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Global Adapted Solution of One-Dimensional Backward Stochastic Riccati Equations, with Application to the Mean-Variance Hedging

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  • Michael Kohlmann

    ()
    (Center of Finance and Econometrics)

  • Shanjian Tang

    ()
    (Center of Finance and Econometrics)

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    Abstract

    We obtain the global existence and uniqueness result for a one-dimensional back- ward stochastic Riccati equation, whose generator contains a quadratic term of L (the second unknown component). This solves the one-dimensional case of Bismut- Peng's problem which was initially proposed by Bismut (1978) in the Springer yellow book LNM 649. We use an approximation technique by constructing a sequence of monotone generators and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean-variance hedging problem with stochastic market conditions.

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    File URL: http://cofe.uni-konstanz.de/Papers/dp00_26.pdf
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    Bibliographic Info

    Paper provided by Center of Finance and Econometrics, University of Konstanz in its series CoFE Discussion Paper with number 00-26.

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    Length: 29 Pages
    Date of creation: Aug 2000
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    Handle: RePEc:knz:cofedp:0026

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    1. Martin Schweizer & HuyËn Pham & (*), Thorsten RheinlÄnder, 1998. "Mean-variance hedging for continuous processes: New proofs and examples," Finance and Stochastics, Springer, vol. 2(2), pages 173-198.
    2. Christian Gourieroux & Jean Paul Laurent & Huy�n Pham, 1998. "Mean-Variance Hedging and Numéraire," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 179-200.
    3. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146.
    4. HuyËn Pham & Jean Paul Laurent, 1999. "Dynamic programming and mean-variance hedging," Finance and Stochastics, Springer, vol. 3(1), pages 83-110.
    5. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    6. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
    7. Martin Schweizer & Christophe Stricker & Freddy Delbaen & Pascale Monat & Walter Schachermayer, 1997. "Weighted norm inequalities and hedging in incomplete markets," Finance and Stochastics, Springer, vol. 1(3), pages 181-227.
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