• Bernoulli
  • Volume 23, Number 4A (2017), 2784-2807.

Conditional convex orders and measurable martingale couplings

Lasse Leskelä and Matti Vihola

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Strassen’s classical martingale coupling theorem states that two random vectors are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analysing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for random vectors conditioned on a random element taking values in a general measurable space. We provide an analogue of the conditional martingale coupling theorem in the language of probability kernels, and discuss how it can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.

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Bernoulli, Volume 23, Number 4A (2017), 2784-2807.

Received: June 2014
Revised: October 2015
First available in Project Euclid: 9 May 2017

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conditional coupling convex stochastic order increasing convex stochastic order martingale coupling pointwise coupling probability kernel


Leskelä, Lasse; Vihola, Matti. Conditional convex orders and measurable martingale couplings. Bernoulli 23 (2017), no. 4A, 2784--2807. doi:10.3150/16-BEJ827.

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  • [1] Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser.
  • [2] Andrieu, C. and Roberts, G.O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Statist. 37 697–725.
  • [3] Andrieu, C. and Vihola, M. (2015). Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms. Ann. Appl. Probab. 25 1030–1077.
  • [4] Andrieu, C. and Vihola, M. (2016). Establishing some order amongst exact approximations of MCMCs. Ann. Appl. Probab. To appear.
  • [5] Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-independent bounds for option prices – A mass transport approach. Finance Stoch. 17 477–501.
  • [6] Beiglböck, M. and Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44 42–106.
  • [7] Borglin, A. and Keiding, H. (2002). Stochastic dominance and conditional expectation – An insurance theoretical approach. Geneva Pap. Risk Insur., Theory 27 31–48.
  • [8] Caracciolo, S., Pelissetto, A. and Sokal, A.D. (1990). Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints. J. Stat. Phys. 60 1–53.
  • [9] Dolinsky, Y. and Soner, H.M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
  • [10] Fontbona, J., Guérin, H. and Méléard, S. (2010). Measurability of optimal transportation and strong coupling of martingale measures. Electron. Commun. Probab. 15 124–133.
  • [11] Galichon, A., Henry-Labordère, P. and Touzi, N. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24 312–336.
  • [12] Himmelberg, C.J. (1975). Measurable relations. Fund. Math. 87 53–72.
  • [13] Hirsch, F., Profeta, C., Roynette, B. and Yor, M. (2011). Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series 3. Milan: Springer.
  • [14] Hirshberg, A. and Shortt, R.M. (1998). A version of Strassen’s theorem for vector-valued measures. Proc. Amer. Math. Soc. 126 1669–1671.
  • [15] Hobson, D. and Neuberger, A. (2012). Robust bounds for forward start options. Math. Finance 22 31–56.
  • [16] Hobson, D.G. (1998). The maximum maximum of a martingale. In Séminaire de Probabilités XXXII. Lecture Notes in Math. 1686 250–263. Berlin: Springer.
  • [17] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications. New York: Springer.
  • [18] Kellerer, H.G. (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198 99–122.
  • [19] Kertz, R.P. and Rösler, U. (2000). Complete lattices of probability measures with applications to martingale theory. In Game Theory, Optimal Stopping, Probability and Statistics. Institute of Mathematical Statistics Lecture Notes – Monograph Series 35 153–177. Beachwood, OH: IMS.
  • [20] Kuratowski, K. and Ryll-Nardzewski, C. (1965). A general theorem on selectors. Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 13 397–403.
  • [21] Leisen, F. and Mira, A. (2008). An extension of Peskun and Tierney orderings to continuous time Markov chains. Statist. Sinica 18 1641–1651.
  • [22] Leskelä, L. (2010). Stochastic relations of random variables and processes. J. Theoret. Probab. 23 523–546.
  • [23] Lowther, G. (2008). Fitting martingales to given marginals. Preprint. Available at arXiv:0808.2319v1.
  • [24] Marshall, A.W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering 143. New York: Academic Press.
  • [25] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [26] Peskun, P.H. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60 607–612.
  • [27] Rüschendorf, L. (1991). On conditional stochastic ordering of distributions. Adv. in Appl. Probab. 23 46–63.
  • [28] Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. Springer Series in Statistics. New York: Springer.
  • [29] Shortt, R.M. (1983). Strassen’s marginal problem in two or more dimensions. Z. Wahrsch. Verw. Gebiete 64 313–325.
  • [30] Srivastava, S.M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180. New York: Springer.
  • [31] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat. 36 423–439.
  • [32] Stroock, D.W. (1993). Probability Theory, an Analytic View. Cambridge: Cambridge Univ. Press.
  • [33] Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics 97. New York: Springer.
  • [34] Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Probab. 8 1–9.
  • [35] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Berlin: Springer.
  • [36] Whitt, W. (1980). Uniform conditional stochastic order. J. Appl. Probab. 17 112–123.
  • [37] Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. J. Appl. Probab. 22 619–633.