We design a class of numerical schemes for backward stochastic differential equation driven by G-Brownian motion (G-BSDE), which is related to a fully nonlinear PDE. Based on Peng's central limit theorem, we employ the CLT method to approximate G-distributed. Rigorous stability and convergence analysis are also carried out. It is shown that the θ-scheme admits a half order convergence rate in the general case. In particular, for the case of θ 1 ∈ [0, 1] and θ 2 = 0, the scheme can reach first-order in the deterministic case. Several numerical tests are given to support our theoretical results.
This paper formulates a model of utility for a continuous time framework that captures the decision-maker's concern with ambiguity about both volatility and drift. Corresponding extensions of some basic results in asset pricing theory are presented. First, we derive arbitrage-free pricing rules based on hedging arguments. Ambiguous volatility implies market incompleteness that rules out perfect hedging. Consequently, hedging arguments determine prices only up to intervals. However, sharper predictions can be obtained by assuming preference maximization and equilibrium. Thus we apply the model of utility to a representative agent endowment economy to study equilibrium asset returns. A version of the C-CAPM is derived and the effects of ambiguous volatility are described.
In this paper, we study comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of the following BSDE driven by a G-Brownian motion. Yt = ξ + T t f (s, Ys, Zs)ds + T t g(s, Ys, Zs)d B s − T t ZsdBs − (KT − Kt), where K is a decreasing G-martingale.
Abstract. This paper is concerned with a stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation (FBSDE), while the forward state is constrained in a convex set at the terminal time. An equivalent backward control problem is introduced. By using Ekeland's variational principle, a stochastic maximum principle is obtained. Applications to state constrained stochastic linear-quadratic control models and a recursive utility optimization problem are investigated.
This paper formulates a model of utility for a continuous time framework that captures the decision-maker's concern with ambiguity about both the drift and volatility of the driving process. At a technical level, the analysis requires a significant departure from existing continuous time modeling because it cannot be done within a probability space framework. This is because ambiguity about volatility leads invariably to a set of nonequivalent priors, that is, to priors that disagree about which scenarios are possible.
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