Representation theorems for generators of BSDEs with monotonic and convex growth generators

In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short), whose generators are monotonic and convex growth in $y$ and quadratic growth in $z$. We also obtain a converse comparison theorem for such BSDEs.Comment: 12 pages, revised version. arXiv admin note: text overlap with arXiv:1405.478

Representation theorems for generators of Reflected BSDEs with continuous and linear-growth generators

In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDE), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDE). As some applications, a general converse comparison theorem for RBSDE is obtained and some properties of RBSDE are discussed.Comment: 13 pages. Final versio

Representation theorem for generators of quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in $z$. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic $g$-expectation.Comment: 16 pages. The proof of Lemma 3.3 is rewritten. Comments are welcom

Anticipated backward stochastic differential equations

In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/08-AOP423 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the representation for dynamically consistent nonlinear evaluations: uniformly continuous case

A system of dynamically consistent nonlinear evaluation (${\cal{F}}$-evaluation) provides an ideal characterization for the dynamical behaviors of risk measures and the pricing of contingent claims. The purpose of this paper is to study the representation for the ${\cal{F}}$-evaluation by the solution of a backward stochastic differential equation (BSDE). Under a general domination condition, we prove that any ${\cal{F}}$-evaluation can be represented by the solution of a BSDE with a generator which is Lipschitz in $y$ and uniformly continuous in $z$.Comment: 34 pages. minor corrections. Comments are welcom

Representation for filtration-consistent nonlinear expectations under a general domination condition

In this paper, we consider filtration-consistent nonlinear expectations which satisfy a general domination condition (dominated by ${\cal{E}}^{\phi}$). We show that this kind of nonlinear expectations can be represented by $g$-expectations defined by the solutions of backward stochastic differential equations, whose generators are independent on $y$ and uniformly continuous in $z$.Comment: 21 pages, Remark 3.1 is rewritten. Assumption (H2) in the results are eliminated. Comments are welcom

Non-exponential Sanov and Schilder theorems on Wiener space: BSDEs, Schr\"odinger problems and Control

We derive new limit theorems for Brownian motion, which can be seen as non-exponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits of backward stochastic differential equations and their related partial differential equations. As a second application, we extend prior results on the small-noise limit of the Schr\"odinger problem as an optimal transport cost, unifying the control-theoretic and probabilistic approaches initiated respectively by T. Mikami and C. L\'eonard. Lastly, our results suggest a new scheme for the computation of mean field optimal control problems, distinct from the conventional particle approximation. A key ingredient in our analysis is an extension of the classical variational formula (often attributed to Borell or Bou\'e-Dupuis) for the Laplace transform of Wiener measure

Multidimensional dynamic risk measure via conditional g-expectation

This paper deals with multidimensional dynamic risk measures induced by conditional $g$-expectations. A notion of multidimensional $g$-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional $g$-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic $g$-risk measure is nonincreasingly convex if and only if the generator $g$ satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex $g$-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries; optimal risk sharing for \protect\gamma -tolerant $g$-risk measures is investigated. Insurance $g$-risk measure and other ways to induce $g$-risk measures are also studied at the end of the paper.Comment: 37 page

Convexity bounds for BSDE solutions, with applications to indifference valuation

We consider backward stochastic differential equations (BSDEs) with a particular quadratic generator and study the behaviour of their solutions when the probability measure is changed, the filtration is shrunk, or the underlying probability space is transformed. Our main results are upper bounds for the solutions of the original BSDEs in terms of solutions to other BSDEs which are easier to solve. We illustrate our results by applying them to exponential utility indifference valuation in a multidimensional Itô process settin

The Bellman equation for power utility maximization with semimartingales

We study utility maximization for power utility random fields with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. We show that any optimal strategy leads to a solution of the corresponding Bellman equation. The optimal strategies are described pointwise in terms of the opportunity process, which is characterized as the minimal solution of the Bellman equation. We also give verification theorems for this equation.Comment: Published in at http://dx.doi.org/10.1214/11-AAP776 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org