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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.

MSC:

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
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References:

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