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Register a new userG-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
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Shige Peng
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We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Itos type (Itos integral, Itos formula, Itos equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem.
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In this paper we obtain a Wiener-Hopf type factorization for a time-inhomogeneous arithmetic Brownian motion with deterministic time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener-Hopf type factorizations for (real-valued) time-inhomogeneous L{e}vy processes. In particular, we argue that the classical Wiener-Hopf factorization for time-homogeneous L{e}vy processes quite likely does not carry over to the case of time-inhomogeneous L{e}vy processes.
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