (ω) = 0. Proof: Similarly, we can prove that for any • ) be the completion of (L 0 ip ( ), • ), then G-expectation and related conditional expectation can be continuously extended to the Banach space-(L 1 G ( ), • ).G-expectation satisfies properties (a) ∼ (d), and conditional G-expectation satisfies properties (a) ∼ (g).In the completion space, property ( f ) holds for any bounded random variable X , but we have , and X ≥ 0. Similarly, we can define for any 1 ≤ p ≤ p , and it holds for any L p G ( t ) (the proof can be found in [23]).
For p ≥ 1 and 0 < T < ∞ ( fixed T ).Consider the following type of simple processes G (0, T ), the related Bochner's integral is defined as and We can easily check that

G-Expectation and Related Properties
G-expectation is a kind of time consistent nonlinear expectations, which has the properties of linear expectation in Weiner space except linearity.In this section, we prove some fundamental properties of G-expectation.
We set First we will prove that an important class of random variable-bounded continuous functions belongs to the completion space-L 1 G ( ) [9]).

and this convergence is uniform with respect to P v . We have E G
Next we will prove the tightness of Λ. Lemma 3.2.Λ is tight, that is, for any > 0, there exists a compact set K ⊂ C([0, T ]) ⊂ Ω, such that for any P v ∈ Λ, P v (K c ) < , where K c is the complement of K.
Proof: For any continuous function x(t), t ∈ [0, T ], define By Arzela-Ascoli theorem and Prokhrov theorem (see Theorem 4.4.11 in [4]), to prove the tightness of Λ, we only need to prove that for any α > 0, for any P v ∈ Λ, we have For α > 0, by Proposition 7.2 in the Appendix we know that , such that f n monotonically converges to f .As Λ is tight, for any > 0, there exists a compact set K ∈ T , such that sup where K c is the complement of K. Since a compact set is closed in any metric space, we know that And by Dini's theorem on any compact set K, f n converges to f uniformly,

Capacity under G-Framework
Since the publication of Kolomogrov's famous book on probability, the study of "the nonlinear probability" theory named "capacity" has been studied intensively in the past decades, see [5], [8], [12], [16], [18], [22], [26].Hence it is meaningful to extend such a theory to G-framework, and this section contributes to such an extension.We shall now define a nonlinear measure through G-expectation and investigate its properties.(1) For any Borel set A, 0 ≤ P G (A) ≤ 1; (2) If A ⊂ B, then P G (A) ≤ P G (B); (4) If A n is an increasing sequence of Borel sets, then Remark 3.6.Here property (4) dose not hold for the intersection of decreasing sets.There are two ways to define capacity under G framework, see [10].
Let Λ be the closure of Λ under weak topology. ( P G (A) = sup P∈ Λ P v (A).
Then P G and P G all satisfy the properties in Remark 3.2.
We use the standard capacity related vocabulary: A set A is polar if P G (A) = 0, a property holds ąřquasisurely ąś (q.s.), if it holds outside a polar set.Here P G quasi-surely is equivalent to P G quasi-surely.Even in general, P G (A) Then by Theorem 59 in [10](page 24), P G (A) = P G (A) = 0. Thus, a property holds P G -quasi-surely if and only if it holds P G -quasi-surely.

but we can not define G conditional expectation. So far we can only define G conditional expectation for the random variables in L 1
G ( ), in the rest of the paper, we denote First we give the property of this Choquet capacity-P G : Proposition 3.8.Let p ≥ 1.
(1) If A is a polar set, then for any we get the result.
(3) By ( 2), we know that for any > 0, Then for every positive integer k, there exists n k > 0, such that = 0, then by Proposition 3.8, for any > 0, we have Let ↓ 0, for P G is a Choquet capacity, we know Remark 3.10.After defining the capacity, one important issue is whether I A ∈ L 1 G ( ), for any Borel set A ∈ .Here we give a counter example that there exists a sequence of Borel sets which do not belong to L 1 G ( ).Example 3.11.
For any sequences of random variables Then we can not define conditional G-expectation for I A , where A is any Borel set, and even for any open set, that means we can not define conditional G-capacity, and that is why we claim that G-expectation is not filtration consistent.

SYMMETRIC MARTINGALE IN G-FRAMEWORK
We begin with the definition of martingale in G-framework.
In this section, when the corresponding G-heat equation is uniformly parabolic, which means σ 0 > 0, we prove that the symmetric martingale is a martingale under each probability measure P v , and give the Doob's martingale inequality for symmetric martingales.

Path analysis
In this section, we give some path properties of quadratic variation process 〈B〉 t and the related stochastic integral From the definition of E G [•], we know that the canonical process B t is a quadratic integrable martingale under each P v .So they have a universal version of "quadratic variation process of B t ", and by the definition of stochastic integral with respect to G-Brownian motion, for any η ∈ M 2 G (0, T ), Proof: For we have sup

Representation theorem
In this section, we are concerned with the G-martingale when the corresponding G-heat equation is uniformly parabolic (σ 0 > 0).In the following, for any X ∈ L 0 ip ( T ), we will give X a representation in terms of stochastic integral.For this part, Peng gives the conjecture for representation theorem of G-martingales.Soner et al [21] prove this theorem for a large class of G-martingales by BSDE method.Here for the ease of exposition, we prove the theorem separately for some special martingales.
First we prove a lemma.Lemma 4.4.Suppose σ 0 > 0, if u is the solution of G-heat equation, then we have Proof: The proof follows that of Itô's formula.Due to the regularity of parabolic equation, see [18], we know u, u x , u x x are all uniformly continuous.Since Lipschitz continuous functions are dense in uniform continuous functions, we assume that u, u x , u x x are Lipschitz continuous. Let where and θ 1 , θ 2 are constants in [0, 1], which depend on ω, t and n.Hence, Here, K is the Lipschitz constant of u x x , and by similar argument we get Then, we have Similarly we get and Since u solves G-heat equation, we get Theorem 4.5.When σ 0 > 0, then for any X ∈ L 0 ip ( T ), we have Proof: When m = 1, for the regularity of the u, lim r→0 u(r, B t−r (ω)) = ϕ(B t (ω)).
By Lemma 4.4, and path analysis in Section 3, we know Due to the definition of G-expectation, E G [ϕ(B t )] = u(t, 0), so the result holds when m = 1.
By similar argument, we get When m = 2, for each By continuous dependence estimate theorem in [13], we know for each fixed t, u(T − t, x, 0) is lipschitz continuous and bounded with respect to x, then there exist That is Here η and z are different from (4.2).
Then by induction, we know the result is true for any X ∈ L 0 ip ( T ).

Properties for the Symmetric Martingale
From [23], Proof: and Then there exists a subsequence η k ⊂ η n such that T 0 (η k − η)d〈B〉 s −→ 0, q.s., By Theorem 4.1.42in [24], we have Since t 0 z s d B s is a quadratic integrable martingale for each P v , then sup On the other hand sup Therefore sup Hence, sup On the other hand sup The next corollary states that there is a universal version of the conditional expectation for the symmetric random variables under each P v .
Then M is a G-Brownian motion in the sense that M has the same finite distribution as the G-Brownian motion B.
Remark 5.2.The Lèvy's martingale characterization of Brownian motion in a probability space states that, B t is a continuous martingale with respect to t , and B 2 t − t is a t martingale, iff B t is a Brownian motion.Our martingale characterization of G-Brownian motion covers Lèvy's martingale characterization of Browninan motion, when σ 0 = 1.In a probability space, the quadratic process 〈B〉 t of Brownian motion B t almost sure equals to t with respect to the probability measure P.But in the G-framework, the quadratic process 〈B〉 t of G -Brownian motion B t is not a fixed function any more.Instead, it is a stochastic process in G-framework.The criteria (I I I) is the description of the nonsymmetric property for 〈B〉 t .Condition (IV) is reasonable, thanks to [10], for any G-Brownian motion, it has continuous path.
In a probability space, thanks to the characteristic function of normal distribution, Lèvy's martingale characteristic theorem of Brownian motion holds.But in G-framework, there is no characteristic function of "G-normal distribution"(see the definition in [23]), we have to find a different method to solve this problem.
Next we shall prove Theorem 5.1 in 4 steps.

Proof of Theorem 5.1:
Step 1. ∀η ∈ M 2 G (0, T ), we define the stochastic integral of Itô's type By conditions (I) ∼ (I V ) in Theorem 5.1, we have the following proposition.
Proposition 5.3.for any t j ≤ t j+1 < t i ≤ t i+1 , Step 2: The quadratic variation process 〈M 〉 t of M t exists.Defining the stochastic integral with respect to 〈M 〉 t in M 1 G (0, T ), we can get the isometric formula in the G-framework.Now we give the isometry formula in the G-framework.
First, we give a proposition.

Proposition 5.5. For all
Remark 5.7.We can prove Lemma 5.4 and 5.6 by the similar method to that in [23] and [29].Thus we omit the proof.
Next, we investigate the property of the quadratic process 〈M 〉 t .
Proof: We first prove that 〈M 〉 t is continuous in [0, T ] quasi-surely, which means 〈M 〉 t has continuous path outside a polar set.To prove the continuity of 〈M 〉 t , we only need to prove the continuity So the result holds.

Lemma 5.11. If u(t, x) is the viscosity solution to the G-heat equation, and σ
Proof: For 0 < σ 0 ≤ 1, then by Theorem 4.13 in [17], u(t, x) ∈ C 1,2 ((0, T ]×R), and u(t, x), u t (t, x) and u x x (t, x) are all uniformly continuous, u x (t, x) is bounded and continuous, see in [28].Then by Lemma 5.10, for any 0 < < v ≤ T , we have We also have where C is the Lipschitz constant of ϕ.
Without loss of generality, by the definition of G-Brownian motion, see [23], we only need to prove the case for m = 2: Based on Step 1 ∼ Step 4, M has the same finite distribution with G-Brownian motion B, we complete the proof of the Theorem 5.1.

DISCUSSION
In this paper, we investigate the properties of capacity defined by G-expectation, and prove that Gexpectation is not filtration consistent.Meanwhile by path-wise analysis, we prove that symmetric G-martingales are martingales under each P v .Based on these arguments, we obtain the martingale characterization theorem for a G-Brownian motion.The application of this framework in finance can be found in [9].Our martingale characterization theorem of G-Brownian motion includes Lèvy's martingale characterization theorem of Brownian motion.In the proof of classical Lèvy's martingale characterization theorem, the characterization function of normal distribution plays an important role.But in G-framework the characterization function of "G-normal" distribution dose not hold.
Our proof of martingale characterization of G-Brownian motion gives a totally different method to prove Lèvy's martingale characterization of Brownian motion.In our proof the compactness of Λ is essential.Based on our result one can investigate some elementary problems such as the martingale representation with respect to G-Brownian motion in G-framework.Proof: The proof is trivial, and hence omitted.

Remark 2 . 3 .
By properties (c) and (d), E G [•] satisfies E G [c] = c where c is a constant.
see Theorem 26 in[10].So the sets A n do not belong to L 1 G ( ).Actually, the next lemma tells us even not all the open Borel sets belong to L 1 G ( ).Lemma 3.12.There exists an open set A ∈ T , such that I A dose not belongs to L 1 G ( T ).Proof: We prove this result by contradiction.If for any open setA ∈ T , I A ∈ L 1 G ( T ) holds.For all the open sets satisfying A n ↓ φ, we have P G (A n ) ↓ 0. Because for any Borel set B, there exists compact sets {F n } ⊂ B satisfying P G (B \ F n ) ↓ 0, see[16].Then we have for any Borel set B ∈ T , I B ∈ L 1 G ( T ).But Example 3.11 shows that not all the Borel sets belong to L 1 G ( T ), which is a contradiction.

T 0 ηd M t , and prove that t 0 ηd
M s is a symmetric martingale under E G [•].
Definition 3.5.P G (A) = sup P v ∈Λ P v (A), for any Borel set A, where P v is the distribution of [10]is well defined, which means T 0 η s d B s is a P v local martingale.Similar arguments can be found in Lemma 2.10 in[10].Then, due to the Doob's martingale inequality for each P v , the following result holds.