## The Annals of Probability

### An iterated Azéma–Yor type embedding for finitely many marginals

#### Abstract

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azéma and Yor [In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$.

In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordère et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

#### Article information

Source
Ann. Probab., Volume 45, Number 4 (2017), 2210-2247.

Dates
Revised: October 2015
First available in Project Euclid: 11 August 2017

https://projecteuclid.org/euclid.aop/1502438426

Digital Object Identifier
doi:10.1214/16-AOP1110

Mathematical Reviews number (MathSciNet)
MR3693961

Zentralblatt MATH identifier
1380.60048

#### Citation

Obłój, Jan; Spoida, Peter. An iterated Azéma–Yor type embedding for finitely many marginals. Ann. Probab. 45 (2017), no. 4, 2210--2247. doi:10.1214/16-AOP1110. https://projecteuclid.org/euclid.aop/1502438426

#### References

• [1] Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (C. Dellacherie, P. A. Meyer and M. Weil, eds.). Lecture Notes in Math. 721 90–115. Springer, Berlin.
• [2] Azéma, J., Gundy, R. F. and Yor, M. (1980). Sur l’intégrabilité uniforme des martingales continues. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 53–61. Springer, Berlin.
• [3] Beiglböck, M., Cox, A. M. G. and Huesmann, M. (2015). Optimal transport and Skorokhod embedding. Available at arXiv:1307.3656v3.
• [4] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). The maximum maximum of a martingale constrained by an intermediate law. Probab. Theory Related Fields 119 558–578.
• [5] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). Robust hedging of barrier options. Math. Finance 11 285–314.
• [6] Carraro, L., El Karoui, N. and Obłój, J. (2012). On Azéma–Yor processes, their optimal properties and the Bachelier-drawdown equation. Ann. Probab. 40 372–400.
• [7] Cox, A. M. G. and Obłój, J. (2011). Robust hedging of double touch barrier options. SIAM J. Financial Math. 2 141–182.
• [8] Cox, A. M. G. and Obłój, J. (2011). Robust pricing and hedging of double no-touch options. Finance Stoch. 15 573–605.
• [9] Cox, A. M. G., Obłój, J. and Touzi, N. (2015). The Root solution to the multi-marginal embedding problem: An optimal stopping and time-reveral approach. Available at arXiv:1505.03169.
• [10] Cox, A. M. G. and Wang, J. (2013). Root’s barrier: Construction, optimality and applications to variance options. Ann. Appl. Probab. 23 859–894.
• [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] Guo, G., Tan, X. and Touzi, N. (2017). On the monotonicity principle of optimal Skorokhod embedding problem. SIAM J. Control Optim. 54 2478–2489.
• [13] Henry-Labordère, P., Obłój, J., Spoida, P. and Touzi, N. (2016). The maximum maximum of a martingale with given $n$ marginals. Ann. Appl. Probab. 26 1–44.
• [14] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris–Princeton Lectures on Mathematical Finance 2010 (R. A. Carmona, I. Çinlar, E. Ekeland, E. Jouini, J. A. Scheinkman and N. Touzi, eds.). Lecture Notes in Math. 2003 267–318. Springer, Berlin.
• [15] Hobson, D. and Neuberger, A. (2012). Robust bounds for forward start options. Math. Finance 22 31–56.
• [16] Hobson, D. G. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
• [17] Loynes, R. M. (1970). Stopping times on Brownian motion: Some properties of Root’s construction. Z. Wahrsch. Verw. Gebiete 16 211–218.
• [18] Madan, D. B. and Yor, M. (2002). Making Markov martingales meet marginals: With explicit constructions. Bernoulli 8 509–536.
• [19] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Stat. 43 1293–1311.
• [20] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
• [21] Obłój, J. (2010). Skorokhod Embedding. In Encyclopedia of Quantitative Finance (R. Cont, ed.) 1653–1657. Wiley, Chichester.
• [22] Obłój, J., Spoida, P. and Touzi, N. (2015). Martingale inequalities for the maximum via pathwise arguments. In In Memoriam Marc Yor—Séminaire n Probabilités XLVII (C. Donati-Martin, A. Lejay and A. Rouault, eds.). Lecture Notes in Math. 2137 227–247. Springer, Cham.
• [23] Rogers, L. C. G. (1989). A guided tour through excursions. Bull. Lond. Math. Soc. 21 305–341.
• [24] Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.
• [25] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat. 36 423–439.