Stopping times and related Itô's calculus with G-Brownian motion

By Xinpeng Li, Shige Peng. 2009

204 citations recorded by the source corpus.

Under the framework of G-expectation and G-Brownian motion, we introduce Ito's integral for stochastic processes without assuming quasi-continuity. Then we can obtain Ito's integral on stopping time interval. This new formulation permits us to obtain Ito's formula for a general C1,2-function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010) [21], [22], [23], [24] and [25] as well as those of Gao (2009) [8] and Zhang et al. (2010) [27].

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/69bd18ded0b893f5c3b9dda51d4b4b2ed427fe1a

How researchers see this paper

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In Economics and Econometrics

1. Marcel Nutz
2. Mingshang Hu
3. Yongsheng Song
4. Falei Wang
5. Guoqiang Zheng
6. Yiqing Lin
7. Shaolin Ji
8. Ramon van Handel

In Finance

1. Marcel Nutz
2. Mingshang Hu
3. Shige Peng
4. Yongsheng Song
5. Shaolin Ji
6. Falei Wang
7. Guoqiang Zheng
8. Joerg Vorbrink

In Management Science and Operations Research

1. Marcel Nutz
2. Yongsheng Song
3. Shige Peng
4. Mingshang Hu
5. Joerg Vorbrink
6. Shaolin Ji
7. Falei Wang
8. Guoqiang Zheng

Citations in the corpus

• Financial markets with volatility uncertainty (2010, background): “Therefore, together with Itô’s formula (Theorem 5.4 in Li and Peng (2009)) we obtain û(t, S0,xt )−û(0, x) = ∫ t 0 ∂tû(s, S 0,x s ) + rS 0,x s ∂xû(s, S 0,x s )ds + ∫ t 0 Ss∂xû(s, S 0,x s )dBs + ∫ t 0 1 2 S2s∂xxû(s, S 0,x s )d〈B〉s (6) = ∫ t 0 Ss∂xû(s, S 0,x s )dBs + 1 2 ∫ t 0 S2s∂xxû(s, S…”
• Constructing Sublinear Expectations on Path Space (2012, background): “Stopping times, which tend to be discontinuous functions of ω, could not be treated directly (see [12, 15, 24] for related partial results) and the existence of conditional G-expectations beyond quasi-continuous random variables remained open.”
• Comparison Theorem, Feynman-Kac Formula and Girsanov Transformation for BSDEs Driven by G-Brownian Motion (2012, background): “Recently, Peng systemically established a time-consistent fully nonlinear expectation theory (see [17], [18] and [23]).”
• Pathwise construction of stochastic integrals (2011, background): “A natural extension of the classical construction to a family P, carried out in [2, 11, 8] under speci c regularity assumptions, is to consider the upper expectation EP [ ⋅ ] = supP∈P EP [ ⋅ ] instead of the usual expectation and de ne the stochastic integral along the lines of the usual closure ope”
• Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, result): “The following result of Li and Peng (2009) significantly improved the previous ones.”
• Backward stochastic differential equations driven by G-Brownian motion (2012, methodology): “Following the framework in Li and Peng [9], we take hi (x) = I[−n+ i n ,−n+ i+1 n ) (x), i = 0, .”
• Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion (2014, methodology): “At the end of this section, we generalize the BDG type inequality in [8] for η ∈ M(2) w([0, T ]).”
• Quasi-continuous random variables and processes under the G-expectation framework (2014, background): “Applying G-Itô’s formula given in [8] yields that u(Xt∧τR) =u (x0) + ∫ t∧τR”