Optimal transportation under controlled stochastic dynamics

By Xiaolu Tan, Nizar Touzi. 2013

145 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/a972e0a73839ece3bd2af0400713ba6db46609db

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Citations in the corpus

Complete Duality for Martingale Optimal Transport on the Line (2015, background): “, [1, 8, 12, 13, 14, 19, 33] for models in discrete time and [6, 10, 15, 17, 18, 21, 22, 29, 31, 32, 35, 37] for continuous-time models.”
Applications of weak transport theory (2020, unknown): “This is a stochastic mass transport problem (or optimal transport of semimartingales) as introduced by Tan and Touzi [50].”
Causal transport plans and their Monge–Kantorovich problems (2013, background): “Recently these ideas have also received a growing interest through their implications in financial mathematics (for instance see [35], [6] and the references therein).”
An Explicit Martingale Version of the One-Dimensional Brenier's Theorem with Full Marginals Constraint (2016, unknown): “We finally mention the work by Beiglböck, Cox & Huesmann [3] which derives new results on the Skorohod embedding problem by using the martingale transport approach, see also Beiglböck, Cox, Huesmann, Perkovsky & Promel [4], and Guo, Tan & Touzi [35] [36].”
Kantorovich duality for general transport costs and applications (2014, methodology): “We refer to Mikami [25], Léonard [19], Tan-Touzi [35] for similar strategies.”
Stability of martingale optimal transport and weak optimal transport (2019, unknown): “tter class of problems from this viewpoint. 1.1. Stability of martingale optimal transport. The martingale optimal transport problem is a variant of (OT) stemming from robust mathematical ﬁnance (cf. [29, 14, 38, 21, 20, 17, 12, 8, 31, 32, 35, 19, 30, 22, 27] among many others). In order to deﬁne th”
The maximum maximum of a martingale with given $\mathbf{n}$ marginals (2012, background): “The ﬁrst one, reported in Section 4, consists in a short pathwise argument, based on a guess of the form of the optimal superhedging strategy for a simple one-touch barrier option, combined with the Skorokhod embedding results of [33].”
Model-free superhedging duality (2015, unknown): “The reformulation of the superhedging duality in the framework of optimal mass transport led to important results both in discrete and continuous time as in [3, 11, 12, 14, 15, 20, 23].”
Capacities, Measurable Selection and Dynamic Programming Part I: Abstract Framework (2013, methodology): “Using the same techniques, the DPP has been proved for some speciﬁc control problems, e.g. in Tan and Touzi [31], Neufeld and Nutz [26] etc.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “[23] A. Galichon, P. Henry-Labordère, and N. Touzi.”
Option pricing with linear market impact and nonlinear Black–Scholes equations (2018, unknown): “There is also a clear connection between this work and the work by Soner, Touzi and Zhang [35] about dual formulation of second-order target problems, and also with the problem of optimal transport by controlled diffusions studied in [36].”
A MODEL‐FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER‐REPLICATION THEOREM (2013, result): “eoryof optimaltransport; see [GHLT11, TT11, BHLP12]. In [GHLT11] Galichon,Henry-Laborde`re and Touzi systematically use a controlled stochastic dynamics approach, building on results of Tan and Touzi [TT11]. This enables the authors to derive the equation pM(Φ) = pR(Φ) in the context of the lookback”
Arbitrage and duality in nondominated discrete-time models (2013, background): “The authors thank Mathias Beiglböck and Jan Obłój for their constructive comments.”