Electronic Journal of Probability

Discretionary stopping of stochastic differential equations with generalised drift

Mihail Zervos, Neofytos Rodosthenous, Pui Chan Lon, and Thomas Bernhardt

Full-text: Open access


We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem’s value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem’s data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case shows that the so-called “principle of smooth fit” does not hold in general for optimal stopping problems involving solutions to SDEs with generalised drift.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 140, 39 pp.

Received: 30 March 2019
Accepted: 16 October 2019
First available in Project Euclid: 5 December 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

optimal stopping stochastic differential equations with generalised drift skew Brownian motion variational inequalities perpetual American options

Creative Commons Attribution 4.0 International License.


Zervos, Mihail; Rodosthenous, Neofytos; Lon, Pui Chan; Bernhardt, Thomas. Discretionary stopping of stochastic differential equations with generalised drift. Electron. J. Probab. 24 (2019), paper no. 140, 39 pp. doi:10.1214/19-EJP377. https://projecteuclid.org/euclid.ejp/1575514915

Export citation


  • [1] Alvarez, L. H. R. and Salminen, P.: Timing in the presence of directional predictability: Optimal stopping of Skew Brownian Motion. Math. Methods Oper. Res. 86, (2017), 377–400.
  • [2] Assing, S. and Schmidt, W. M.: Continuous strong Markov processes in dimension one. A stochastic calculus approach. Springer-Verlag, Berlin, 1998. xii+137 pp.
  • [3] Bassan, B. and Ceci, C.: Optimal stopping problems with discontinuous reward: regularity of the value function and viscosity solutions, Stoch. Stoch. Rep. 72, (2002), 55–77.
  • [4] Beneš, V. E.: Some combined control and stopping problems. Paper presented at the CRM Workshop on Stochastic Systems, Montréal, November 1992.
  • [5] Bensoussan, A. and Lions, J. L.: Applications of variational inequalities in stochastic control. Studies in Mathematics and its Applications 12. North-Holland Publishing Co., Amsterdam-New York, 1982. xi+564 pp.
  • [6] Borodin, A. N. and Salminen, P.: Handbook of Brownian motion – facts and formulae. Second edition. Probability and its Applications. Birkhäuser Verlag, Basel, 2002. xvi+672 pp.
  • [7] Corns, T. R. A. and Satchell, S. E.: Skew Brownian motion and pricing European options. The European Journal of Finance 13, (2007), 523–544.
  • [8] Cox, A. M. G., Obloj, J. and Touzi, N.: The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach. Probab. Theory Related Fields 173, (2019), 211–259.
  • [9] Cox, A. M. G. and Wang, J.: Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23, (2013), 859–894.
  • [10] Crocce, F. and Mordecki, E.: Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions. Stochastics 86, (2014), 491–509.
  • [11] Davis, M. H. A. and Zervos, M.: A problem of singular stochastic control with discretionary stopping. Ann. Appl. Probab. 4, (1994), 226–240.
  • [12] Decamps, M., Goovaerts, M. and Schoutens, W.: Self exciting threshold interest rates models. Int. J. Theor. Appl. Finance 9, (2006), 1093–1122.
  • [13] Decamps, M., Goovaerts, M. and Schoutens, W.: Asymmetric skew Bessel processes and their applications to finance. J. Comput. Appl. Math. 186, (2006), 130–147.
  • [14] Engelbert, H. J. and Schmidt, W.: Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. III. Math. Nachr. 151, (1991), 149–197.
  • [15] Friedman, A.: (2006), Stochastic differential equations and applications. Two volumes bound as one. Reprint of the 1975 and 1976 original published in two volumes. Dover Publications, Inc., Mineola, NY, 2006. xvi+531 pp.
  • [16] Hämäläinen, J.: Portfolio selection with directional return estimates, SSRN: http://ssrn.com/abstract=2279823.
  • [17] Harrison, J. M. and Shepp, L. A.: On skew Brownian motion. Ann. Probab. 9, (1981), 309–313.
  • [18] Itô, K. and McKean, H. P. Jr.: (1974), Diffusion processes and their sample paths, Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp.
  • [19] Karatzas, I. and Sudderth, W. D.: Control and stopping of a diffusion process on an interval. Ann. Appl. Probab. 9, (1999), 188–196.
  • [20] Krylov, N. V.: Controlled Diffusion Processes. Translated from the Russian by A. B. Aries. Applications of Mathematics, 14. Springer-Verlag, New York-Berlin, 1980. xii+308 pp.
  • [21] Lamberton, D.: Optimal stopping with irregular reward functions. Stochastic Process. Appl. 119, (2009), 3253–3284.
  • [22] Lamberton, D. and Zervos, M.: On the optimal stopping of a one-dimensional diffusion. Electron. J. Probab. 18, (2013), no. 34, 49 pp.
  • [23] Lejay, A.: On the constructions of the skew Brownian motion. Probab. Surv. 3, (2006), 413–466.
  • [24] Lon, P. C.: Two explicitly solvable problems with discretionary stopping. PhD thesis, The London School of Economics and Political Science (LSE), 2011, http://etheses.lse.ac.uk/id/eprint/337.
  • [25] Lon, P. C., Rodosthenous, N. and Zervos, M.: On the optimal stopping of a skew geometric Brownian motion. In Modern trends in controlled stochastic processes: theory and applications, volume II (A. B. Piunovskiy, ed.). Luniver Press, 2015, 231–245.
  • [26] Murphy, J. J.: Technical analysis of the financial markets: a comprehensive guide to trading methods and applications. New York Institute of Finance. Penguin, 1999, New York.
  • [27] Nilsen, W. and Sayit, H.: No arbitrage in markets with bounces and sinks. International Review of Applied Financial Issues & Economics 3, (2011), 696–699.
  • [28] Øksendal, B.: Stochastic Differential equations. An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. xxiv+360 pp.
  • [29] Øksendal, B. and Reikvam, K.: Viscosity solutions of optimal stopping problems. Stochastics Stochastics Rep. 62, (1998), 285–301.
  • [30] Peskir, G.: Principle of smooth fit and diffusions with angles. Stochastics 79, (2007), 293–302.
  • [31] Peskir, G. and Shiryaev, A. N.: Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2006. xxii+500 pp.
  • [32] Revuz, D. and Yor, M.: Continuous martingales and Brownian motion. Third edition. Fundamental Principles of Mathematical Sciences 293. Springer-Verlag, Berlin, 1999. xiv+602 pp.
  • [33] Rossello, D.: Arbitrage in skew Brownian motion models. Insurance Math. Econom. 50, (2012), 50–56.