Nonlinear Expectations and Stochastic Calculus under Uncertainty

By Shige Peng. Probability Theory and Stochastic Modelling, 2010

665 citations recorded by the source corpus.

In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/f81ec57f603d36b8f91e7e1b316c6a1bb0c05128

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Citations in the corpus

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017, methodology): “nd nonlinear second-order backward stochastic dierential equations 2BSDEs. Our method in particular can be used to sample from Shige Peng’s nonlinear expectation in high space-dimensions (cf., e.g., [89, 90, 91, 92]). The proposed algorithm exploits a connection between PDEs and 2BSDEs (cf., e.g., C”
G-Lévy processes under sublinear expectations (2009, background): “The distribution of can be seen as the pure uncertainty of mean (see [17–20]).”
Limit theorems with rate of convergence under sublinear expectations (2017, methodology): “Peng (2010) formulated a law of large numbers (LLN) under the sublinear expectation (1.1) as follows.”
A semigroup approach to nonlinear Lévy processes (2017, background): “tion. The G-Brownian motion due to [24, 25] is a special case of a nonlinear L´evy process, see also Dolinsky et al. [7] or Denis et al. [5]. For an introduction to nonlinear expectations we refer to [17]. Since we do not require any path regularity we call a process (Xt)t≥0 an E-L´evy process with ”
The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion (2017, background): “Introduction Since Peng [1] introduced the theory of G-expectation and its related stochastic calculus, they have attracted researchers’ great interest for their potential applications in uncertainty problems, risk measures, superhedging in finance and so on.”