We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence. We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge-Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.Comment: Published in at http://dx.doi.org/10.1214/12-AOP797 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).1 generally based on the stability of the controls with respect to conditioning and concatenation, together with a measurable selection argument, which, roughly speaking, allow to prove the measurability of the associated value function, as well as constructing almost optimal controls through "pasting". This is exactly the approach followed by Bertsekas and Shreve [6], and Dellacherie [25] for discrete time stochastic control problems. In continuous time, a comprehensive study of the dynamic programming principle remained more elusive. Thus, El Karoui,in [34], established the dynamic programming principle for the optimal stopping problem in a continuous time setting, using crucially the strong stability properties of stopping times, as well as the fact that the measurable selection argument can be avoided in this context, since an essential supremum over stopping times can be approximated by a supremum over a countable family of random variables. Later, for general controlled Markov processes (in continuous time) problems, El Karoui, Huu Nguyen and Jeanblanc [36] provided a framework to derive the dynamic programming principle using the measurable selection theorem, by interpreting the controls as probability measures on the canonical trajectory space (see e.g. Theorems 6.2, 6.3 and 6.4 of [36]). Another commonly used approach to derive the DPP was to bypass the measurable selection argument by proving, under additional assumptions, a priori regularity of the value function. This was the strategy adopted, among others, by Fleming and Soner [43], and in the so-called weak DPP of Bouchard and Touzi [15], which has then been extended by Bouchard and Nutz [10,12] and Bouchard, Moreau and Nutz [9] to optimal control problems wit...

We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrödinger equations.

We give a study to the algorithm for semi-linear parabolic PDEs in Henry-Labordère [11] and then generalize it to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang [5] and extended in Ekren, Touzi and Zhang [6,7].

Background: The unique properties of wheat flour primarily depend on gluten, which is the most important source of protein for human being. γ-Gliadins have been considered to be the most ancient of the wheat gluten family. The complex family structure of γ-gliadins complicates the determination of their function. Moreover, γ-gliadins contain several sets of celiac disease epitopes. However, no systematic research has been conducted yet.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC "Processus Croissant pour l'Ordre Convexe," see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem in Beiglböck, Cox & Huesmann [1] to the case of finitely-many marginal constraints 1 . Using the classical convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than [1]. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.