Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T02:12:03.342Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of solvable multidimensional stopping problems in the presence of Knightian uncertainty

Part of: Stochastic processes Markov processes Sequential methods Mathematical finance

Published online by Cambridge University Press:  01 July 2021

Luis H. R. Alvarez E. Open the ORCID record for Luis H. R. Alvarez E. [Opens in a new window]  and
Sören Christensen
Luis H. R. Alvarez E.*
Affiliation:
University of Turku
Sören Christensen*
Affiliation:
Christian-Albrechts-Universität zu Kiel
*
*Postal address: Department of Accounting and Finance, Turku School of Economics, FIN-20014 University of Turku, Finland. Email address: lhralv@utu.fi
**Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany. Email address: christensen@math.uni-kiel.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity-averse decision-maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst-case measure in terms of a family of explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may lead to stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.

Keywords

κ-ambiguitymultidimensional Brownian motiondiffusion processesBessel processes

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 400 - 424
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, E., L. H. R. (2007). Knightian uncertainty, $\kappa$ -ignorance, and optimal timing. Tech. Rep. 25, Aboa Center of Economics, University of Turku.Google Scholar
Alvarez, E., L. H. R. and Christensen, S. (2019). A solvable two-dimensional optimal stopping problem in the presence of ambiguity. Preprint. Available at .Google Scholar
Bayraktar, E., Karatzas, I. and Yao, S. (2010). Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54, 1025–1067.CrossRefGoogle Scholar
Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations—Part I. Stoch. Process. Appl. 121, 185–211.CrossRefGoogle Scholar
Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations—Part II. Stoch. Process. Appl. 121, 212–264.CrossRefGoogle Scholar
Beibel, M. and Lerche, H. R. (1997). A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 7, 93–108.Google Scholar
Bewley, T. F. (2002). Knightian decision theory. Part I. Decisions Econom. Finance 25, 79–110.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2015). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.Google Scholar
Chen, Z. and Epstein, L. (2002). Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403–1443.CrossRefGoogle Scholar
Cheng, X. and Riedel, F. (2013). Optimal stopping under ambiguity in continuous time. Math. Financial Econom. 7, 29–68.CrossRefGoogle Scholar
Christensen, S. (2013). Optimal decision under ambiguity for diffusion processes. Math. Meth. Operat. Res. 77, 207–226.CrossRefGoogle Scholar
Christensen, S., Crocce, F., Mordecki, E. and Salminen, P. (2019). On optimal stopping of multidimensional diffusions. Stoch. Process. Appl. 129, 2561–2581.CrossRefGoogle Scholar
Christensen, S. and Irle, A. (2011). A harmonic function technique for the optimal stopping of diffusions. Stochastics 83, 347–363.CrossRefGoogle Scholar
Ekren, I., Touzi, N. and Zhang, J. (2014). Optimal stopping under nonlinear expectation. Stoch. Process. Appl. 124, 3277–3311.CrossRefGoogle Scholar
Epstein, L. G. and Ji, S. (2019). Optimal learning under robustness and time-consistency. To appear in Operat. Res.Google Scholar
Epstein, L. G. and Miao, J. (2003). A two-person dynamic equilibrium under ambiguity. J. Econom. Dynam. Control 27, 1253–1288.CrossRefGoogle Scholar
Epstein, L. G. and Schneider, M. (2003). Recursive multiple-priors. J. Econom. Theory 113, 1–31.CrossRefGoogle Scholar
Epstein, L. G. and Wang, T. (1994). Intertemporal asset pricing under Knightian uncertainty. Econometrica 62, 283–322.CrossRefGoogle Scholar
Franceschi, S. (2019). Green’s functions with oblique Neumann boundary conditions in a wedge. Preprint. Available at .Google Scholar
Gapeev, P. V. and Lerche, H. R. (2011). On the structure of discounted optimal stopping problems for one-dimensional diffusions. Stochastics 83, 537–554.CrossRefGoogle Scholar
Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with nonunique prior. J. Math. Econom. 18, 141–153.CrossRefGoogle Scholar
Harrison, J. M. and Reiman, M. I. (1981). On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41, 345–361.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1977). A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619–633.Google Scholar
Klibanoff, P., Marinacci, M. and Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892.CrossRefGoogle Scholar
Knight, F. (1921). Risk, Uncertainty, and Profit. Houghton Miffin, Boston, MA.Google Scholar
Lerche, H. R. and Urusov, M. (2007). Optimal stopping via measure transformation: the Beibel–Lerche approach. Stochastics 79, 275–291.CrossRefGoogle Scholar
Linetsky, V. (2004). The spectral representation of Bessel processes with constant drift: applications in queueing and finance. J. Appl. Prob. 41, 327–344.CrossRefGoogle Scholar
Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498.CrossRefGoogle Scholar
Miao, J. and Wang, N. (2011). Risk, uncertainty, and option exercise. J. Econom. Dynam. Control 35, 442–461.CrossRefGoogle Scholar
Mordecki, E. and Salminen, P. (2019). Optimal stopping of Brownian motion with broken drift. High Frequency 2, 113–120.CrossRefGoogle Scholar
Nishimura, K. G. and Ozaki, H. (2004). Search and Knightian uncertainty. J. Econom. Theory 119, 299–333.CrossRefGoogle Scholar
Nishimura, K. G. and Ozaki, H. (2006). An axiomatic approach to $\epsilon$ -contamination. Econom. Theory 27, 333–340.CrossRefGoogle Scholar
Nishimura, K. G. and Ozaki, H. (2007). Irreversible investment and Knightian uncertainty. J. Econom. Theory 136, 668–694.CrossRefGoogle Scholar
Nutz, M. and Zhang, J. (2015). Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Prob. 25, 2503–2534.CrossRefGoogle Scholar
Peng, S. (1997). Backward SDE and related g-expectation. In Backward Stochastic Differential Equations (Paris, 1995–1996), Longman, Harlow, pp. 141–159.Google Scholar
Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Prob. Theory Relat. Fields 113, 473–499.CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Riedel, F. (2009). Optimal stopping with multiple priors. Econometrica 77, 857–908.Google Scholar
Williams, R. J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Prob. 13, 758–778.CrossRefGoogle Scholar