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January  2019, 4: 2 doi: 10.1186/s41546-019-0036-4

## Piecewise constant martingales and lazy clocks

 1. Université d'Évry, France; 2. Louvain Finance Center(LFIN) and Center for Operations Research and Econometrics(CORE), Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium

Received  March 2018 Revised  January 17, 2019 Published  February 2019

Conditional expectations (like, e.g., discounted prices in financial applications) are martingales under an appropriate filtration and probability measure. When the information flow arrives in a punctual way, a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates. Providing a way to find and construct piecewise constant martingales evolving in a connected subset of $\mathbb{R}$ is the purpose of this paper. After a brief review of possible standard techniques, we propose a construction scheme based on the sampling of latent martingales $\tilde Z$ with lazy clocks θ. These θ are time-change processes staying in arrears of the true time but that can synchronize at random times to the real (calendar) clock. This specific choice makes the resulting time-changed process Zt = $\tilde Z$θt a martingale (called a lazy martingale) without any assumption on $\tilde Z$, and in most cases, the lazy clock θ is adapted to the filtration of the lazy martingale Z, so that sample paths of Z on [0, T ] only requires sample paths of (θ, $\tilde Z$) up to T. This would not be the case if the stochastic clock θ could be ahead of the real clock, as is typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (interval of) $\mathbb{R}$.
Citation: Christophe Profeta, Frédéric Vrins. Piecewise constant martingales and lazy clocks. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 2-. doi: 10.1186/s41546-019-0036-4
##### References:
 [1] Aksamit, A. and M. Jeanblanc. (2017). Enlargement of Filtrations with Finance in View, Springer, Switzerland. [2] Altman, E., B. Brady, A. Resti, and A. Sironi. (2003). The link between defaults and recovery rates:theory, empirical evidence, and implications. Technical report, Stern School of Business. [3] Amraoui, S., L. Cousot, S. Hitier, and J.-P. Laurent. (2012). Pricing CDOs with state-dependent stochastic recovery rates, Quant. Finan. 12, no. 8, 1219-1240. [4] Andersen, L. and J. Sidenius. (2004). Extensions to the gaussian copula:random recovery and random factor loadings, J. Credit Risk 1, no. 1, 29-70. [5] Baldi, P. (2017). Stochastic Calculus, Universitext. Springer, Switzerland. [6] Bertoin, J. (1996). Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge. [7] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. I. Representation results, SIAM J. Control. 13, no. 5, 999-1021. [8] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. II. Applications, SIAM. J. Control 13, no. 5, 1022-1061. [9] Cont, R. and P. Tankov. (2004). Financial Modelling with Jump Processes, Chapman & Hall, USA. [10] Dellacherie, C., B. Maisonneuve, and P.-A. Meyer. (1992). Probabilités et Potentiel-Processus de Markov, Hermann, France. [11] Gaspar, R. and I. Slinko. (2008). On recovery and intensity's correlation-a new class of credit models, J. Credit Risk 4, no. 2, 1-33. [12] Gradinaru, M., B. Roynette, P. Vallois, and M. Yor. (1999). Abel transform and integrals of Bessel local times, Ann. Inst. H. Poincaré Probab. Statist. 35, no. 4, 531-572. [13] Gradshteyn, I.S. and I.M. Ryzhik. (2007). Table of integrals, series, and products, seventh edition, Elsevier/Academic Press, Amsterdam. [14] Herdegen, M. and S. Herrmann. (2016). Single jump processes and strict local martingales, Stoch. Process. Appl. 126, no. 2, 337-359. [15] Jacod, J. and A.V. Skorohod. (1994). Jumping filtrations and martingales with finite variation, Springer, Berlin. [16] Jeanblanc, M. and F. Vrins. (2018). Conic martingales from stochastic integrals, Math. Financ. 28, no. 2, 516-535. [17] Jeanblanc, M., M. Yor, and M. Chesney. (2007). Martingale Methods for Financial Markets, Springer Verlag, Berlin. [18] Kahale, N. (2008). Analytic crossing probabilities for certain barriers by Brownian motion, Ann. Appl. Probab. 18, no. 4, 1424-1440. [19] Karatzas, I. and S. Shreve. (2005). Brownian Motion and Stochastic Calculus, Springer, New York. [20] Mansuy, R. and M. Yor. (2006). Random Times and Enlargement of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, Springer, Berlin Heidelberg. [21] Protter, P. (2005). Stochastic Integration and Differential Equations, Second edition, Springer, Berlin. [22] Rainer, C. (1996). Projection d'une diffusion sur sa filtration lente, Springer, Berlin. [23] Revuz, D. and M. Yor. (1999). Continuous martingales and Brownian motion, Springer-Verlag, New-York. [24] Salminen, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Probab. 20, no. 1, 411-426. [25] Salminen, P. (1997). On last exit decompositions of linear diffusions, Studia. Sci. Math. Hungar. 33, no. 1-3, 251-262. [26] Shreve, S.E. (2004). Stochastic Calculus for Finance vol. II-Continuous-time models, Springer, New York. [27] Vrins, F. (2016). Characteristic function of time-inhomogeneous Lévy-driven Ornstein-Uhlenbeck pro-cesses, Stat. Probab. Lett. 116, 55-61.

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##### References:
 [1] Aksamit, A. and M. Jeanblanc. (2017). Enlargement of Filtrations with Finance in View, Springer, Switzerland. [2] Altman, E., B. Brady, A. Resti, and A. Sironi. (2003). The link between defaults and recovery rates:theory, empirical evidence, and implications. Technical report, Stern School of Business. [3] Amraoui, S., L. Cousot, S. Hitier, and J.-P. Laurent. (2012). Pricing CDOs with state-dependent stochastic recovery rates, Quant. Finan. 12, no. 8, 1219-1240. [4] Andersen, L. and J. Sidenius. (2004). Extensions to the gaussian copula:random recovery and random factor loadings, J. Credit Risk 1, no. 1, 29-70. [5] Baldi, P. (2017). Stochastic Calculus, Universitext. Springer, Switzerland. [6] Bertoin, J. (1996). Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge. [7] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. I. Representation results, SIAM J. Control. 13, no. 5, 999-1021. [8] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. II. Applications, SIAM. J. Control 13, no. 5, 1022-1061. [9] Cont, R. and P. Tankov. (2004). Financial Modelling with Jump Processes, Chapman & Hall, USA. [10] Dellacherie, C., B. Maisonneuve, and P.-A. Meyer. (1992). Probabilités et Potentiel-Processus de Markov, Hermann, France. [11] Gaspar, R. and I. Slinko. (2008). On recovery and intensity's correlation-a new class of credit models, J. Credit Risk 4, no. 2, 1-33. [12] Gradinaru, M., B. Roynette, P. Vallois, and M. Yor. (1999). Abel transform and integrals of Bessel local times, Ann. Inst. H. Poincaré Probab. Statist. 35, no. 4, 531-572. [13] Gradshteyn, I.S. and I.M. Ryzhik. (2007). Table of integrals, series, and products, seventh edition, Elsevier/Academic Press, Amsterdam. [14] Herdegen, M. and S. Herrmann. (2016). Single jump processes and strict local martingales, Stoch. Process. Appl. 126, no. 2, 337-359. [15] Jacod, J. and A.V. Skorohod. (1994). Jumping filtrations and martingales with finite variation, Springer, Berlin. [16] Jeanblanc, M. and F. Vrins. (2018). Conic martingales from stochastic integrals, Math. Financ. 28, no. 2, 516-535. [17] Jeanblanc, M., M. Yor, and M. Chesney. (2007). Martingale Methods for Financial Markets, Springer Verlag, Berlin. [18] Kahale, N. (2008). Analytic crossing probabilities for certain barriers by Brownian motion, Ann. Appl. Probab. 18, no. 4, 1424-1440. [19] Karatzas, I. and S. Shreve. (2005). Brownian Motion and Stochastic Calculus, Springer, New York. [20] Mansuy, R. and M. Yor. (2006). Random Times and Enlargement of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, Springer, Berlin Heidelberg. [21] Protter, P. (2005). Stochastic Integration and Differential Equations, Second edition, Springer, Berlin. [22] Rainer, C. (1996). Projection d'une diffusion sur sa filtration lente, Springer, Berlin. [23] Revuz, D. and M. Yor. (1999). Continuous martingales and Brownian motion, Springer-Verlag, New-York. [24] Salminen, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Probab. 20, no. 1, 411-426. [25] Salminen, P. (1997). On last exit decompositions of linear diffusions, Studia. Sci. Math. Hungar. 33, no. 1-3, 251-262. [26] Shreve, S.E. (2004). Stochastic Calculus for Finance vol. II-Continuous-time models, Springer, New York. [27] Vrins, F. (2016). Characteristic function of time-inhomogeneous Lévy-driven Ornstein-Uhlenbeck pro-cesses, Stat. Probab. Lett. 116, 55-61.
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