Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Dynamic robust duality in utility maximization

Abstract: A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth $X^*(T) : = X_{\varphi^*}(T)$ of the problem to maximize the expected $U$-utility of the terminal wealth $X_{\varphi}(T)$ generated by admissible portfolios $\varphi(t), 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$, where $V$ is the convex conjugate function of the concave function $U$. In this paper we consider markets modeled by It\^o-L\'evy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a \emph{dynamic} relation, valid for all $t \in [0,T]$. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, $0 \leq t \leq T$. In the terminal time case $t=T$ we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio $\varphi^*$ and the optimal measure $Q^*$. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related $T$-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.

Supportability of network cost functions

Abstract: This paper considers a network supply problem in which flows between any pair of nodes are possible. It is assumed that users place a value on connection to other users in the network, and (possibly) on access to an external source. Cost on each link is an arbitrary concave function of link capacity. The objective is to study coalitional stability in this situation, when collections of flows can be served by competing suppliers. In contrast to other network games, this approach focuses on the cost of serving flows rather than the cost of attaching nodes to the network. The network is said to be stable if the derived cost function is supportable. Supportable cost functions, defined by Sharkey and Telser [9], are cost functions for which there exists a price vector which covers total cost, and simultaneously deters entry at any lower output by a rival firm with the same cost function. If the minimal cost network includes a link between every pair of nodes, then the cost function is shown to be supportable. In the special case in which link cost is independent of capacity, the cost function is also supportable. The paper also considers “Steiner” networks in which new nodes may be created in order to minimize total cost, or in which access may be obtained at more than one source location. When link costs are independent of capacity in such a network, it is argued that the cost function is approximately supportable in a well defined sense.

On the convex Poincaré inequality and weak transportation inequalities

Abstract: We prove that for a probability measure on $\mathbb{R}^{n}$, the Poincare inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov–Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar, and Tetali.