Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation

DOI10.1016/j.spa.2007.10.015zbMath1158.60023arXivmath/0601699OpenAlexW2062114931WikidataQ96624829 ScholiaQ96624829MaRDI QIDQ2518615

Shige Peng

Publication date: 16 January 2009

Published in: Stochastic Processes and their Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0601699

zbMATH Keywords

stochastic differential equationBrownian motionItô integralJensen's inequality\(g\)-expectationbackward stochastic differential equation\(G\)-expectationItô calculus\(G\) -normal distribution

Mathematics Subject Classification ID

Stochastic ordinary differential equations (aspects of stochastic analysis) (60H10) Applications of stochastic analysis (to PDEs, etc.) (60H30) Stochastic integrals (60H05)

Related Items (only showing first 100 items - show all)

Distributional Uncertainty of the Financial Time Series Measured by $G$-Expectation ⋮ Strong convergence for weighted sums of END random variables under the sub-linear expectations ⋮ Delay feedback stabilisation of stochastic differential equations driven by G-Brownian motion ⋮ Maximally Distributed Random Fields under Sublinear Expectation ⋮ Probabilistic Solutions for a Class of Path-Dependent Hamilton-Jacobi-Bellman Equations ⋮ Strong law of large numbers under moment restrictions in sublinear expectation spaces ⋮ Upper bounds for ruin probabilities under model uncertainty ⋮ Complete integral convergence for arrays of row-wise extended independent random variables under Sub-linear expectations ⋮ Complete convergence for weighted sums of negatively dependent random variables under the sub-linear expectations ⋮ Complete and complete moment convergence for weighted sums of arrays of rowwise negatively dependent random variables under the sub-linear expectations ⋮ Strong laws of large numbers for weighted sums of extended negatively dependent random variables under sub-linear expectations ⋮ An invariance principle of strong law of large numbers under nonadditive probabilities ⋮ Complete and complete integral convergence for arrays of row wise widely negative dependent random variables under the sub-linear expectations ⋮ Complete convergence theorem for negatively dependent random variables under sub-linear expectations ⋮ Multi-dimensional BSDEs driven by G-Brownian motion and related system of fully nonlinear PDEs ⋮ Robust Consumption-Investment with Return Ambiguity: A Dual Approach with Volatility Ambiguity ⋮ Capacity inequalities and strong laws for \(m\)-widely acceptable random variables under sub-linear expectations ⋮ Discrete‐time approximation for stochastic optimal control problems under the G‐expectation framework ⋮ Markov chains under nonlinear expectation ⋮ Convex monotone semigroups on lattices of continuous functions ⋮ A strong law of large numbers under sublinear expectations ⋮ Complete integral convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations ⋮ Delay feedback control of highly nonlinear neutral stochastic delay differential equations driven by G-Brownian motion ⋮ Nonlinear semigroups built on generating families and their Lipschitz sets ⋮ Controllability discussion for fractional stochastic Volterra-Fredholm integro-differential systems of order \(1<r<2\) ⋮ Complete convergence theorems for arrays of row-wise extended negatively dependent random variables under sub-linear expectations ⋮ Complete moment convergence for ND random variables under the sub-linear expectations ⋮ Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problem under volatility uncertainty ⋮ Quadratic BSDEs with mean reflection driven by G-brownian motion ⋮ Rough path analysis for local time of G-Brownian motion ⋮ Forward-backward stochastic differential equations driven by \(G\)-Brownian motion under weakly coupling condition ⋮ A dynamic analytic method for risk-aware controlled martingale problems ⋮ Practical stability in relation to a part of variables for stochastic reaction–diffusion systems driven by G-Brownian motion ⋮ Strong limit theorems of weighted sums for extended negatively dependent random variables under sub-linear expectations ⋮ A deviation inequality for increment of a \(G\)-Brownian motion under \(G\)-expectation and applications ⋮ Equivalent conditions of complete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers for i.i.d. sequences under sub-linear expectations ⋮ Complete and Complete Integral Convergence for Arrays of Rowwise Extended Negatively Dependent Random Variables under Sublinear Expectations ⋮ Stability with respect to a part of the variables of stochastic nonlinear systems driven by G-Brownian motion ⋮ Nonlinear continuous semimartingales ⋮ \(G\)-stochastic maximum principle for risk-sensitive control problem and its applications ⋮ G-Gaussian processes under sublinear expectations and \(q \)-Brownian motion in quantum mechanics ⋮ Central limit theorem for linear processes generated by m-dependent random variables under the sub-linear expectation ⋮ A generalized stochastic process: fractional \(G\)-Brownian motion ⋮ Nonparametric estimation of trend for stochastic processes driven by \(G\)-Brownian motion with small noise ⋮ On the capacity for degenerated \(G\)-Brownian motion and its application ⋮ Survey on path-dependent PDEs ⋮ The Carathéodory approximation scheme for stochastic differential equations with G‐Lévy process ⋮ Multi-dimensional mean-reflected BSDEs driven by \(G\)-Brownian motion with time-varying non-Lipschitz coefficients ⋮ Boundedness analysis of neutral stochastic systems driven by \(G\)-Brownian motion ⋮ Stochastic differential equations for orthogonal eigenvectors of (G,ε)-Wishart process related to multivariate G-fractional Brownian motion ⋮ Stability analysis of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion ⋮ Unnamed Item ⋮ Non-linear expectations in spaces of Colombeau generalized functions ⋮ The application of multi-dimensional Jensen’s inequality for G-martingale ⋮ Reflected stochastic differential equations driven by G-Brownian motion in non-convex domains ⋮ Complete convergence for weighted sums of extended negatively dependent random variables under sub-linear expectations ⋮ A complete convergence theorem for weighted sums under the sub-linear expectations ⋮ General Martingale Characterization ofG-Brownian Motion ⋮ Asymptotical boundedness for stochastic coupled systems on networks with time-varying delay driven by G-Brownian motion ⋮ On complete convergence for extended independent random variables under sub-linear expectations ⋮ Unnamed Item ⋮ Stabilisation of multi-weights stochastic complex networks with time-varying delay driven by G-Brownian motion via aperiodically intermittent adaptive control ⋮ Distributed Consensus and Convergence Rate Analysis of Multiagent Systems with Noises under $G$-Expectation ⋮ Stability of solutions of Caputo fractional stochastic differential equations ⋮ Stabilisation of stochastic multi-group models driven by G-Brownian motion via delay feedback control ⋮ Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems ⋮ Exponential inequalities under sub-linear expectations with applications to strong law of large numbers ⋮ Dynamic programming principle and Hamilton-Jacobi-Bellman equation under nonlinear expectation ⋮ Complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations ⋮ Complete convergence for arrays of row-wise ND random variables under sub-linear expectations ⋮ Mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients ⋮ The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion ⋮ Stability analysis of Hopfield neural networks with unbounded delay driven by G-Brownian motion ⋮ Complete and complete integral convergence for weighted sums of arrays of rowwise widely negative dependent random variables under the sub-linear expectations ⋮ Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by G-Brownian motion with state-feedback control ⋮ Existence of relaxed stochastic optimal control for G-SDEs with controlled jumps ⋮ Complete convergence and complete moment convergence for arrays of rowwise negatively dependent random variables under sub-linear expectations ⋮ Quadratic backward stochastic differential equations driven by \(G\)-Brownian motion: discrete solutions and approximation ⋮ Gradient estimates for nonlinear diffusion semigroups by coupling methods ⋮ Strong laws of large numbers for sublinear expectation under controlled 1st moment condition ⋮ Quasi-continuous random variables and processes under the \(G\)-expectation framework ⋮ Stabilization of stochastic differential equations driven by \(G\)-Brownian motion with feedback control based on discrete-time state observation ⋮ Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications ⋮ Strong laws of large numbers for sub-linear expectations ⋮ Super-replication with nonlinear transaction costs and volatility uncertainty ⋮ Uncertainty orders on the sublinear expectation space ⋮ \(p\)-moment stability of solutions to stochastic differential equations driven by \(G\)-Brownian motion ⋮ Asymptotical boundedness for stochastic coupled systems on networks driven by \(G\)-Brownian motion ⋮ Numerical simulations for \(G\)-Brownian motion ⋮ Robust valuation, arbitrage ambiguity and profit \& loss analysis ⋮ Dynamic programming principle for stochastic recursive optimal control problem driven by a \(G\)-Brownian motion ⋮ Duality and convergence for binomial markets with friction ⋮ Supermartingale decomposition theorem under \(G\)-expectation ⋮ Numerical scheme for Dynkin games under model uncertainty ⋮ Multiple \(G\)-Itō integral in \(G\)-expectation space ⋮ Random \(G\)-expectations ⋮ Strict comparison theorems under sublinear expectations ⋮ Large deviations for stochastic differential equations driven by \(G\)-Brownian motion ⋮ Central limit theorem for capacities ⋮ On representation theorem of sublinear expectation related to \(G\)-Lévy process and paths of \(G\)-Lévy process

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Dynamical evaluations
• Filtration consistent nonlinear expectations and evaluations of contingent claims
• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
• Espaces de Sobolev gaussiens. (Gaussian Sobolev spaces)
• On a non-linear semi-group attached to stochastic optimal control
• Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type
• Jensen's inequality for \(g\)-expectation. I
• A converse comparison theorem for BSDEs and related properties of \(g\)-expectation
• On Jensen's inequality for \(g\)-expectation
• Filtration-consistent nonlinear expectations and related \(g\)-expectations
• Some results on the uniqueness of generators of backward stochastic differential equations
• A result on the probability measures dominated by \(g\)-expectation
• A general downcrossing inequality for \(g\)-martingales
• Nonlinear expectations and nonlinear Markov chains
• A general converse comparison theorem for backward stochastic differential equations
• Coherent Measures of Risk
• A Generalized dynamic programming principle and hamilton-jacobi-bellman equation
• User’s guide to viscosity solutions of second order partial differential equations
• A property of backward stochastic differential equations
• Uncertain volatility and the risk-free synthesis of derivatives
• Ambiguity, Risk, and Asset Returns in Continuous Time
• Robust Statistics
• Stochastic differential equations. An introduction with applications.

This page was built for publication: Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation