Integration with respect to the G-Brownian local time

Yan, Litan; Sun, Xichao; Gao, Bo

doi:10.1016/j.jmaa.2014.11.046

Cited by 6 publications

(3 citation statements)

References 22 publications

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“…The purpose of this section is to obtain the distribution of sup s≤t B s − B t , for this we need to use the Itô-Tanaka formula. In [7,18], the authors obtained the Itô-Tanaka formula and related properties for G-Brownian motion. Here we use the representation theorem for G-expectation to study the Itô-Tanaka formula.…”

Section: Reflection Principle Of G-brownian Motionmentioning

confidence: 99%

Lévy's martingale characterization and reflection principle of G-Brownian motion

Liu

2019

Journal of Mathematical Analysis and Applications

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Section: Reflection Principle Of G-brownian Motionmentioning

confidence: 99%

Lévy's martingale characterization and reflection principle of G-Brownian motion

Liu

2019

Journal of Mathematical Analysis and Applications

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“…We present some basic notions and results of nonlinear and sublinear expectations in this section. More details can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Preliminariesmentioning

confidence: 99%

Product space for two processes with independent increments under nonlinear expectations

Gao¹,

Hu²,

Ji³

et al. 2017

Electron. Commun. Probab.

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In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments. IntroductionPeng [6,7] introduced the notions of distribution and independence under nonlinear expectation spaces.Under sublinear case, Peng [10] obtained the corresponding central limit theorem for a sequence of i.i.d. random vectors. The limit distribution is called G-normal distribution. Based on this distribution, Peng [8,9] gave the definition of G-Brownian motion, which is a kind of process with stationary and independent increments, and then discussed the Itô stochastic analysis with respect to G-Brownian motion.It is well-known that the existence for a sequence of i.i.d. random vectors is important for central limit theorem. In the nonlinear case, Peng [12] introduced the product space technique to construct a sequence of i.i.d. random vectors. But this product space technique does not hold in the continuous time case. More precisely, let (M t ) t≥0 and (N t ) t≥0 be two d-dimensional processes with independent increments defined respectively on nonlinear expectation spaces (Ω 1 , H 1 ,Ê 1 ) and (Ω 2 , H 2 ,Ê 2 ), we want to construct a 2d-dimensional process (M t ,Ñ t ) t≥0 with independent increments defined on a nonlinear expectation space. Different from linear expectation case, the independence is not mutual under nonlinear case (see [4]). So this (M t ,Ñ t ) t≥0 is not a process with independent increments.In this paper, we introduce a discretization method, which can overcome the problem of independence.More precisely, for each given D n = {i2 −n : i ≥ 0}, we can construct a nonlinear expectationÊ n under which (M t ,Ñ t ) t∈Dn posesses independent increments. ButÊ n , n ≥ 1, are not consistent, i.e., the values of the same random variable underÊ n are not equal. Fortunately, we can prove that the limit ofÊ n exisits by using the notion of tightness, which was introduced by Peng in [13] to prove central limit theorem under sublinear case. Denote the limit ofÊ n byÊ, we show that (M t ,Ñ t ) t≥0 is the process with independent increments underÊ. This paper is organized as follows: In Section 2, we recall some basic notions and results of nonlinear expectations. The main theorem is stated and proved in Section 3. PreliminariesWe present some basic notions and results of nonlinear and sublinear expectations in this section. More details can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Let Ω be a given nonempty set and H be a linear space of real-valued functions on Ω such that ifdenotes the set of all bounded and Lipschitz functions on R d . H is considered as the space of random variables. Similarly, {X = (X 1 , . . . , X d ) : X i ∈ H, i ≤ d} denotes the space of d-dimensional random vectors. Definition 2.1 A sublinear expectationÊ on H is a functionalÊ : H → R satisfying th...

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“…For more aspects on these material we refer to Peng [4], Denis et al [8], and Hu and Peng [9]. More works for -Brownian motion can be found in Hu and Li [10], Lin [11], Peng et al [12], Song [13], Xu et al [14], Yan et al [15,16], and the references therein.…”

Section: Preliminariesmentioning

confidence: 99%