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On SDEs with marginal laws evolving in finite-dimensional exponential families

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  • Brigo, Damiano

Abstract

In the present paper, given a diffusion coefficient and a curve in an exponential family, we define a drift such that the density of the resulting diffusion process evolves in the prescribed exponential family according to the given curve. As an application to mathematical finance, we construct a family of stock price processes that are equivalent in discrete time while implying arbitrary prices for options written on them. As an application to nonlinear filtering, we construct nonlinear filtering problems admitting a finite-dimensional filter.

Suggested Citation

  • Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
    • Handle: RePEc:eee:stapro:v:49:y:2000:i:2:p:127-134
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    References listed on IDEAS

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    1. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    2. Damiano Brigo & Fabio Mercurio, 2000. "Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices," Finance and Stochastics, Springer, vol. 4(2), pages 147-159.
    3. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
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    Cited by:

    1. Damiano Brigo, 2008. "The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation," Papers 0812.4052, arXiv.org.
    2. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    3. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.
    4. D. Brigo, 2023. "Probability-Free Models in Option Pricing: Statistically Indistinguishable Dynamics and Historical vs Implied Volatility," World Scientific Book Chapters, in: David Gershon & Alexander Lipton & Mathieu Rosenbaum & Zvi Wiener (ed.), Options — 45 years since the Publication of the Black–Scholes–Merton Model The Gershon Fintech Center Conference, chapter 4, pages 47-61, World Scientific Publishing Co. Pte. Ltd..

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