Martingale representation theorem for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>G</mml:mi></mml:math>-expectation

Soner, H. Meté; Touzi, Nizar; Zhang, Jianfeng

doi:10.1016/j.spa.2010.10.006

Cited by 165 publications

(28 citation statements)

References 19 publications

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“…Other prominent examples of nonlinear expectations include the g-expectation, see Coquet et al [11], and the G-expectation introduced by Peng [27], [28], see also Dolinsky et al [18] or Denis et al [17]. We also refer to Cheridito et al [9] and Soner et al [29], [30] for the connection of the latter to fully non-linear PDEs and 2BSDEs.…”

Section: Introductionmentioning

confidence: 99%

Kolmogorov-type and general extension results for nonlinear expectations

Denk¹,

Kupper²,

Nendel³

2018

Banach J. Math. Anal.

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We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this paper is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell-Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels. 1 2 R. DENK, M. KUPPER, and M. NENDELthe help of Choquet's capacibility theorem [10] we obtain the uniqueness of such an extension in a certain class of expectations. Thus, our extension result can be viewed as a generalization of the Daniell-Stone extension theorem, which states that a linear expectation E which is continuous from above on a Riesz subspace M has a unique linear extensionĒ to L ∞ over the σ-algebra σ(M) generated by M. While for linear expectations the extension is still continuous from above, the same does not hold for convex expectations. Note that the continuity from above of a convex expectation E on L ∞ is a very strong condition which, in particular, implies that E(X) = E(Y ) whenever X = Y µ-almost surely for some probability measure µ, and that the representing probability measures in the dual representation of E are dominated by µ as well. However, nonlinear expectations are continuous from above on certain subspaces of L ∞ , see e.g. Cheridito et al. [7] and the references therein. Hence, nonlinear expectations can be constructed by defining them on a subspace M and extending them to L ∞ , the space of bounded σ(M)-measurable functions.In the second part of the paper we illustrate this extension procedure in a Kolmogorov type setting. That is, for an arbitrary index set I we construct nonlinear expectations on L ∞ (S I ), where S I is the I-th product of a Polish space S. To that end, we first consider a family of expectations E J on linear subsetsindexed by the set H of all finite subsets of I. In line with Peng [26], under the natural consistency condition E K (f ) = E J (f • pr JK ) for every f ∈ M K and all J, K ∈ H with K ⊂ J, where pr JK denotes the projection from M J to M K , the family (E J ) can be extended to the space M := {f • pr J : f ∈ M J , J ∈ H }. Moreover, if each E J is convex and continuous from above on M J the same also holds for the extension on M. Hence, by the general extension result, Theorem 3.10, from the first part, there exists a convex expectationĒ on L ∞ which is continuous from below, such thatĒ(f • pr J ) = E J (f ) for all f ∈ M J and J ⊂ I finite, see Theorem 4.6. The corresponding dual version in the sublinear case leads to Theorem 4.7, which is a robust version of Kolmogorov's extension theor...

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Section: Introductionmentioning

confidence: 99%

Kolmogorov-type and general extension results for nonlinear expectations

Denk¹,

Kupper²,

Nendel³

2018

Banach J. Math. Anal.

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“…As noted in the proof of Proposition 3.4 of Soner-Touzi-Zhang [12], the validity of (5.1) for X ∈ C b,Lip (Ω) follows from [1]; the assertion for X ∈ L 1 G (Ω) is then seen to hold by approximations as done in the proof of their proposition. Since there seems to be an inadequacy in its approximating argument and the family {P θ : θ ∈ A Θ 0,T } of probability measures is strictly smaller than the one in their proposition, we give a proof of this lemma for the sake of self-containedness of the paper.…”

Section: A Characterization Of Symmetric G-martingalesmentioning

confidence: 83%

“…The keys to the proof of our main result are: (i) the representation of the upper expectation for G-expectation due to Denis-Hu-Peng [1], with an enlargement of the associated class of martingale measures as given in Soner-Touzi-Zhang [12]; and (ii) Girsanov's formula for martingales in the classical stochastic analysis. Our methodology is different from that of Xu-Shang-Zhang [13], in which they obtained Girsanov's formula for one-dimensional G-Brownian motion; their proof relies on the martingale characterization of one-dimensional G-Brownian motion in [14], which restricts their argument to one dimension, whereas the method we employ in this paper equally works for multidimensional G-Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Girsanov’s formula forG-Brownian motion

Osuka

2013

Stochastic Processes and their Applications

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In this paper, we establish Girsanov's formula for G-Brownian motion. Peng (2007, 2008) constructed G-Brownian motion on the space of continuous paths under a sublinear expectation called G-expectation; as obtained by Denis et al. (2011), G-expectation is represented as the supremum of linear expectations with respect to martingale measures of a certain class. Our argument is based on this representation with an enlargement of the associated class of martingale measures, and on Girsanov's formula for martingales in the classical stochastic analysis. The methodology differs from that of Xu et al. (2011), and applies to the multidimensional G-Brownian motion.

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“…The following lemma provides a representation for solution Y T,ξ of equation (4). [13] (see also Proposition 3.4 in [28]) and noting that (K T,ξ t ) is a decreasing G-martingale, we have…”

Section: Some Results Of Classical Penalized Bsdesmentioning

confidence: 99%

Supermartingale decomposition theorem under $G$-expectation

Li¹,

Péng²,

Song³

2018

Electron. J. Probab.

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The objective of this paper is to establish the decomposition theorem for supermartingales under the G-framework. We first introduce a g-nonlinear expectation via a kind of G-BSDE and the associated supermartingales. We have shown that this kind of supermartingales have the decomposition similar to the classical case. The main ideas are to apply the uniformly continuous property of S β G (0, T ), the representation of the solution to G-BSDE and the approximation method via penalization.where ξ is a given random variable in L 2 (Ω, F T , P ) and A · is a given continuous and increasing process with A 0 = 0 and A t ∈ L 2 (Ω, F t , P ) for each t ∈ (0, T ]. We call y · a g-supersolution. If A · ≡ 0 then y · is called a g-solution. For the later case, since for each given t ≤ T , the F t measurable random variable y t is uniquely determined by the terminal condition y T = ξ ∈ L 2 (Ω, F T , P ), we then can define a backward semigroup [19,21] This semiproup gives us a generalized notion of nonlinear expectation with corresponding F t -conditional expectation, called g-expectation [19]. By applying the comparison theorem of BSDE we know that any g-supersolution Y · is also a g-supermartingale (i.e., we haveBut the proof of the inverse claim, namely, a g-supermartingale is a g-supersolution, is not at all trivial (we refer to [20] for detailed proof). In fact this is a generalization of the classical Doob-Meyer decomposition to the case of nonlinear expectations, and the linear situation corresponds to the case g ≡ 0.Moreover, this nonlinear Doob-Meyer decomposition theorem plays a key role to obtain the following representation theorem of nonlinear expectations: for a given arbitrary F t -conditional nonlinear expectation (E s,t [ξ]) 0≤s≤t<∞ with certain regularity, there exists a unique function g = g(·, y, z) satisfying the usual condition of BSDE, such that,We refer to [4], [21], [23] for the proof of this very deep result, also to [7] a wide class of time consistent risk measures are identified to be g-expectations.It is known that volatility model uncertainty (VMU) involves essentially non-dominated family of probability measures P on (Ω, F ). This is a main reason why many risk measures, and pricing operators cannot be well-defined within a framework of probability space such as Wiener space (Ω, F T , P ).[22] introduced the framework of (fully nonlinear) time consistent G-expectation space (Ω, L 1 G (Ω),Ê) such that all probability measures in P are dominated by this sublinear expectation and such that the canonical process B · (ω) = ω(·) becomes a nonlinear Brownian motion, called G-Brownian. Many random variables, negligible under the probability measure P ∈ P, as well as under other measures in P, can be clearly distinguished in this new framework. The corresponding theory of stochastic integration and stochastic calculus of Itô's type have been established in [22,25]. In particular, the existence and uniqueness of BSDE driven by G-Brownian motion (G-BSDE) have been established in [10]. Roughly speaking (see ...

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Martingale representation theorem for theG-expectation

Cited by 165 publications

References 19 publications

Kolmogorov-type and general extension results for nonlinear expectations

Kolmogorov-type and general extension results for nonlinear expectations

Girsanov’s formula forG-Brownian motion

Supermartingale decomposition theorem under $G$-expectation

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