Mean-variance hedging and stochastic control: beyond the Brownian setting

We show for continuous semimartingales in a general filtration how the mean-variance hedging problem can be treated as a linear-quadratic stochastic control problem. The adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process, and we give necessary and sufficient conditions for the solvability of these generalized stochastic Riccati equations. Motivated from mathematical finance, this paper takes a first step toward linear-quadratic stochastic control in more general than Brownian settings.

[1]  F. Delbaen,et al.  The fundamental theorem of asset pricing for unbounded stochastic processes , 1998 .

[2]  Andrew E. B. Lim,et al.  LQ control and mean-variance portfolio selec-tions: The stochastic parameter case , 2001 .

[3]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .

[4]  Michael Kohlmann,et al.  Multidimensional Backward Stochastic Riccati Equations and Applications , 2002, SIAM J. Control. Optim..

[5]  Xun Yu Zhou,et al.  Indefinite Stochastic Riccati Equations , 2003, SIAM J. Control. Optim..

[6]  Shanjian Tang,et al.  Global Adapted Solution of One-Dimensional Backward Stochastic Riccati Equations, with Application to the Mean-Variance Hedging , 2002 .

[7]  Thorsten Rheinländer,et al.  On L2-projections on a space of stochastic integrals , 1997 .

[8]  Walter Schachermayer,et al.  The Existence of Absolutely Continuous Local Martingale Measures (1995) , 1995 .

[9]  Dimitris Bertsimas,et al.  Hedging Derivative Securities and Incomplete Markets: An Formula-Arbitrage Approach , 2001, Oper. Res..

[10]  M. Mania,et al.  A Semimartingale Bellman Equation and the Variance-Optimal Martingale Measure , 2000 .

[11]  Michael Mania,et al.  A Semimartingale Backward Equation and the Variance-Optimal Martingale Measure under General Information Flow , 2003, SIAM J. Control. Optim..

[12]  Walter Schachermayer,et al.  The variance-optimal martingale measure for continuous processes , 1996 .

[13]  Huyên Pham,et al.  On quadratic hedging in continuous time , 2000, Math. Methods Oper. Res..

[14]  R. J. Williams.  Review: Philip Protter, Stochastic integration and differential equations— a new approach , 1991 .

[15]  Michael Kohlmann,et al.  Minimization of Risk and Linear Quadratic Optimal Control Theory , 2003, SIAM J. Control. Optim..

[16]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[17]  Huyên Pham,et al.  Dynamic programming and mean-variance hedging , 1999, Finance Stochastics.

[18]  M. Schweizer.  On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition , 1995 .

[19]  M. Mania,et al.  BACKWARD STOCHASTIC PDE AND IMPERFECT HEDGING , 2003 .

[20]  P. Protter.  Stochastic integration and differential equations , 1990 .

[21]  Shanjian Tang,et al.  General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations , 2003, SIAM J. Control. Optim..

[22]  Marc C. Steinbach,et al.  Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis , 2001, SIAM Rev..

[23]  Walter Schachermayer,et al.  Weighted norm inequalities and hedging in incomplete markets , 1997, Finance Stochastics.

[24]  Xun Yu Zhou,et al.  Relationship Between Backward Stochastic Differential Equations and Stochastic Controls: A Linear-Quadratic Approach , 2000, SIAM J. Control. Optim..

[25]  M. Schweizer.  Approximation pricing and the variance-optimal martingale measure , 1996 .