The Annals of Applied Probability

A control problem with fuel constraint and Dawson–Watanabe superprocesses

Alexander Schied

Full-text: Open access

Abstract

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson–Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2472-2499.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447695

Digital Object Identifier
doi:10.1214/12-AAP908

Mathematical Reviews number (MathSciNet)
MR3127942

Zentralblatt MATH identifier
1288.60100

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 93E20: Optimal stochastic control
Secondary: 60H20: Stochastic integral equations 60H30: Applications of stochastic analysis (to PDE, etc.) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Dawson–Watanabe superprocess $J$-functional log-Laplace equation optimal stochastic control with fuel constraint optimal trade execution

Citation

Schied, Alexander. A control problem with fuel constraint and Dawson–Watanabe superprocesses. Ann. Appl. Probab. 23 (2013), no. 6, 2472--2499. doi:10.1214/12-AAP908. https://projecteuclid.org/euclid.aoap/1382447695


Export citation

References

  • Almgren, R. (2012). Optimal trading with stochastic liquidity and volatility. SIAM J. Financial Math. 3 163–181.
  • Almgren, R. and Chriss, N. (2000). Optimal execution of portfolio transactions. Journal of Risk 3 5–39.
  • Beneš, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics 4 39–83.
  • Björk, T. and Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. Preprint.
  • Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 iv+179.
  • Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984–1053.
  • Dynkin, E. B. (1991a). A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields 89 89–115.
  • Dynkin, E. B. (1991b). Path processes and historical superprocesses. Probab. Theory Related Fields 90 1–36.
  • Dynkin, E. B. (1992). Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942–962.
  • Dynkin, E. B. (1994). An Introduction to Branching Measure-valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI.
  • Dynkin, E. B. (2004). Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series 34. Amer. Math. Soc., Providence, RI.
  • Ekeland, I. and Lazrak, A. (2006). Being serious about non-commitment: Subgame perfect equilibrium in continuous time. Preprint. Available at arXiv:math/0604264.
  • Engländer, J. and Pinsky, R. G. (1999). On the construction and support properties of measure-valued diffusions on $D\subseteq\mathbf{R}^d$ with spatially dependent branching. Ann. Probab. 27 684–730.
  • Fleischmann, K. and Mueller, C. (1997). A super-Brownian motion with a locally infinite catalytic mass. Probab. Theory Related Fields 107 325–357.
  • Forsyth, P., Kennedy, J., Tse, T. S. and Windclif, H. (2012). Optimal trade execution: A mean-quadratic-variation approach. Journal of Economic Dynamics and Control 36 1971–1991.
  • Gatheral, J. and Schied, A. (2013). Dynamical models of market impact and algorithms for order execution. In Handbook on Systemic Risk (J.-P. Fouque and J. Langsam, eds.) 579–602. Cambridge Univ. Press, Cambridge.
  • Karatzas, I. (1985). Probabilistic aspects of finite-fuel stochastic control. Proc. Natl. Acad. Sci. USA 82 5579–5581.
  • Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
  • Schied, A. (1996). Sample path large deviations for super-Brownian motion. Probab. Theory Related Fields 104 319–347.
  • Schied, A. (1999). Existence and regularity for a class of infinite-measure $(\xi,\psi,K)$-superprocesses. J. Theoret. Probab. 12 1011–1035.
  • Schöneborn, T. (2008). Trade execution in illiquid markets. Optimal stochastic control and multi-agent equilibria. Ph.D. thesis, TU Berlin.
  • Tse, T. S., Forsyth, P. A., Kennedy, J. S. and Windclif, H. (2011). Comparison between the mean variance optimal and the mean quadratic variation optimal trading strategies. Preprint.