Electronic Communications in Probability

An Exponential Martingale Equation

Revaz Tevzadze and Mikhael Mania

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We prove an existence of a unique solution of an exponential martingale equation in the class of BMO martingales. The solution is used to characterize optimal martingale measures.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 22, 206-216.

Accepted: 27 September 2006
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90A09
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 90C39: Dynamic programming [See also 49L20]

Backward stochastic differential equation exponential martingale martingale measures

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Tevzadze, Revaz; Mania, Mikhael. An Exponential Martingale Equation. Electron. Commun. Probab. 11 (2006), paper no. 22, 206--216. doi:10.1214/ECP.v11-1220. https://projecteuclid.org/euclid.ecp/1465058865

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  • Biagini, Francesca; Guasoni, Paolo; Pratelli, Maurizio. Mean-variance hedging for stochastic volatility models. Math. Finance 10 (2000), no. 2, 109–123.
  • Delbaen, Freddy; Schachermayer, Walter. The variance-optimal martingale measure for continuous processes. Bernoulli 2 (1996), no. 1, 81–105.
  • Dellacherie, Claude; Meyer, Paul-André. Probabilités et potentiel. Chapitres V à VIII. Industrial Topics], 1385. Hermann, Paris, 1980. xviii+476 pp. ISBN: 2-7056-1385-4
  • De Donno, M.; Guasoni, P.; Pratelli, M. Super-replication and utility maximization in large financial Stochastic Process. Appl. 115 (2005), no. 12, 2006–2022.
  • El Karoui, N.; Peng, S.; Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1–71.
  • Grandits, Peter; Rheinländer, Thorsten. On the minimal entropy martingale measure. Ann. Probab. 30 (2002), no. 3, 1003–1038.
  • Hobson, David. Stochastic volatility models, correlation, and the $q$-optimal Math. Finance 14 (2004), no. 4, 537–556.
  • Jacod, Jean. Calcul stochastique et problèmes de martingales. (French) [Stochastic calculus and martingale problems] Lecture Notes in Mathematics, 714. Springer, Berlin, 1979. x+539 pp. ISBN: 3-540-09253-6
  • Kazamaki, Norihiko. Continuous exponential martingales and BMO. Lecture Notes in Mathematics, 1579. Springer-Verlag, Berlin, 1994. viii+91 pp. ISBN: 3-540-58042-5
  • Kobylanski, Magdalena. Backward stochastic differential equations and partial differential Ann. Probab. 28 (2000), no. 2, 558–602.
  • Laurent, Jean Paul; Pham, Huyên. Dynamic programming and mean-variance hedging. Finance Stoch. 3 (1999), no. 1, 83–110.
  • Lepeltier, J.-P.; San Martín, J.. Existence for BSDE with superlinear-quadratic coefficient. Stochastics Stochastics Rep. 63 (1998), no. 3-4, 227–240.
  • Mania, M.; Tevzadze, R. A semimartingale Bellman equation and the variance-optimal martingale Georgian Math. J. 7 (2000), no. 4, 765–792.
  • Mania, Michael; Tevzadze, Revaz. A martingale equation of exponential type. 507–516, Springer, Berlin, 2006.
  • M. Mania, M. Santacroce and R. Tevzadze, A Semimartingale backward equation related to the $p$-optimal martingale measure and the minimal price of contingent claims, Stoch. Processes and Related Topics, Stochastics Monogr., 12 Taylor Francis. (2002), 169-202.
  • Mania, Michael; Tevzadze, Revaz. A semimartingale backward equation and the variance-optimal martingale SIAM J. Control Optim. 42 (2003), no. 5, 1703–1726 (electronic).
  • Rheinländer, Thorsten. An entropy approach to the Stein and Stein model with correlation. Finance Stoch. 9 (2005), no. 3, 399–413.
  • Schweizer, Martin. Approximation pricing and the variance-optimal martingale measure. Ann. Probab. 24 (1996), no. 1, 206–236.