A general stochastic target problem with jump diffusion and an application to a hedging problem for large investors

Nicolas Saintier (University of Buenos Aires)

Abstract


Let $Z(t,z)$ be a $\mathbb{R}^d$-valued controlled jump diffusion starting from the point $z$ at time $t$. The aim of this paper is to characterize the set $V(t)$ of initial conditions $z$ such that $Z(t,z)$ can be driven into a given target at a given time. We do this by proving that the characteristic function of the complement $V(t)$ satisfies some partial differential equation in the viscosity sense. As an application, we study the problem of hedging in a financial market with a large investor.

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Pages: 106-119

Publication Date: April 24, 2007

DOI: 10.1214/ECP.v12-1261

References

  1. 1. J.P.Aubin, I. Ekeland. Applied non linear analysis Pure and applied mathematics (1984) Wiley. Math. Review 0749753
  2. 2. P. Bank, D. Baum. Hedging and portfolio optimization in financial markets with a large trader Mathematical Finance. 14 (2004),1--18. Math. Review 2030833 3. G. Barles. Solutions de viscosit des quations de Hamilton-Jacobi. Mathmatiques et applications 17 (1994) Springer-Verlag. Math. Review 1613876
  3. 4. C. Berge. Espaces topologiques: fonctions multivoques. Collection universitaire de mathmatiques 3 (1959) Dunod. Math. Review 0105663
  4. 5. K. C. Border. Fixed points theorems with applications to economics and game theory (1985) Cambridge University Press. Math. Review 0790845
  5. 6. B. Bouchard. Stochastic targets with mixed diffusion processes and viscosity solutions. Stochastic processes and their applications. 101 (2002), 273--302. Math. Review 1931270
  6. 7. P. Bremaud. Point processs and queues, martingale dynamics.Springer series in statistics (1981) Springer-Verlag. Math. Review 0636252
  7. 8. R. Buckdahn, E. Pardoux. BSDE's with jumps and associated integro-partial differential equations. Preprint .
  8. 9. M.G. Crandall, H. Ishii and P-L.Lions. User's guide to viscosity solutions of second order partial differential equations.Bulletin of the AMS 1 (1992), 1--67. Math. Review 1118699
  9. 10. J. Cvitanic, J. Ma. Hedging options for a large investor and forward-backward SDE's.The annals of applied probability 6 (1996), 370--398. Math. Review 1398050
  10. 11. E. Platen, M. Schweizer. On feedback effects from hedging derivatives.Mathematical Finance.1 (1998), 67--84. Math. Review 1613291
  11. 12. P. Protter. Stochastic integration and differential equations.Applications of Mathematics (1990) Springer-Verlag. Math. Review 1037262
  12. 13. H.M. Soner, N. Touzi. Dynamic programming for stochastic target problems and geometric flows.JEMS.4 (2002), 201--236. Math. Review 1924400
  13. 14. H.M. Soner, N. Touzi. Stochastic target problems, dynamic programming and viscosity solutions. SIAM J. Control Optim.41 (2002), 404--424. Math. Review 1920265
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