MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS

In this paper,we study the martingale inequalities under G-expectation and their applications.To this end,we introduce a new kind of random time,called G-stopping time,and then investigate the properties of a G-martingale(supermartingale)such as the optional sampling theorem and upcrossing inequalities.With the help of these properties,we can show the martingale convergence property under G-expectation.

A Property of g-Expectation

This paper proves that, under the hypothesis g(t, 0, 0)≡0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige’s conjecture.

A Result On the Probability Measures Dominated by g-Expectation

It is proved that a probability measure is dominated by g-expectation if and only if it can be generated by Girsanov transformation via a process which is uniformly bounded by μ.

Extended conditional G-expectations and related stopping times

In this paper,we extend the definition of conditional G-expectation to a larger space on which the dynamical consistency still holds.We can consistently define,by taking the limit,the conditional G-expectation for each random variable X,which is the downward limit(respectively,upward limit)of a monotone sequence (Xi) in L_(G)^(1)(Ω).To accomplish this procedure,some careful analysis is needed.Moreover,we present a suitable definition of stopping times and obtain the optional stopping theorem.We also provide some basic and interesting properties for the extended conditional G-expectation.

ON JENSEN'S INEQUALITY FOR g-EXPECTATION

Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g (t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.

Inf-convolution of G-expectations

In this paper we will discuss the optimal risk transfer problems when risk measures are generated by G-expectations,and we present the relationship between inf-convolution of G-expectations and the infconvolution of drivers G.

g-expectation of distributions

We define g-expectation of a distribution as the infimum of the g-expectations of all the terminal random variables sharing that distribution.We present two special cases for nonlinear g where the g-expectation of distributions can be explicitly derived.As a related problem,we introduce the notion of law-invariant g-expectation and provide its sufficient conditions.Examples of application in financial dynamic portfolio choice are supplied.

Stochastic ordering by g-expectations

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations.Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity,monotonicity,and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations.Applications to contingent claim price comparison under different hedging portfolio constraints are provided.

A NOTE ON g-CONCAVE FUNCTION

An equivalent condition is derived for g-concave function defined by (static) g-expectation. Several extensions including quadratic generators and (g,h)-concavity are also considered.

A General Converse Comparison Theorem for Backward Stochastic Differential Equation with Non-lipschitz Coefficient

In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.

Numerical Scheme for Solving Stochastic Differential Equations with G-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-Itôformula 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.

G-stochastic maximum principle for risk-sensitive control problem and its applications

This study advances the G-stochastic maximum principle(G-SMP)from a risk-neutral framework to a risk-sensitive one.A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion(G-SDEs),where the control variable may influence all terms.We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost.Initially,we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework.Subsequently,we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation.Furthermore,we simplified the G-adjoint process from a dual-component structure to a singular component.Moreover,we explained the necessary optimality conditions for this model by considering a convex set of admissible controls.To describe the main findings,we present two examples:the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.

Jensen's Inequality for Backward Stochastic Differential Equations

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g（t, 0）≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen＇s inequality for gexpectation in [4, 7-9].

g-Variance

In this paper, we introduce the notion of g-variance and study some properties of this operator. We find the nonlinear variance operator, g-variance, does not preserve some basic properties of traditional mathematic variance. We also consider the relationship Dμ[ξ] and supQe T^1 VarQ[ξ] （D-μ [ξ] and infQe T^1 VarQ [ξ]）. The result shows that the maximum （minimum） variance is not always equal to Dμ[·] （D-μ[·]）. If g satisfies some restrictive conditions, we get the uniqueness theorem and the comparison theorem via g-variance.

NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.

Some properties on G-evaluation and its applications to G-martingale decomposition

In this article, a sublinear expectation induced by G-expectation is introduced, which is called G- evaluation for convenience. As an application, we prove that for any ξ∈ L β G (Ω T ) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Ito integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {M t } has a quasi-continuous version.

Limit theorem and uniqueness theorem of backward stochastic differential equations

This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0) = 0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectationεg;this paper also proves that if a filtration consistent expectation S can be represented as a g-expectationεg, then the corresponding generator g must be unique.

On the Existence and Uniqueness of Solutions to Stochastic Differential Equations Driven by G-Brownian Motion with Integral-Lipschitz Coefficients

In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion （GSDEs） with integral-Lipschitz coefficients.

Some Properties of Stochastic Differential Equations Driven by the G-Brownian Motion

In this paper, we study the property of continuous dependence on the parameters of stochastic integrals and solutions of stochastic differential equations driven by the G-Brownian motion. In addition, the uniqueness and comparison theorems for those stochastic differential equations with non-Lipschitz coefficients are obtained.

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.