Robust maximization of asymptotic growth under covariance uncertainty. (English) Zbl 1287.60081

The main quotation is [C. Kardaras and S. Robertson, Ann. Appl. Probab. 22, No. 4, 1576–1610 (2012; Zbl 1262.60040)]. The authors solve the problem how to trade optimally in a market when the investing horizon is long and the dynamics of the underlying assets are uncertain. The 2 page introduction and also systematical remarks, show the connection with the following description of the paper under review.
Let \(\alpha ,\alpha '\in (0,1]\), \(E=\cup_{n\geq 1}E_{n}\subset{\mathbb R}^{d}\), \(\overline {E_{n}}\subset E_{n+1}\), \(E_{n}\) bounded open convex, \(\partial E_{n}\) being \(C^{2,\alpha '}\), \(E^{\wedge }= E\cup \{\Delta \}\) be the one point compactification of \(E\), \(x_{0}\in E\) fixed, \(X_{t}\) the coordinate process on \(\Omega =\{\omega \in C([0,\infty ),E^{\wedge }):\omega (t)=\Delta\,if\, \omega (s)= \Delta \text{ and } t>s\}\), \(S^{d}\) be the set of symmetric d\(\times\)d matrices, \({\mathcal A}(a,b)= \{M \in S^{d}:a\cdot 1\leq M\leq b\cdot 1\}\), \(C\) the set of all \(C_{\mathrm{loc}}^{1,\alpha }\) functions \(c:E\rightarrow S^{d}\), and \(\theta (x)\cdot 1\leq c(x)\leq \Theta (x)\cdot 1\). For \(c\in C\) consider \(L^{c}\), acting on \(f\in C^{2}(E)\), as \[ (L^{c}f) (x)= (1/2)\sum_{i,j}c_{ij}(x)(\partial^{2}f/\partial x_{i}\partial x_{j})(x), \] i.e., as \((1/2)\mathrm{Tr}(c(x)(D^{2}f)(x))\) and the solution \(Q_{x_{0}}^{c}\), starting from \(x_{0}\), of the martingale problem \[ f(X_{\min(s,\zeta_{n})})-\int_{0}^{\min(s,\zeta_{n})}(L^{c}f)(X_{u})du, \] where \(\zeta_{n}=\inf\{t:\omega (t)\notin E_{n}\}\) and \(\Pi\) the set of all probabilities on \(\Omega\) which are locally \({\mathcal Q}_{x_{0}}^{c}\) absolutely continuous for some \(c\in C\). Let \({\mathcal V}\) be the set of all \({\mathbb R}^{d}\)-valued predictable \(\pi_{t}\), \({\mathcal Q}_{x_{0}}^{c}\) integrable for all \(c\) and with \(V_{t}^{\pi }=1+\int_{0}^{t}\pi_{t}^{\ast '}dX_{t}>0\) \({\mathcal Q}_{x_{0}}^{c}\) a.s., and define \[ g(\pi ,\operatorname{P})= \sup\{\gamma : \operatorname{P}\text{-}\liminf_{t\rightarrow \infty }(t^{- 1}\log V_{t}^{\pi })\geq \gamma, \operatorname{P}\text{-a.s.}\} \] The authors solve the problem by introducing a subset \(\Pi^{\ast }\subset \Pi\) and a \(\pi^{\ast }\in {\mathcal V}\) such that \(g(\pi^{\ast },\operatorname{P})\geq \lambda^{\ast }\) for all \(\operatorname{P}\in \Pi^{\ast }\), where \[ \lambda^{\ast }= \sup_{\pi \in {\mathcal V}}\inf_{\operatorname{P}\in \Pi^{\ast }}g(\pi , \operatorname{P})=\inf_{\operatorname{P}\in \Pi^{\ast }}\sup_{\pi \in {\mathcal V}}g(\pi ,\operatorname{P}). \] For \[ x\in E, M\in S^{d}, F(x,M)=(1/2)\sup_{A\in {\mathcal A}(\theta (x),\Theta (x))}\mathrm{Tr}(AM), \] for a domain \(D\subset E\) and \(\lambda \in {\mathbb R}\), let \[ H_{\lambda }^{c}(D)=\{\eta \in C^{2}(D):L^{c}\eta +\lambda \eta = 0,\eta >0\}, \]
\[ H_{\lambda }(D)=\{\eta \in C^{2}(D): F(x,D^{2}\eta )+\lambda \eta \leq 0,\eta >0\} \] and, similarly, \(H_{\lambda }^{+}(D)\) with \(\eta \in C^{2}(\overline {D})\) and the inequality being in the viscosity sense. Let \(\lambda^{\ast ,c}(D)=\sup\{\lambda : H_{\lambda }^{c}(D)\neq \emptyset \}\), \(\lambda^{\ast }(D)=\sup\{\lambda :H_{\lambda }(D)\neq \emptyset \}\) and similarly \(\lambda^{+}(D)\).
The first theorem proved in the paper states that \(\lambda^{\ast }(D)=\inf_{c\in C}\lambda^{\ast ,c}(D)\), for a bounded convex \(D\subset E\) with \(C^{2,\beta }\) boundary, \(\beta \in (0,1]\). The proof of \(\geq \) requires a result about continuous selections, established in an appendix (where some general information is given about the topic), namely that for two continuous \(\gamma ,\Gamma :E\rightarrow (0,+\infty )\), \(\gamma \leq \Gamma\), and \(\eta\) continuous on \(\overline {D}\), \(\eta >0\) on \(D\), \(0\) on \(\partial D\), \(F(x,D^{2}\eta )+\lambda^{+}(D)\eta =0\) in the viscosity sense on \(D\) and for \(m\in {\mathbb N}\), there exists a continuous \(c:D\rightarrow S^{d}\) with \(c(x)\in {\mathcal A}(\gamma (x),\Gamma (x))\), \(F_{\gamma ,\Gamma }(x,D^{2}\eta )\leq L^{c}\eta (x)+(1/m)\).
The main technical result is \(\lambda^{\ast }(E)=\inf_{c\in C}\lambda^{\ast ,c}(E)\), preceded by \(\lambda^{\ast }(E)=\lim\lambda^{\ast }(E_{n})\) (descending). Another appendix is devoted to the proof of one of the steps. The third theorem considers \(\eta^{\ast }\in H_{\lambda^{\ast }(E)}(E)\), \(\eta^{\ast }(x_{0})=1\) and states that \(\pi_{t}^{\ast }=e^{\lambda^{\ast }t}(\nabla \eta^{\ast })(X_{t})\), \[ \Pi^{\ast }=\{\operatorname{P}\in \Pi :\operatorname{P}\text{-}\liminf_{t\rightarrow \infty }(t^{- 1}\log\eta^{\ast }(X_{t}))\geq 0,\operatorname{P}\text{-a.s.}\} \] and \(\lambda^{\ast }=\lambda^{\ast }(E)\) solve the problem.


60H20 Stochastic integral equations
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
49K35 Optimality conditions for minimax problems
Full Text: DOI arXiv Euclid


[1] Arapostathis, A., Borkar, V. S. and Ghosh, M. K. (2012). Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and Its Applications 143 . Cambridge Univ. Press, Cambridge. · Zbl 1236.93137 · doi:10.1017/CBO9781139003605
[2] Avellaneda, M., Levy, A. and Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73-88.
[3] Bayraktar, E. and Huang, Y. J. (2011). On the multi-dimensional controller and stopper games. Technical report, Univ. Michigan. Available at . · Zbl 1268.49045
[4] Birindelli, I. and Demengel, F. (2007). Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6 335-366. · Zbl 1132.35032 · doi:10.3934/cpaa.2007.6.335
[5] Birindelli, I. and Demengel, F. (2010). Eigenfunctions for singular fully nonlinear equations in unbounded domains. NoDEA Nonlinear Differential Equations Appl. 17 697-714. · Zbl 1204.35087 · doi:10.1007/s00030-010-0077-y
[6] Borkar, V. S. (2006). Ergodic control of diffusion processes. In International Congress of Mathematicians. Vol. III 1299-1309. Eur. Math. Soc., Zürich. · Zbl 1108.93076
[7] Brown, A. L. (1989). Set valued mappings, continuous selections, and metric projections. J. Approx. Theory 57 48-68. · Zbl 0675.41037 · doi:10.1016/0021-9045(89)90083-X
[8] Busca, J., Esteban, M. J. and Quaas, A. (2005). Nonlinear eigenvalues and bifurcation problems for Pucci’s operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 187-206. · Zbl 1205.35087 · doi:10.1016/j.anihpc.2004.05.004
[9] Caffarelli, L. A. and Cabré, X. (1995). Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications 43 . Amer. Math. Soc., Providence, RI. · Zbl 0834.35002
[10] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to \(G\)-Brownian motion paths. Potential Anal. 34 139-161. · Zbl 1225.60057 · doi:10.1007/s11118-010-9185-x
[11] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827-852. · Zbl 1142.91034 · doi:10.1214/105051606000000169
[12] Fernholz, D. and Karatzas, I. (2011). Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21 2191-2225. · Zbl 1239.60057 · doi:10.1214/10-AAP755
[13] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions , 2nd ed. Stochastic Modelling and Applied Probability 25 . Springer, New York. · Zbl 1105.60005
[14] Fujisaki, M. (1999). On probabilistic approach to the eigenvalue problem for maximal elliptic operator. Osaka J. Math. 36 981-992. · Zbl 0963.35147
[15] Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order . Springer, Berlin. Reprint of the 1998 edition. · Zbl 1042.35002
[16] Kardaras, C. and Robertson, S. (2012). Robust maximization of asymptotic growth. Ann. Appl. Probab. 22 1576-1610. · Zbl 1262.60040 · doi:10.1214/11-AAP802
[17] Kawohl, B. and Kutev, N. (2007). Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Comm. Partial Differential Equations 32 1209-1224. · Zbl 1185.35084 · doi:10.1080/03605300601113043
[18] Kelley, J. L. (1955). General Topology . Van Nostrand, New York. · Zbl 0066.16604
[19] Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117-133.
[20] McCoy, J. W. (1965). An extension of the concept of \(L_{n}\) sets. Proc. Amer. Math. Soc. 16 177-180. · Zbl 0142.40203 · doi:10.2307/2033838
[21] Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra . SIAM, Philadelphia, PA. · Zbl 0962.15001
[22] Michael, E. (1956). Continuous selections. I. Ann. of Math. (2) 63 361-382. · Zbl 0071.15902 · doi:10.2307/1969615
[23] Nutz, M. (2012). A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 23. · Zbl 1244.93176 · doi:10.1214/EJP.v17-1892
[24] Nutz, M. and Soner, H. M. (2010). Superhedging and dynamic risk measures under volatility uncertainty. Technical report, ETH Zürich. Available at . · Zbl 1263.91026 · doi:10.1137/100814925
[25] Peng, S. (2007). \(G\)-Brownian motion and dynamic risk measure under volatility uncertainty. Technical report, Shandong Univ. Available at . · doi:10.1007/978-3-540-70847-6_25
[26] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45 . Cambridge Univ. Press, Cambridge. · Zbl 0858.31001
[27] Pucci, C. (1966). Maximum and minimum first eigenvalues for a class of elliptic operators. Proc. Amer. Math. Soc. 17 788-795. · Zbl 0149.07601 · doi:10.2307/2036253
[28] Pucci, C. (1966). Operatori ellittici estremanti. Ann. Mat. Pura Appl. (4) 72 141-170. · Zbl 0154.12402 · doi:10.1007/BF02414332
[29] Quaas, A. and Sirakov, B. (2008). Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators. Adv. Math. 218 105-135. · Zbl 1143.35077 · doi:10.1016/j.aim.2007.12.002
[30] Safonov, M. V. (1988). Classical solution of second-order nonlinear elliptic equations. Izv. Akad. Nauk SSSR Ser. Mat. 52 1272-1287, 1328. · Zbl 0682.35048 · doi:10.1070/IM1989v033n03ABEH000858
[31] Soner, H. M., Touzi, N. and Zhang, J. (2011). Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 1844-1879. · Zbl 1245.60062 · doi:10.1214/EJP.v16-950
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.