Risk Sensitive Investment Management with Affine Processes: a Viscosity Approach

In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) In this setting, the Hamilton- Jacobi-Bellman equation is a partial integro-differential PDE. The main result of the paper is to show that the value function of the control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.

[1]  E. Thorp,et al.  The Kelly Capital Growth Investment Criterion: Theory and Practice , 2011 .

[2]  Mark H. A. Davis,et al.  Jump-Diffusion Risk-Sensitive Asset Management , 2009, 0905.4740.

[3]  Mark H. A. Davis,et al.  Risk-sensitive benchmarked asset management , 2008 .

[4]  G. Barles,et al.  Second-order elliptic integro-differential equations: viscosity solutions' theory revisited , 2007, math/0702263.

[5]  E. Jakobsen,et al.  A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations , 2006 .

[6]  Tomasz R. Bielecki,et al.  Risk Sensitive Portfolio Management with Cox--Ingersoll--Ross Interest Rates: The HJB Equation , 2005, SIAM J. Control. Optim..

[7]  Georges Dionne,et al.  Credit Risk: Pricing, Measurement, and Management , 2005 .

[8]  Anna Lisa Amadori*,et al.  Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations , 2004 .

[9]  Tomasz R. Bielecki,et al.  Economic Properties of the Risk Sensitive Criterion for Portfolio Management , 2003 .

[10]  Anna Lisa Amadori Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach , 2003 .

[11]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[12]  H. Nagai,et al.  Risk-sensitive portfolio optimization on infinite time horizon , 2002 .

[13]  Daniel Hernández-Hernández,et al.  Risk Sensitive Asset Management With Constrained Trading Strategies , 2001 .

[14]  Recent Developments in Mathematical Finance , 2001 .

[15]  Darrel,et al.  PDE solutions of stochastic differential utility * , 2001 .

[16]  Tomasz R. Bielecki,et al.  Risk sensitive asset management with transaction costs , 2000, Finance Stochastics.

[17]  A. Amadori The obstacle problem for nonlinear integro-di erential operators arising in option pricing , 2000 .

[18]  S. Pliska,et al.  Risk-Sensitive Dynamic Asset Management , 1999 .

[19]  Huy En Pham Optimal Stopping of Controlled Jump Diiusion Processes: a Viscosity Solution Approach , 1998 .

[20]  G. Barles,et al.  Backward stochastic differential equations and integral-partial differential equations , 1997 .

[21]  Olivier Alvarez,et al.  Viscosity solutions of nonlinear integro-differential equations , 1996 .

[22]  Wendell H. Fleming Optimal investment models and risk sensitive stochastic control , 1995 .

[23]  M. James Controlled markov processes and viscosity solutions , 1994 .

[24]  Mario Lefebvre,et al.  Risk-sensitive optimal investment policy , 1994 .

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

[26]  P. Lions,et al.  PDE solutions of stochastic differential utility , 1992 .

[27]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[28]  P. Whittle Risk-Sensitive Optimal Control , 1990 .

[29]  A. Bensoussan,et al.  Optimal control of partially observable stochastic systems with an exponential-of-integral performance index , 1985 .

[30]  L. Rogers Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[31]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[32]  Rhodes,et al.  Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games , 1973 .

[33]  F. Black Capital Market Equilibrium with Restricted Borrowing , 1972 .