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Conditioned stochastic differential equations: theory, examples and application to finance

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  • Baudoin, Fabrice

Abstract

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a "quantitative" value to this weak information.

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  • Baudoin, Fabrice, 0. "Conditioned stochastic differential equations: theory, examples and application to finance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 109-145, July.
    • Handle: RePEc:eee:spapps:v:100:y::i:1-2:p:109-145
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    References listed on IDEAS

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    1. Amendinger, Jürgen & Becherer, Dirk & Schweizer, Martin, 2000. "Quantifying the value of initial investment information," SFB 373 Discussion Papers 2000,41, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 263-286, July.
    3. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Cited by:

    1. Sottinen, Tommi & Yazigi, Adil, 2014. "Generalized Gaussian bridges," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3084-3105.
    2. D'Auria, Bernardo & Salmerón Garrido, José Antonio, 2019. "Insider information and its relation with the arbitrage condition and the utility maximization problem," DES - Working Papers. Statistics and Econometrics. WS 28805, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Alexander Schied, 2005. "Optimal Investments for Robust Utility Functionals in Complete Market Models," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 750-764, August.
    4. Markussen, Bo, 2009. "Laplace approximation of transition densities posed as Brownian expectations," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 208-231, January.
    5. Pierre Henry-Labordere, 2019. "From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models," Working Papers hal-02090807, HAL.
    6. Bernardo D'Auria & Jos'e Antonio Salmer'on, 2017. "Valuing the anticipative information on the stochastic short interest rates," Papers 1711.03642, arXiv.org, revised Nov 2021.
    7. Bernardo D'Auria & Jos'e Antonio Salmer'on, 2019. "Insider information and its relation with the arbitrage condition and the utility maximization problem," Papers 1909.03430, arXiv.org, revised Dec 2019.
    8. Alili, Larbi & Wu, Ching-Tang, 2009. "Further results on some singular linear stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1386-1399, April.
    9. Alexander Schied & Ching-Tang Wu, 2005. "Duality theory for optimal investments under model uncertainty," SFB 649 Discussion Papers SFB649DP2005-025, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Sep 2005.
    10. Luciano Campi & Matteo del Vigna, 2011. "Weak Insider Trading and Behavioral Finance," Working Papers hal-00566185, HAL.
    11. Anastasis Kratsios, 2017. "Optimal Stochastic Decensoring and Applications to Calibration of Market Models," Papers 1712.04844, arXiv.org, revised Dec 2017.
    12. Nikolas Nüsken & Lorenz Richter, 2021. "Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-48, August.
    13. Pierre Henry-Labordere, 2019. "From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models," Papers 1904.04554, arXiv.org.
    14. Schied Alexander & Wu Ching-Tang, 2005. "Duality theory for optimal investments under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(3/2005), pages 199-217, March.
    15. Hoyle, Edward & Hughston, Lane P. & Macrina, Andrea, 2011. "Lévy random bridges and the modelling of financial information," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 856-884, April.
    16. Bernardo D'Auria & Jos'e A. Salmer'on, 2021. "Anticipative information in a Brownian-Poissonmarket: the binary information," Papers 2111.01529, arXiv.org.
    17. D'Auria, Bernardo & Salmerón Garrido, José Antonio, 2021. "Anticipative information in a Brownian-Poisson market: the binary information," DES - Working Papers. Statistics and Econometrics. WS 33624, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Paolo Guasoni, 2006. "Asymmetric Information in Fads Models," Finance and Stochastics, Springer, vol. 10(2), pages 159-177, April.
    19. Geoff Lindsell, 2022. "Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets in Initially Enlarged Filtrations," Papers 2203.08859, arXiv.org, revised Mar 2022.

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