Second Order Stochastic Target Problems

In this chapter, we extend the class of stochastic target problems of the previous section to the case where the quadratic variation of the control process ν is involved in the optimization problem. This new class of problems is motivated by applications in financial mathematics.

[1]  Nizar Touzi,et al.  A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs , 2009, 0905.1863.

[2]  Thaleia Zariphopoulou,et al.  A solution approach to valuation with unhedgeable risks , 2001, Finance Stochastics.

[3]  R. Tevzadze.  Solvability of backward stochastic differential equations with quadratic growth , 2007, math/0703484.

[4]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[5]  P. Protter,et al.  Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence , 2006 .

[6]  H. Soner,et al.  The multi-dimensional super-replication problem under gamma constraints , 2005 .

[7]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[8]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[9]  H. Soner,et al.  Small time path behavior of double stochastic integrals and applications to stochastic control , 2005, math/0602453.

[10]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[11]  J. Bismut.  Conjugate convex functions in optimal stochastic control , 1973 .

[12]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1991 .

[13]  M. Morlais.  Equations différentielles stochastiques rétrogrades à croissance quadratique et applications , 2007 .

[14]  P. Reny.  On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games , 1999 .

[15]  M. Kobylanski.  Backward stochastic differential equations and partial differential equations with quadratic growth , 2000 .

[16]  J. Wang,et al.  Maximal Use of Central Differencing for Hamilton-Jacobi-Bellman PDEs in Finance , 2008, SIAM J. Numer. Anal..

[17]  Peter A. Forsyth,et al.  Numerical convergence properties of option pricing PDEs with uncertain volatility , 2003 .

[18]  N. Karoui,et al.  Backward Stochastic Differential Equations , 1997 .

[19]  Nizar Touzi,et al.  The Dynamic Programming Equation for Second Order Stochastic Target Problems , 2009, SIAM J. Control. Optim..

[20]  N. Krylov.  Controlled Diffusion Processes , 1980 .

[21]  N. Karoui,et al.  Controle de processus de Markov , 1988 .

[22]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[23]  Guy Barles,et al.  Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations , 2007, Math. Comput..

[24]  H. Soner,et al.  Dynamic programming for stochastic target problems and geometric flows , 2002 .

[25]  Peter Bank,et al.  Hedging and Portfolio Optimization in Financial Markets with a Large Trader , 2004 .

[26]  Nicole El Karoui,et al.  Pricing Via Utility Maximization and Entropy , 2000 .

[27]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[28]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[29]  G. Barles,et al.  CONVERGENCE OF NUMERICAL SCHEMES FOR PARABOLIC EQUATIONS ARISING IN FINANCE THEORY , 1995 .

[30]  Philip Protter,et al.  Liquidity Risk and Arbitrage Pricing Theory , 2004 .

[31]  J. Frédéric Bonnans,et al.  Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation , 2003, SIAM J. Numer. Anal..

[32]  Nizar Touzi,et al.  Wellposedness of second order backward SDEs , 2010, 1003.6053.

[33]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[34]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[35]  P. Forsyth,et al.  PDE methods for pricing barrier options , 2000 .

[36]  Tyrone E. Duncan,et al.  Numerical Methods for Stochastic Control Problems in Continuous Time (Harold J. Kushner and Paul G. Dupuis) , 1994, SIAM Rev..

[37]  P. Imkeller,et al.  Utility maximization in incomplete markets , 2005, math/0508448.

[38]  Nizar Touzi,et al.  Option hedging for small investors under liquidity costs , 2010, Finance Stochastics.

[39]  H. Soner,et al.  Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs , 2005, math/0509295.

[40]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .