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Differentiability of quadratic BSDEs generated by continuous martingales

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  • Peter Imkeller
  • Anthony R'eveillac
  • Anja Richter

Abstract

In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward--backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component.

Suggested Citation

  • Peter Imkeller & Anthony R'eveillac & Anja Richter, 2009. "Differentiability of quadratic BSDEs generated by continuous martingales," Papers 0907.0941, arXiv.org, revised Mar 2012.
    • Handle: RePEc:arx:papers:0907.0941
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    Cited by:

    1. Nam, Kihun, 2021. "Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 376-411.
    2. Benedetti, Giuseppe & Campi, Luciano, 2016. "Utility indifference valuation for non-smooth payoffs with an application to power derivatives," LSE Research Online Documents on Economics 63016, London School of Economics and Political Science, LSE Library.
    3. Umut c{C}etin & Albina Danilova, 2014. "Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems," Papers 1407.2420, arXiv.org, revised Sep 2016.
    4. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    5. Çetin, Umut & Danilova, Albina, 2016. "Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems," LSE Research Online Documents on Economics 63259, London School of Economics and Political Science, LSE Library.

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