Some properties of stochastic differential equations driven by the G-Brownian motion

Lin, Qian

doi:10.1007/s10114-013-0701-y

Cited by 35 publications

(21 citation statements)

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“…In this section, we review some basic notions and results of G-expectation. More relevant details can be found in [6,[15][16][17][20][21][22][23] 2…”

Section: Preliminariesmentioning

confidence: 99%

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Sobolev-type stochastic differential equations driven by G-Brownian motion

Ren

Yin

2019

International Journal of Control

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The objective of this paper is to study the strong Markov property for the stochastic differential equations driven by G-Brownian motion (G-SDEs for short). We first extend the deterministic-time conditional G-expectation to optional times. The strong Markov property for G-SDEs is then obtained by Kolmogorov's criterion for tightness. In particular, for any given optional time τ and G-Brownian motion B, the reflection principle for B holds and (B τ +t − B τ ) t≥0 is still a G-Brownian motion.The strong Markov property for stochastic differential equations (SDEs) is one of the most fundamental results in the theory of classical stochastic processes. It claims that for any given optional time τ we havefor SDEs (X x t ) t≥0 with initial value x. Here E P and E P [·|F τ + ] stands for the expectation and conditional expectation, respectively, related to a probability measure P . It was obtained by K. Itô in his pioneering work [11], and since then, it has been widely applied to stochastic control, mathematical finance and probabilistic method for partial differential equations (PDEs); see, e.g., [1,4,19].Recently, motivated by probabilistic interpretations for fully nonlinear PDEs and financial problems with model uncertainty, Peng [20][21][22] systematically introduced the notion of nonlinear G-expectationÊ [·] by stochastic control and PDE methods. Under the G-expectation framework, a new kind of Brownian motion, called G-Brownian motion, was constructed. The corresponding stochastic calculus of Itô's type was also established. Furthermore, by the contracting mapping theorem, Peng obtained the existence and In this paper, we first construct the conditional G-expectationÊ τ + [·] for any given optional time τ by extending the definition of conditional G-expectationÊ t [·] to optional times. The main tools in this construction are a universal continuity estimate forÊ t [·] (see Lemma 3.3) and a new kind of consistency property (see Proposition 3.9). We also show thatÊ τ + [·] can preserve most useful properties of classical conditional expectations except the linearity. Based on the conditional expectationÊ τ + [·], we then further obtain the strong Markov property (1.3) for G-SDEs by adapting the standard discretization method. In contrast to the linear case, the main difficulty is that in the nonlinear expectation context the dominated convergence theorem does not hold in general. We tackle this problem by using Kolmogorov's criterion for tightness and the properties ofÊ τ + [·]. In particular, for G-Brownian motion B, we obtain that the reflection principle for B holds and (B τ +t − B τ ) t≥0 is still a G-Brownian motion. Finally, with the help of the strong Markov property, the level set of G-Brownian motion is also investigated.We note that problem of constructingÊ τ + [·] was first considered in [18], whereÊ τ + [·] is defined for all upper semianalytic (more general than Borel-measurable) functions by the analytic sets theory. But the corresponding conditional expectation is also upper semianalytic and when...

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“…In this section, we review some basic notions and results of G-expectation. More relevant details can be found in [6,[15][16][17][20][21][22][23] 2…”

Section: Preliminariesmentioning

confidence: 99%

“…Let ϕ be lower semi-continuous, bounded from below on Ω and τ be an optional time. ThenE τ + [ϕ(B τ +· )] =Ê[ϕ(B y · )] y=Bτ ,(4 15). …”

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confidence: 99%

Sobolev-type stochastic differential equations driven by G-Brownian motion

Ren

Yin

2019

International Journal of Control

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“…On that basis, Gao [6] and Peng [23] studied the existence and uniqueness of the solution of G-SDE under a standard Lipschitz condition. For a recent account and development of this theory we refer the reader to [9,10,11,12,14,16,17,18,21,33]. Recently, Lin [13] obtained the existence and uniqueness of the solution of G-SDE with reflecting boundary.…”

Section: Introductionmentioning

confidence: 99%

Reflected stochastic differential equations driven by G-Brownian motion with nonlinear resistance

Luo

2015

Front. Math. China

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“…Moreover, Lin [11] obtained the existence and uniqueness of the solution of G-SDE with reflecting boundary. For a recent account and development of this theory we refer the reader to [1,7,8,9,10,12,20].…”

Section: Introductionmentioning

confidence: 99%