(MARTINGALE) OPTIMAL TRANSPORT AND ANOMALY DETECTION WITH NEURAL NETWORKS: A PRIMAL-DUAL ALGORITHM

In this paper, we introduce a primal-dual algorithm for solving (martingale) optimal transportation problems, with cost functions satisfying the twist condition, close to the one that has been used recently for training generative adversarial networks. As some additional applications, we consider anomaly detection and automatic generation of financial data

The maximum maximum of a martingale with given nn marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A dual algorithm for stochastic control problems : Applications to Uncertain Volatility Models and CVA

We derive an algorithm in the spirit of Rogers [SIAM J. Control Optim., 46 (2007), pp. 1116--1132] and Davis and Burstein [Stochastics Stochastics Rep., 40 (1992), pp. 203--256] that leads to upper bounds for stochastic control problems. Our bounds complement lower biased estimates recently obtained in Guyon and Henry-Labordère [J. Comput. Finance, 14 (2011), pp. 37--71]. We evaluate our estimates in numerical examples motivated by mathematical finance. Read More: http://epubs.siam.org/doi/10.1137/15M101994

An Explicit Martingale Version of the Onedimensional Brenier’s Theorem with Full Marginals Constraint

Abstract We provide an extension to the infinitely-many marginals case of the martingale version of the Fréchet-Hoeffding coupling (which corresponds to the one-dimensional Brenier theorem). In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi 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 As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps

Gravitational couplings of orientifold planes

We reanalyse the gravitational couplings of the perturbative orientifold planes $Op^-$, $Op^+$ (and D-branes). We first compute their $D_{-1}$ instantonic corrections for $p=3$. Then, by using U-dualities, we obtain the Wess-Zumino terms of orientifolds with RR flux for $p \leq 5$. The expressions for the effective actions can be partially checked via M-theory. We point out a previous oversimplification and we show in fact that the difficulty still stands in the way of the full computation of 7 Brane instanton corrections.Comment: 13 pages, 1 figure, 2 tables. 3 references adde

Borcherds symmetries in M-theory

It is well known but rather mysterious that root spaces of the $E_k$ Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on $T^k$ corresponds to blow-up of $k$ points in general position with respect to each other. All theories of the Magic triangle that reduce to the $E_n$ sigma model in three dimensions correspond to singular del Pezzo surfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively aris

An Explicit Martingale Version of Brenier’s Theorem ∗

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in [3, 24]. In this paper, we extend the one-dimensional Brenier’s theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in [4] by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in [3], and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.