## The Annals of Applied Probability

### On dynamic deviation measures and continuous-time portfolio optimization

#### Abstract

In this paper, we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency, we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of stochastic differential equations. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure, we formulate a dynamic mean-deviation portfolio optimization problem in a jump-diffusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3342-3384.

Dates
Revised: January 2017
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aoap/1513328703

Digital Object Identifier
doi:10.1214/17-AAP1282

Mathematical Reviews number (MathSciNet)
MR3737927

Zentralblatt MATH identifier
1382.60089

#### Citation

Pistorius, Martijn; Stadje, Mitja. On dynamic deviation measures and continuous-time portfolio optimization. Ann. Appl. Probab. 27 (2017), no. 6, 3342--3384. doi:10.1214/17-AAP1282. https://projecteuclid.org/euclid.aoap/1513328703

#### References

• Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
• Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. and Ku, H. (2007). Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152 5–22.
• Barrieu, P. and El Karoui, N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9 269–298.
• Barrieu, P. and El Karoui, N. (2009). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In Indifference Pricing: Theory and Applications (R. Carmona, ed.). 77–146. Princeton Univ. Press, Princeton, NJ.
• Basak, S. and Chabakauri, G. (2010). Dynamic mean-variance asset allocation. Rev. Financ. Stud. 23 2970–3016.
• Bensoussan, A., Wong, K. C., Yam, S. C. P. and Yung, S. P. (2014). Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting. SIAM J. Financial Math. 5 153–190.
• Bion-Nadal, J. and Kervarec, M. (2012). Risk measuring under model uncertainty. Ann. Appl. Probab. 22 213–238.
• Björk, T. and Murgoci, A. (2010). A general theory of Markovian Time Inconsistent Stochastic Control Problems. Working paper, Stockholm School of Economics.
• Björk, T., Murgoci, A. and Zhou, X. Y. (2014). Mean-variance portfolio optimization with state-dependent risk aversion. Math. Finance 24 1–24.
• Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Polit. Econ. 81 637–654.
• Chen, Z. and Epstein, L. (2002). Ambiguity, risk, and asset returns in continuous time. Econometrica 70 1403–1443.
• Cheng, S., Liu, Y. and Wang, S. (2004). Progress in risk measurement. Adv. Model. Optim. 6 1–20.
• Cheridito, P. and Kupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance 14 137–162.
• Coquet, F., Hu, Y., Mémin, J. and Peng, S. (2002). Filtration-consistent nonlinear expectations and related $g$-expectations. Probab. Theory Related Fields 123 1–27.
• Czichowsky, C. (2013). Time-consistent mean-variance portfolio selection in discrete and continuous time. Finance Stoch. 17 227–271.
• Delbaen, F. (2006). The structure of m-stable sets and in particular of the set of risk neutral measures. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Math. 1874 215–258. Springer, Berlin.
• Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145–192.
• Delbaen, F., Peng, S. and Rosazza Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14 449–472.
• Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
• Ekeland, I. and Pirvu, T. A. (2008). Investment and consumption without commitment. Math. Financ. Econ. 2 57–86.
• El Karoui, N. and Ravanelli, C. (2009). Cash subadditive risk measures and interest rate ambiguity. Math. Finance 19 561–590.
• Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, 3rd extended ed. de Gruyter, Berlin.
• Grechuk, B., Molyboha, A. and Zabarankin, M. (2009). Maximum entropy principle with general deviation measures. Math. Oper. Res. 34 445–467.
• Grechuk, B., Molyboha, A. and Zabarankin, M. (2013). Cooperative games with general deviation measures. Math. Finance 23 339–365.
• Grechuk, B. and Zabarankin, M. (2014). Inverse portfolio problem with mean-deviation model. European J. Oper. Res. 234 481–490.
• Hu, Y., Ma, J., Peng, S. and Yao, S. (2008). Representation theorems for quadratic $F$-consistent nonlinear expectations. Stochastic Process. Appl. 118 1518–1551.
• Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• Jiang, L. (2008). Convexity, translation invariance and subadditivity for $g$-expectations and related risk measures. Ann. Appl. Probab. 18 245–258.
• Klöppel, S. and Schweizer, M. (2007). Dynamic indifference valuation via convex risk measures. Math. Finance 17 599–627.
• Krätschmer, V., Ladkau, M., Laeven, R. A., Schoenmakers, J. and Stadje, M. (2015). Optimal stopping under drift and jump uncertainty. Preprint, available at https://sites.google.com/site/mstadje/.
• Li, Z., Zeng, Y. and Lai, Y. (2012). Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance Math. Econom. 51 191–203.
• Märkert, A. and Schultz, R. (2005). On deviation measures in stochastic integer programming. Oper. Res. Lett. 33 441–449.
• Markowitz, H. M. (1952). Portfolio selection. J. Finance 7 77–91.
• Pelsser, A. and Stadje, M. (2014). Time-consistent and market-consistent evaluations. Math. Finance 24 25–65.
• Riedel, F. (2004). Dynamic coherent risk measures. Stochastic Process. Appl. 112 185–200.
• Rockafellar, R. T., Uryasev, S. and Zabarankin, M. (2006a). Generalized deviations in risk analysis. Finance Stoch. 10 51–74.
• Rockafellar, R. T., Uryasev, S. and Zabarankin, M. (2006b). Optimality conditions in portfolio analysis with general deviation measures. Math. Program. 108 515–540.
• Rockafellar, R. T., Uryasev, S. P. and Zabarankin, M. (2006c). Master funds in portfolio analysis with general deviation measures. J. Bank. Financ. 30 743–778.
• Rockafellar, R. T., Uryasev, S. P. and Zabarankin, M. (2007). Equilibrium with investors using a diversity of deviation measures. J. Bank. Financ. 31 3251–3268.
• Rosazza Gianin, E. (2006). Risk measures via $g$-expectations. Insurance Math. Econom. 39 19–34.
• Royer, M. (2006). Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 1358–1376.
• Ruszczyński, A. and Shapiro, A. (2006). Conditional risk mappings. Math. Oper. Res. 31 544–561.
• Stoyanov, S. V., Rachev, S. T., Ortobelli, S. and Fabozzi, F. J. (2008). Relative deviation metrics and the problem of strategy replication. J. Bank. Financ. 32 199–206.
• Wang, J. and Forsyth, P. A. (2011). Continuous time mean variance asset allocation: A time-consistent strategy. European J. Oper. Res. 209 184–201.
• Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ.
• Zeidler, E. (1995). Applied Functional Analysis: Applications to Mathematical Physics. Applied Mathematical Sciences 108. Springer, New York.