arXiv subjects

Çağın Ararat

Publications and source records attributed to Çağın Ararat.

28 records · Page 2Linked to original sources

arXiv2021

End-of-Life Inventory Management Problem: Results and Insights

We consider a manufacturer who manages the end-of-life phase and takes one of the three actions at each period: (1) place an order, (2) use existing inventory, (3) stop holding inventory and use an outside/alternative source. Two examples of this source are discounts for a new generation product and delegating operations. Demand is described by a non-homogeneous Poisson process, and the decision to stop holding inventory is described by a stopping time. After formulating this problem as an optimal stopping problem with additional decisions and presenting its dynamic programming algorithm, we use martingale theory to facilitate the calculation of the value function. Moreover, we show analytical results to understand the additional difficulties of the problem solved, as well as structural results on optimal stopping times. Furthermore, we devise an expandable taxonomy and categorize the models in the literature. Analytical insights from the models as well as an extensive numerical analysis show the value of our approach. The results indicate that the loss can be high in case the manufacturer does not exploit flexibility in placing orders or use an outside source. Several managerial insights are obtained through numerical analysis as well as structural results to facilitate decision-making during the end-of-life horizon.

math.OC↗

arXiv2021

Set-Valued Backward Stochastic Differential Equations

In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann-Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with non-singleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.

math.PR↗

arXiv2020

Portfolio optimization with two quasiconvex risk measures

We study a static portfolio optimization problem with two risk measures: a principle risk measure in the objective function and a secondary risk measure whose value is controlled in the constraints. This problem is of interest when it is necessary to consider the risk preferences of two parties, such as a portfolio manager and a regulator, at the same time. A special case of this problem where the risk measures are assumed to be coherent (positively homogeneous) is studied recently in a joint work of the author. The present paper extends the analysis to a more general setting by assuming that the two risk measures are only quasiconvex. First, we study the case where the principal risk measure is convex. We introduce a dual problem, show that there is zero duality gap between the portfolio optimization problem and the dual problem, and finally identify a condition under which the Lagrange multiplier associated to the dual problem at optimality gives an optimal portfolio. Next, we study the general case without the convexity assumption and show that an approximately optimal solution with prescribed optimality gap can be achieved by using the well-known bisection algorithm combined with a duality result that we prove.

q-fin.PM↗

arXiv2020

Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as 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.

q-fin.RM↗

arXiv2020

Portfolio optimization with two coherent risk measures

We provide analytical results for a static portfolio optimization problem with two coherent risk measures. The use of two risk measures is motivated by joint decision-making for portfolio selection where the risk perception of the portfolio manager is of primary concern, hence, it appears in the objective function, and the risk perception of an external authority needs to be taken into account as well, which appears in the form of a risk constraint. The problem covers the risk minimization problem with an expected return constraint and the expected return maximization problem with a risk constraint, as special cases. For the general case of an arbitrary joint distribution for the asset returns, under certain conditions, we characterize the optimal portfolio as the optimal Lagrange multiplier associated to an equality-constrained dual problem. Then, we consider the special case of Gaussian returns for which it is possible to identify all cases where an optimal solution exists and to give an explicit formula for the optimal portfolio whenever it exists.

q-fin.PM↗

arXiv2020

Constructive covers of a finite set

Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ and they are not subsets of each other. We refer to such choices of sets as constructive $k$-covers and provide a semi-analytic summation formula to calculate the exact number of constructive $k$-covers of $\{1,\ldots,n\}$. Each term in the summation is the product of a new variant of Stirling numbers of the second kind, referred to as integrated Stirling numbers, and the cardinality of a certain set which we calculate by an optimization-based procedure with no-good cuts for binary variables.

math.CO↗

arXiv2019

Dual representations for systemic risk measures

The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg-Noe model as well as those of the resource allocation and network flow models.

q-fin.RM↗

arXiv2017

Multi-objective risk-averse two-stage stochastic programming problems

We consider a multi-objective risk-averse two-stage stochastic programming problem with a multivariate convex risk measure. We suggest a convex vector optimization formulation with set-valued constraints and propose an extended version of Benson's algorithm to solve this problem. Using Lagrangian duality, we develop scenario-wise decomposition methods to solve the two scalarization problems appearing in Benson's algorithm. Then, we propose a procedure to recover the primal solutions of these scalarization problems from the solutions of their Lagrangian dual problems. Finally, we test our algorithms on a multi-asset portfolio optimization problem under transaction costs.

math.OC↗

arXiv2017

Set-valued shortfall and divergence risk measures

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.

q-fin.RM↗

arXiv2014

A Characterization Theorem for Aumann Integrals

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be written as the Aumann integral with respect to some $σ$-finite measure. These conditions include the analog of the conlinearity and monotone convergence properties of the classical Daniell-Stone theorem for the Lebesgue integral, and three geometric properties that are peculiar to the set-valued case as they are redundant in the one-dimensional setting.

math.FA↗