G-expectation Framework Research Articles

Robust investment for insurers with correlation ambiguity

This paper investigates the investment decision of insurers when there is ambiguous correlation between the financial market and the insurance business. The robust decision model that accommodates correlation ambiguity between a risky financial asset and the insurer’s non-tradable surplus is solved under the G-expectation framework. We find that correlation ambiguity leads to a more conservative investment strategy in financial assets, providing a plausible explanation for insurers’ under- or zero investment in the financial market during normal economic times. We also show that the range of priors set of correlation coefficients can be statistically inferred, and insurers will quit the financial market when the range of priors set exceeds a certain level, which is more likely to happen when the remaining investment horizon is long.

Option Pricing Model Driven by G-Lévy Process under the G-Expectation Framework

In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.

Numerical Scheme for Solving Stochastic Differential Equations with <i>G</i>-Lévy Process

In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itô formula and G-expectation property, we propose Euler scheme and Milstein scheme which have order-1.0 convergence rate. And two numerical experiments including Ornstein-Uhlenbeck and Black-Scholes cases are given.

Local wellposedness of coupled backward stochastic differential equations driven by G-Brownian motions

We investigate coupled forward backward stochastic differential equations driven by the G-Brownian motion (Bt)t≥0 which take the form,(0.1)Xt=x+∫0tb(s,Xs,Ys)ds+∫0tσ(s,Xs,Ys)dBsYt=ψ(XT)+∫tTg(s,Xs,Ys,Zs)ds−∫tTZsdBs−(KT−Kt), where K being part of the solution, is required to be some decreasing G-martingale. Under the Lipschitz assumptions, the existence and uniqueness of the solution (X,Y,Z,K) in small time duration is proved by using the contraction mapping principle. Different from the classical case, the integrability orders for components X,Y,Z,K of solutions are not homogeneous, which is the main characteristic for this problem under the G-expectation framework.

Egoroff’s Theorem and Lusin’s Theorem for Capacities in the Framework of g-Expectation

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. The σ-additivity of measures plays a crucial role in the proofs of these theorems. Later, many researchers have carried out lots of studies on Egoroff’s theorem and Lusin’s theorem when the measure is monotone and nonadditive (see, e.g., Li and Yasuda (2004) and Li and Mesiar (2011)). In this paper, we study Egoroff’s theorem and Lusin’s theorem for capacities in the framework of g-expectation. We give some different assumptions that provide Egoroff’s theorem and Lusin’s theorem in the framework of g-expectation.

Quasi-sure exponential stability and stabilisation of stochastic delay differential equations under G-expectation framework

In this work, we obtain the sufficient conditions to examine quasi-sure exponential stability of stochastic delay differential equation under G-expectation framework (in short, G-SDDE). Our argument is based on quasi-sure exponential stability of G-SDE and we should point out that the keys to the proof of our main result are the fact that there exists a positive number such that the G-SDDE is also quasi-surely exponentially stable as long as the time delay constant . Furthermore, this approach allows to design the delay feedback control induced by G-Brownian motion so as to stabilise the given deterministic systems in terms of quasi-sure exponential stability.

Jensen Inequality of Bivariate Function in the G-Expectation Framework

In the G-expectation framework, Wang [1] first obtained the Jensen inequality of one-dimensional function. In this paper, under some stronger conditions, we obtain the Jensen inequality of bivariate function based on Wang’s proof method. And we give some examples to illustrate the application of Jensen inequality of bivariate function.

Upper bounds for ruin probabilities under model uncertainty

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.

Robust mean–variance hedging via [formula omitted]-expectation

In this paper we study mean–variance hedging under the G-expectation framework. Our analysis is carried out by exploiting the G-martingale representation theorem and the related probabilistic tools, in a continuous financial market with two assets, where the discounted risky one is modeled as a symmetric G-martingale. By tackling progressively larger classes of contingent claims, we are able to explicitly compute the optimal strategy under general assumptions on the form of the contingent claim.

Multiple G-Stratonovich Integral Driven by G-Brownian Motion

In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-It? formula, we obtain the relationship between Hermite polynomials and multiple G-Stratonovich integrals by using mathematical induction method.

Strict comparison theorems under sublinear expectations

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class \(Lip(\Omega )\). Then we prove generalized strict comparison theorems on the enlarged space \(L_G^1(\Omega )\), which is the Banach completion of \(Lip(\Omega )\) under the G-expectation.

Fubini theorem for non additive measures in the framework of g-expectation

ABSTRACTSince the seminal paper of Ghirardato (1997), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997), Chateauneuf and Lefort (2008) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.

On the pth moment estimates of solutions to stochastic functional differential equations in the G-framework.

The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis–Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.

Dynamic programming principle for stochastic recursive optimal control problem driven by a [formula omitted]-Brownian motion

In this paper, we study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related fully nonlinear HJB equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

An optimal insurance design problem under Knightian uncertainty

This paper solves an optimal insurance design problem in which both the insurer and the insured are subject to Knightian uncertainty about the loss distribution. The Knightian uncertainty is modeled in a multi-prior g-expectation framework. We obtain an endogenous characterization of the optimal indemnity that extends classical theorems of Arrow (Essays in the Theory of Risk Bearing. Markham, Chicago 1971) and Raviv (Am Econ Rev 69(1):84–96, 1979) in the classical situation. In the presence of Knightian uncertainty, it is shown that the optimal insurance contract is not only contingent on the realized loss but also on another source of uncertainty coming from the ambiguity.