This article establishes a Girsanov type theorem under the G-Framework of Peng [ 15 Peng , S. 2006 . G-expectation, G-Brownian motion and related calculus of Ito's type. Available at: http://abelsymposium.no/symp2005/preprints/peng.pdf [Google Scholar] ]. Our result generalizes the classical Girsanov theorem for Brownian motion [ 10 Girsanov , I.V. 1960 . On transforming a certain class of stochastic processes by absolutely continuous substitution of measures . Theory Probability and Its Applications 5 : 285 – 301 . [Crossref] [Google Scholar] ]. As an application, we price the European call option when the underlying asset's price follows the Geometric G-Brownian motion.

In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Itô calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Itô formula for C ²-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Itô formula to a slightly more general class of functions (C ²-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general odd powers of the G-Brownian motion.

The main purpose of this article is to study the symmetric martingale property and capacity defined by G-expectation introduced by Peng (cf. http://arxiv.org/PS_cache/math/pdf/0601/0601035v2.pdf) in 2006. We show that the G-capacity can not be dynamic, and also demonstrate the relationship between symmetric G-martingale and the martingale under linear expectation. Based on these results and path-wise analysis, we obtain the martingale characterization theorem for G Brownian motion without Markovian assumption. This theorem covers the Levy's martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy's theorem.

This work is devoted to a continuous time dynamic pension funding model in a defined benefit plan of an employment system. We extend the analysis of some standard models by incorporating a source of uncertainty in the benefit outgo. The key assumption is that the random benefits increase on average at an exponential rate. We model the preference of the manager with the main objective of minimizing both the contribution rate risk and the solvency risk. Two different situations are studied regarding the investment decisions. In the first case, the fund is invested at a constant, risk-free rate of interests; in the second case, the promoter invests in a portfolio with a risky asset and a risk-free bond. We provide, in both cases, explicit expressions for the actuarial liability, normal costs, value function, and the supplementary contribution rate.