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International audience; We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples.

International audience; We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multi-Level Monte-Carlo method. By using and adapting some results from Zhang [22], together with the Garsia-Rodemich-Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multi-Level) Monte-Carlo method, and compute their complexity. We finally provide some numerical simulation results.

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

We consider a discrete time financial market with proportional transaction cost under model uncertainty, and study a super-replication problem. We recover the duality results that are well known in the classical dominated context. Our key argument consists in using a randomization technique together with the minimax theorem to convert the initial problem to a frictionless problem set on an enlarged space. This allows us to appeal to the techniques and results of Bouchard and Nutz (2015) to obtain the duality result.

We prove a robust super-hedging duality result for path-dependent options on assets with jumps, in a continuous time setting. It requires that the collection of martingale measures is rich enough and that the payoff function satisfies some continuity property. It is a by-product of a quasi-sure version of the optional decomposition theorem, which can also be viewed as a functional version of It{\^o}'s Lemma, that applies to non-smooth functionals (of c{\`a}dl{\`a}g processes) which are only concave in space and non-increasing in time, in the sense of Dupire.

We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an $(H)$-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classical optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls, when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem, under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean-Vlasov control problem, that is to say proving that it consists in the limit of a large population control problem with common noise., Comment: 48 pages

International audience; This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Oblój, and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

We propose a new numerical scheme for Backward Stochastic Differential Equations based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm., 28 pages

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo [32] for viscosity solutions of path-dependent PDEs (PPDE), which extends the seminal work of Barles and Souganidis [1] on the viscosity solution of PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in [1]. These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the convergence of numerical schemes for non-Markovian stochastic control problems, second order BSDEs, stochastic differential games etc.

International audience; The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of c`adì ag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 .

We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions., 33 pages

To evaluate the state of centrifugal compressor unit, this paper presents a method based on Logistic Regression Model and Gaussian Model. First, the improved failure mode and effect analysis (FMEA) is used to select important unit parameters on the basis of subsystem of centrifugal compressor unit. And weights of selected important parameters are determined by means of analytic hierarchy process (AHP). Thus, the state evaluation index is established. Then, the distribution models of the parameters in state evaluation index are determined according to its characteristics. Meanwhile, the corresponding parameter distribution model is constructed based on the alarm range of each parameter. Finally, the quantitative evaluation model of centrifugal compressor unit is established using the evaluation value of each parameter distribution model instead of the expert experience score. We applied the model to a motor-driven centrifugal compressor unit in a gas transmission station, and the evaluation result demonstrates the model can evaluate the state of compressor unit accurately.

Gastric cancer is an important cancer type worldwide, the anti‑angiogenic agent BC001 can target the vascular endothelial growth factor receptor 2 (VEGFR2), and significantly suppresses the growth of gastric cancer BGC823 cells in vitro and in vivo. However, numerous results indicated that antiangiogenic drugs could induce autophagy, and the inhibition of autophagy enhanced the anticancer effects of antiangiogenic agents. In the present study, hydroxychloroquine (HCQ), an inhibitor of autophagy, enhanced the antiproliferative and proapoptotic effects of BC001 in vitro. Furthermore, HCQ enhanced the antitumor effects of BC001 on BGC823 xenograft tumors in vivo. Of note, BC001 neither induced nor inhibited autophagy. RNA‑sequencing results revealed that HCQ regulated autophagy or lysosomal‑associated genes, such as tumor protein p53‑inducible nuclear protein 1, interleukin (IL)1B, tumor necrosis factor (TNF), Mediterranean fever, ubiquitin specific peptidase 36, IL6, neuraminidase (NEU)1, ATP‑binding cassette subfamily A member 1, proprotein convertase subtilisin/kexin type 9, myelin basic protein and NEU3. Importantly, HCQ was determined to affect multiple pathways, including 'negative regulation of endothelial cell proliferation', 'blood vessel remodeling', 'cell surface receptor signaling pathways' and 'notch receptor processing' associated with 'signal transduction', 'cancers' and 'immune system', through regulating C‑X‑C motif chemokine ligand 8, TNF, IL6, intercellular adhesion molecule 1 and FOS genes. In summary, HCQ was proposed to enhance the anticancer effects of BC001 in gastric cancer via complex mechanisms.

We extend the martingale version of the one-dimensional Brenier's theorem (Fr echet-Hoeffding coupling), established in Henry-Labord ere and Touzi to the infinitely-many marginals case. In short, their results give an explicit characterization of the optimal martingale transference plans as well as the optimal dual components of a two marginals discrete-time martingale transportation (MT) problem for a large class of reward functions. We consider here the limiting continuous-time case, which leads to an infinitely-many marginals MT problem. By approximation technique, we show that for a class of reward functions, the optimal martingale transference plan is provided by a pure downward jump local L evy model. In particular, it provides a new construction of the martingale peacock process (PCOC "Processus Croissant pour l'Ordre Convexe," see Hirsch, Profeta, Roynette and Yor), and a new remarkable example of discontinuous fake Brownian motions. Further, as in, we also provide a duality result together with dual optimizer in explicit form. Finally, as an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

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 Beiglbock, Cox, and Huesmann [Optimal Transport and Skorokhod Embedding, Preprint, 2013] to the case of finitely many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than Beiglbock, Cox, and Huesmann. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.

We consider a discrete time financial market with proportional transaction costs under model uncertainty, and study a num\'eraire-based semi-static utility maximization problem with an exponential utility preference. The randomization techniques recently developed in \cite{BDT17} allow us to transform the original problem into a frictionless counterpart on an enlarged space. By suggesting a different dynamic programming argument than in \cite{bartl2016exponential}, we are able to prove the existence of the optimal strategy and the convex duality theorem in our context with transaction costs. In the frictionless framework, this alternative dynamic programming argument also allows us to generalize the main results in \cite{bartl2016exponential} to a weaker market condition. Moreover, as an application of the duality representation, some basic features of utility indifference prices are investigated in our robust setting with transaction costs., Comment: Final version, to appear in Mathematics of Operations Research

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)., Comment: 48 pages. Minor modifications compared to the published version

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordere, Obloj, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509–536] peacock under their “increasing mean residual value” condition. We also discuss the associated martingale inequality.

We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [Theory Probab. Appl. 9 (1964) 445–449], Watanabe [J. Math. Kyoto Univ. 4 (1965) 385–398] and McKean [Comm. Pure Appl. Math. 28 (1975) 323–331], by allowing for polynomial nonlinearity in the pair $(u,Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to “small maturity” or “small nonlinearity” of the PDE. Our main ingredient is the Malliavin automatic differentiation technique as in [Ann. Appl. Probab. 27 (2017) 3305–3341], based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

International audience; We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.

We give a study to the algorithm for semi-linear parabolic PDEs in Henry-Labordere (2012) 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 et al. (to appear) [5] and extended in Ekren et al. (2012) [6] , [7] .