On Jensen’s inequality and Hölder’s inequality for g-expectation

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1–27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143–217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.     

Jensen's inequality for g-expectation: part 1

Briand et al. (Electron. Comm. Probab. 5 (2000) 101–117) gave a counterexample and proposition to show that given g,g-expectations usually do not satisfy Jensen's inequality for most of convex functions. This yields a natural question, under which conditions on g, do g-expectations satisfy Jensen's inequality for convex functions? In this paper, we shall deal with this question in the case that g is convex and give a necessary and sufficient condition on g under which Jensen's inequality holds for convex functions. To cite this article: Z. Chen et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

反射倒向随机微分方程的逆比较问题(Ⅰ)

在Briand,Coquet,Hu,Memin,Peng[1],Coquet,Hu,Memin,Peng[2],Chen[3],Jiang [8]等中,研究了倒向随机微分方程的逆比较定理,就是通过比较倒向随机微分方程的解来比较倒向随机微分方程的生成元问题．在文[9]中Li和Tang首次研究了反射倒向随机微分方程的逆比较问题．本文考虑在更一般的条件下,反射倒向随机微分方程的生成元的逆比较问题．     

On Jensen’s inequality for g-expectation and for nonlinear expectation

In this paper, we give a necessary and sufficient condition for g under which Jensen’s inequality holds for g-expectation. In particular, we show that if Jensen’s inequality holds for g-expectation, then g is independent of y and g is superhomogeneous. We also establish a necessary and sufficient condition under which Jensen’s inequality holds for a general filtration-consistent nonlinear expectation. Received: 18 January 2005     

Lévy过程驱动的BSDE的反比较定理与Jensen不等式

本文研究了由一维Lévy过程驱动的倒向随机微分方程(BSDE)的反比较定理.利用一般g-期望下BSDE的反比较定理的证明方法,推导出了一般f-期望下BSDE的反比较定理,并给出了一般f-期望下Jensen不等式成立的充分必要条件.     

Principes de grandes déviations pour des processus de « coarse graining »

We improve the Large Deviations Principle satisfied by a Coarse Grained process already analyzed by Boucher, Ellis and Turkington [Ann. Probab. 27 (1999) 297–324]. To cite this article: J. Trashorras, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Representation theorems for generators of backward stochastic differential equations

It is proved that the generator g of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs if and only if g is a Lebesgue generator. To cite this article: L. Jiang, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Distance riemannienne,théorème de Rademacher et inégalité de transport sur le groupe des lacets

In this Note, we shall consider the Riemannian distance on loop groups, which will be identified to one introduced by Hino and Ramirez [M. Hino, J.A. Ramirez, Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab. 31 (2003) 1254–1295]. A transportation cost inequality is established. To cite this article: S. Fang, J. Shao, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Some results on the uniqueness of generators of backward stochastic differential equations

It is proved that the generator g of a backward stochastic differential equation (BSDE) can be uniquely determined by the initial values of the corresponding BSDEs with all terminal conditions. The main results also confirm and extend Peng's conjecture. To cite this article: L. Jiang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Extension and Application of Itô's Formula Under G-Framework

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.     

条件g-期望的矩不等式

找到了几个使条件g-期望的矩不等式在一般意义下成立的关于g和g-期望的充分条件.     

Utility maximization under g*-expectation

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp?-utility and the power utility, when g(t, z) = β_t|z|² + γ_tz.     

Jensen's inequality for multivariate medians

Given a probability measure μ on Borel sigma-field of Rd, and a function f:Rd?R, the main issue of this work is to establish inequalities of the type f(m)?M, where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure μf=μ○f−1. For the most popular choice of halfspace depth, we prove that the Jensen's inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in [Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461-482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in Rd and we present a version of Jensen's inequality for such medians. Replacing means in classical Jensen's inequality with medians gives rise to applications in the framework of Pitman's estimation.     

A limit theorem for solutions to BSDEs in the space of processes

In this paper, under the most elementary conditions on stochastic differential equations (SDEs in short) and the most elementary conditions on backward stochastic differential equations (BSDEs in short) introduced by Peng, in the space of processes, a limit theorem for solutions to BSDEs with its terminal data being solutions of the SDEs is obtained, based on some recent results of Jiang in the space of random variables in [Jiang, L., 2005a. Converse comparison theorems for backward stochastic differential equations. Statist. Probab. Lett. 71, 173–183; Jiang, L., 2005b. Representation theorems for generators of backward stochastic differential equations. C.R. Acad. Sci. Paris 340 (Ser. I), 161–166; Jiang, L., 2005c. Representation theorems for generators of backward stochastic differential equations and their applications. Stochastic Process. Appl. 115 (12), 1883–1903; Jiang, L., 2005d. Nonlinear expectation—g-expectation theory and its applications in finance. Ph.D Thesis, ShanDong University, China; Jiang, L., 2006. Limit theorem and uniqueness theorem for backward stochastic differential equations. Sci. China Ser. A 49 (10), 1353–1362]. This result generalizes the known results on the limit theorem for solutions to BSDEs in [Jiang, L., 2005a. Converse comparison theorems for backward stochastic differential equations. Statist. Probab. Lett. 71, 173–183; Jiang, L., 2005b. Representation theorems for generators of backward stochastic differential equations. C.R. Acad. Sci. Paris 340 (Ser. I), 161–166; Jiang, L., 2005c. Representation theorems for generators of backward stochastic differential equations and their applications. Stochastic Process. Appl. 115 (12), 1883–1903; Jiang, L., 2005d. Nonlinear expectation—g-expectation theory and its applications in finance. Ph.D Thesis, ShanDong University, China; Jiang, L., 2006. Limit theorem and uniqueness theorem for backward stochastic differential equations. Sci. China Ser. A 49 (10), 1353–1362; Fan, S., 2007. A relationship between the conditional g-evaluation system and the generator g and its applications. Acta Math. Sin. (Engl. Ser.) 23 (8), 1427–1434; Fan, S., 2006. Jensen’s inequality for g-expectation on convex (concave) function. Chinese Ann. Math. Ser. A 27 (5), 635–644 (in Chinese)].     

Two-point Correlation Function and Feynman-Kac Formula for the Stochastic Heat Equation

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on \(\mathbb {R}\). The bounds for p-th moments proved in Chen and Dalang (Ann. Probab. 2015) are simplified. We validate the Feynman-Kac formula for the p-point correlation function of the solutions to this equation with measure-valued initial data.     

Splitting of liftings in products of probability spaces II

For a probability measure R on a product of two probability spaces that is absolutely continuous with respect to the product measure we prove the existence of liftings subordinated to a regular conditional probability and the existence of a lifting for R with lifted sections which satisfies in addition a rectangle formula. These results improve essentially some of the results from the former work of the authors [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408], by weakening considerably the assumptions and by presenting more direct and shorter proofs. In comparison with [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408] it is crucial for applications intended that we can now prescribe one of the factor liftings completely freely. We demonstrate the latter by applications to τ-additive measures, transfer of strong liftings, and stochastic processes.     

Sufficiency and Jensen's inequality for conditional expectations

For finite sets of probability measures, sufficiency is characterized by means of certain positively homogeneous convex functions. The essential tool is a discussion of equality in Jensen's inequality for conditional expectations. In particular, it is shown that characterizations of sufficiency by Csiszár's f-divergence (1963, Publ. Math. Inst. Hung. Acad. Sci. Ser. A, 8, 85–107) and by optimal solutions of a Bayesian decision problem used by Morse and Sacksteder (1966, Ann. Math. Statist., 37, 203–214) can be proved by the same method.     

Vitesse de convergence uniforme presque sûre de l'estimateur linéaire par méthode d'ondelettes

We establish an uniform law of the iterated logarithm for the linear wavelet density estimator. A key tool in the proof of this result is the functional law of the iterated logarithm for the increments of the empirical process proved by Deheuvels and Mason (Ann. Probab. 20 (1992) 1248–1287). To cite this article: A. Massiani, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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 minimal members of convex expectations

In this paper, we show that for a convex expectation E[⋅] defined on L¹(Ω,F,P), the following statements are equivalent:

(i)

E is a minimal member of the set of all convex expectations defined on L¹(Ω,F,P);

(ii)

E is linear;

(iii)

two-dimensional Jensen inequality for E holds.

In addition, we prove a sandwich theorem for convex expectation and concave expectation.