Browsing by Subject "Time-consistency"

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Open Access
Dynamic mean-variance problem: recovering time-consistency
(2021-08) Düzoylum, Seyit Emre
As the foundation of modern portfolio theory, Markowitz’s mean-variance port-folio optimization problem is one of the fundamental problems of ﬁnancial math-ematics. The dynamic version of this problem in which a positive linear com-bination of the mean and variance objectives is minimized is known to be time-inconsistent, hence the classical dynamic programming approach is not applicable. Following the dynamic utility approach in the literature, we consider a less re-strictive notion of time-consistency, where the weights of the mean and variance are allowed to change over time. Precisely speaking, rather than considering a ﬁxed weight vector throughout the investment period, we consider an adapted weight process. Initially, we start by extending the well-known equivalence be-tween the dynamic mean-variance and the dynamic mean-second moment prob-lems in a general setting. Thereby, we utilize this equivalence to give a complete characterization of a time-consistent weight process, that is, a weight process which recovers the time-consistency of the mean-variance problem according to our deﬁnition. We formulate the mean-second moment problem as a biobjective optimization problem and develop a set-valued dynamic programming principle for the biobjective setup. Finally, we retrieve back to the dynamic mean-variance problem under the equivalence results that we establish and propose a backward-forward dynamic programming scheme by the methods of vector optimization. Consequently, we compute both the associated time-consistent weight process and the optimal solutions of the dynamic mean-variance problem.
Open Access
Set-valued risk measures as backward stochastic difference inclusions and equations
(Springer, 2021-01) Ararat, Çağın; Feinstein, Z.
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.