Time Consistency of the Mean-Risk Problem

Kováčová, Gabriela; Rudloff, Birgit

doi:10.1287/opre.2020.2002

Cited by 14 publications

(20 citation statements)

References 29 publications

(47 reference statements)

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“…In contrast, a set-valued equation appears to be the appropriate method if we care about the "mass" of the set itself over time, rather than any specific value in that set. Such concepts are important beyond the immediate study of risk measures, for instance for the mean-risk problem in Kováčová and Rudloff [38], where the dynamic programming principle holds for the multiobjective version but not the traditional scalar approach. These results indicate that the BSDI is likely to be the appropriate approach for that problem; this is left for future study.…”

Section: Discussionmentioning

confidence: 99%

“…The mean-variance and mean-risk problems are well known to generally be time-inconsistent and do not follow the dynamic programming principle, as noted in Karnam et al [34]. However, by considering a multiobjective formulation, Kováčová and Rudloff [38] have found that the dynamic programming principle in the mean-risk problem is satisfied in discrete time. Therefore, the insights we gain on expanding the BSDE representation of the dynamic programming principle to multivariate problems through either backward stochastic differential inclusions or set-valued backward stochastic differential equations is of wider interest and importance.…”

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Set-valued risk measures as backward stochastic difference inclusions and equations

Ararat

Feinstein

2020

Finance Stoch

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Set-valued risk measures as backward stochastic difference inclusions and equations

Ararat

Feinstein

2020

Finance Stoch

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“…The general convex vector optimization problem and its connection to convex projections will be treated in Sect. 4.…”

Section: Convex Vector Optimization Problemmentioning

confidence: 99%

“…Let us start with a short motivation, describing a field of research where convex projection problems, the main object of this paper, arise. A dynamic programming principle, called a set-valued Bellman principle, for a particular vector optimization problem has been proposed in [4], and has also been applied in [3,5] to other multivariate problems. Often, dynamic optimization problems depend on some parameter, in the above examples the initial capital.…”

Section: Introductionmentioning

confidence: 99%

Convex projection and convex multi-objective optimization

Kováčová

Rudloff

2021

J Glob Optim

Self Cite

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In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations.

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“…Further, [13] introduced a "dynamic utility" under which the original time-inconsistent problem became a time-consistent one, and investigated possible approaches to study a general time-inconsistent optimization problem. Reference [14] used a time-consistent dynamic convex risk measure to evaluate the risk of a portfolio and showed that the dynamic mean-risk problem satisfies a set-valued Bellman's principle.…”

Section: Introductionmentioning

confidence: 99%

Multi-time state mean-variance model in continuous time

Yang

2021

ESAIM: COCV

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The objective of the continuous time mean-variance model is to minimize the variance (risk) of an investment portfolio with a given mean at the terminal time. However, the investor can stop the investment plan at any time before the terminal time. To solve this problem, we consider to minimize the variances of the investment portfolio in the multi-time state. The advantage of this multi-time state mean-variance model is the minimization of the risk of the investment portfolio within the investment period. To obtain the optimal strategy of the model, we introduce a sequence of Riccati equations, which are connected by jump boundary conditions. In addition, we establish the relationships between the means and variances in the multi-time state mean-variance model. Furthermore, we use an example to verify that the variances of the multi-time state can affect the average of Maximum-Drawdown of the investment portfolio.

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Time Consistency of the Mean-Risk Problem

Cited by 14 publications

References 29 publications

Set-valued risk measures as backward stochastic difference inclusions and equations

Set-valued risk measures as backward stochastic difference inclusions and equations

Convex projection and convex multi-objective optimization

Multi-time state mean-variance model in continuous time

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