Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time

Jeremy Thane Clark (Michigan State University)


I consider a stochastic optimization problem for a one-dimensional continuous martingale whose diffusion rate is constrained to be between two positive values $r_{1}<r_{2}$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equation depending on the ratio $\frac{r_{2}}{r_{1}}$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-19

Publication Date: July 26, 2014

DOI: 10.1214/ECP.v19-2846


  • Abramowitz, M. and Stegun, I. A.: Handbook of mathematical functions, phDover Publications, Inc., New York, 1972, xiv+1046pp. MR1225604
  • Bayraktar, Erhan; Huang, Yu-Jui. Robust maximization of asymptotic growth under covariance uncertainty. Ann. Appl. Probab. 23 (2013), no. 5, 1817--1840. MR3114918
  • Birindelli, I.; Demengel, F. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6 (2007), no. 2, 335--366. MR2289825
  • Birindelli, I.; Demengel, F. Eigenfunctions for singular fully nonlinear equations in unbounded domains. NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 6, 697--714. MR2740536
  • Borkar, Vivek S. Controlled diffusion processes. Probab. Surv. 2 (2005), 213--244. MR2178045
  • Busca, Jérôme; Esteban, Maria J.; Quaas, Alexander. Nonlinear eigenvalues and bifurcation problems for Pucci's operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 2, 187--206. MR2124162
  • Fleming, Wendell H.; Soner, H. Mete. Controlled Markov processes and viscosity solutions. Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. xviii+429 pp. ISBN: 978-0387-260457; 0-387-26045-5 MR2179357
  • Göing-Jaeschke, Anja; Yor, Marc. A survey and some generalizations of Bessel processes. Bernoulli 9 (2003), no. 2, 313--349. MR1997032
  • Kardaras, Constantinos; Robertson, Scott. Robust maximization of asymptotic growth. Ann. Appl. Probab. 22 (2012), no. 4, 1576--1610. MR2985170
  • Kato, Tosio. Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp. ISBN: 3-540-58661-X MR1335452
  • Pinsky, Ross G. Positive harmonic functions and diffusion. Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press, Cambridge, 1995. xvi+474 pp. ISBN: 0-521-47014-5 MR1326606
  • Pucci, Carlo. Maximum and minimum first eigenvalues for a class of elliptic operators. Proc. Amer. Math. Soc. 17 1966 788--795. MR0199576
  • Quaas, Alexander; Sirakov, Boyan. Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators. Adv. Math. 218 (2008), no. 1, 105--135. MR2409410
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.