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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

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] Baldi, P. (2017). Stochastic Calculus, Universitext. Springer, Switzerland. Google Scholar [6] Bertoin, J. (1996). Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge. Google Scholar [7] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. I. Representation results, SIAM J. Control. 13, no. 5, 999-1021. Google Scholar [8] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. II. Applications, SIAM. J. Control 13, no. 5, 1022-1061. Google Scholar [9] Cont, R. and P. Tankov. (2004). Financial Modelling with Jump Processes, Chapman & Hall, USA. Google Scholar [10] Dellacherie, C., B. Maisonneuve, and P.-A. Meyer. (1992). Probabilités et Potentiel-Processus de Markov, Hermann, France. Google Scholar [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. Google Scholar [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. Google Scholar [13] Gradshteyn, I.S. and I.M. Ryzhik. (2007). Table of integrals, series, and products, seventh edition, Elsevier/Academic Press, Amsterdam. Google Scholar [14] Herdegen, M. and S. Herrmann. (2016). Single jump processes and strict local martingales, Stoch. Process. Appl. 126, no. 2, 337-359. Google Scholar [15] Jacod, J. and A.V. Skorohod. (1994). Jumping filtrations and martingales with finite variation, Springer, Berlin. Google Scholar [16] Jeanblanc, M. and F. Vrins. (2018). Conic martingales from stochastic integrals, Math. Financ. 28, no. 2, 516-535. Google Scholar [17] Jeanblanc, M., M. Yor, and M. Chesney. (2007). Martingale Methods for Financial Markets, Springer Verlag, Berlin. Google Scholar [18] Kahale, N. (2008). Analytic crossing probabilities for certain barriers by Brownian motion, Ann. Appl. Probab. 18, no. 4, 1424-1440. Google Scholar [19] Karatzas, I. and S. Shreve. (2005). Brownian Motion and Stochastic Calculus, Springer, New York. Google Scholar [20] Mansuy, R. and M. Yor. (2006). Random Times and Enlargement of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, Springer, Berlin Heidelberg. Google Scholar [21] Protter, P. (2005). Stochastic Integration and Differential Equations, Second edition, Springer, Berlin. Google Scholar [22] Rainer, C. (1996). Projection d'une diffusion sur sa filtration lente, Springer, Berlin. Google Scholar [23] Revuz, D. and M. Yor. (1999). Continuous martingales and Brownian motion, Springer-Verlag, New-York. Google Scholar [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. Google Scholar [25] Salminen, P. (1997). On last exit decompositions of linear diffusions, Studia. Sci. Math. Hungar. 33, no. 1-3, 251-262. Google Scholar [26] Shreve, S.E. (2004). Stochastic Calculus for Finance vol. II-Continuous-time models, Springer, New York. Google Scholar [27] Vrins, F. (2016). Characteristic function of time-inhomogeneous Lévy-driven Ornstein-Uhlenbeck pro-cesses, Stat. Probab. Lett. 116, 55-61. Google Scholar

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##### References:
 [1] Aksamit, A. and M. Jeanblanc. (2017). Enlargement of Filtrations with Finance in View, Springer, Switzerland. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] Baldi, P. (2017). Stochastic Calculus, Universitext. Springer, Switzerland. Google Scholar [6] Bertoin, J. (1996). Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge. Google Scholar [7] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. I. Representation results, SIAM J. Control. 13, no. 5, 999-1021. Google Scholar [8] Boel, R., P. Varaiya, and E. Wong. (1975). Martingales on jump processes. II. Applications, SIAM. J. Control 13, no. 5, 1022-1061. Google Scholar [9] Cont, R. and P. Tankov. (2004). Financial Modelling with Jump Processes, Chapman & Hall, USA. Google Scholar [10] Dellacherie, C., B. Maisonneuve, and P.-A. Meyer. (1992). Probabilités et Potentiel-Processus de Markov, Hermann, France. Google Scholar [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. Google Scholar [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. Google Scholar [13] Gradshteyn, I.S. and I.M. Ryzhik. (2007). Table of integrals, series, and products, seventh edition, Elsevier/Academic Press, Amsterdam. Google Scholar [14] Herdegen, M. and S. Herrmann. (2016). Single jump processes and strict local martingales, Stoch. Process. Appl. 126, no. 2, 337-359. Google Scholar [15] Jacod, J. and A.V. Skorohod. (1994). Jumping filtrations and martingales with finite variation, Springer, Berlin. Google Scholar [16] Jeanblanc, M. and F. Vrins. (2018). Conic martingales from stochastic integrals, Math. Financ. 28, no. 2, 516-535. Google Scholar [17] Jeanblanc, M., M. Yor, and M. Chesney. (2007). Martingale Methods for Financial Markets, Springer Verlag, Berlin. Google Scholar [18] Kahale, N. (2008). Analytic crossing probabilities for certain barriers by Brownian motion, Ann. Appl. Probab. 18, no. 4, 1424-1440. Google Scholar [19] Karatzas, I. and S. Shreve. (2005). Brownian Motion and Stochastic Calculus, Springer, New York. Google Scholar [20] Mansuy, R. and M. Yor. (2006). Random Times and Enlargement of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, Springer, Berlin Heidelberg. Google Scholar [21] Protter, P. (2005). Stochastic Integration and Differential Equations, Second edition, Springer, Berlin. Google Scholar [22] Rainer, C. (1996). Projection d'une diffusion sur sa filtration lente, Springer, Berlin. Google Scholar [23] Revuz, D. and M. Yor. (1999). Continuous martingales and Brownian motion, Springer-Verlag, New-York. Google Scholar [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. Google Scholar [25] Salminen, P. (1997). On last exit decompositions of linear diffusions, Studia. Sci. Math. Hungar. 33, no. 1-3, 251-262. Google Scholar [26] Shreve, S.E. (2004). Stochastic Calculus for Finance vol. II-Continuous-time models, Springer, New York. Google Scholar [27] Vrins, F. (2016). Characteristic function of time-inhomogeneous Lévy-driven Ornstein-Uhlenbeck pro-cesses, Stat. Probab. Lett. 116, 55-61. Google Scholar
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