on the one dimensional optimal switching problem we explicitly solve the optimal switching problem for one dimensional diffusions by directly using the dynamic programming principle and the excessive characterization of the value function the shape of the value function and the smooth fit principle then can be proved using the properties of concave functions

the skew energy of random oriented graphs abstract given a graph g let g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing wigner s semicircle law moreover we consider the skew energy of random regular oriented graphs g n d and get an exact estimate of the skew energy for almost all regular oriented graphs

the laplacian energy of random graphs abstract gutman et al introduced the concepts of energy e g and laplacian energy e l g for a simple graph g and furthermore they proposed a conjecture that for every graph g e g is not more than e l g unfortunately the conjecture turns out to be incorrect since liu et al and stevanovic et al constructed counterexamples however so et al verified the conjecture for bipartite graphs in the present paper we obtain for a random graph the lower and upper bounds of the laplacian energy and show that the conjecture is true for almost all graphs

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

the spectral distribution of random mixed graphs abstract let g be a mixed graph with n vertices h g the hermitian adjacency matrix of g and 1 g 2 g n g the eigenvalues of h g the hermitian energy of g is defined as e h g i 1 n i g in this paper we characterize the limiting spectral distribution of the hermitian adjacency matrices of random mixed graphs and as an application we give an estimation of the hermitian energy for almost all mixed graphs

valuation of investments in real assets with implications for the stock prices a general model for the valuation of natural resource investments is formulated and analyzed within a stochastic control theoretic framework using dynamic programming the value of such an investment with a general payoff function is determined under the assumption that the commodity price process is given by a stochastic differential equation the analysis results in closed form analytic solutions which can easily be computed and exhibits qualitatively different optimal behaviors depending on parameter values implications for stocks and options are also considered

optimal exit strategies for investment projects abstract we study the problem of an optimal exit strategy for an investment project which is unprofitable and for which the liquidation costs evolve stochastically the firm has the option to keep the project going while waiting for a buyer or liquidating the assets at immediate liquidity and termination costs the liquidity and termination costs are governed by a mean reverting stochastic process whereas the rate of arrival of buyers is governed by a regime shifting markov process we formulate this problem as a multidimensional optimal stopping time problem with random maturity we characterize the objective function as the unique viscosity solution of the associated system of variational hamilton jacobi bellman inequalities we derive explicit solutions and numerical examples in the case of power and logarithmic utility functions when the liquidity premium factor follows a mean reverting cir process

a model for investments in the natural resource industry with switching costs we consider a model for investment decisions in the natural resource industry with switching costs this model gives rise to a problem combining features of both absolutely continuous and impulse stochastic control that we explicitly solve the solution takes qualitatively different forms depending on parameter values

a finite horizon optimal multiple switching problem 1we consider the problem of optimal multiple switching in a finite horizon when the state of the system including the switching costs is a general adapted stochastic process the problem is formulated as an extended impulse control problem and solved using probabilistic tools such as the snell envelope of processes and reflected backward stochastic differential equations finally when the state of the system is a markov process we show that the associated vector of value functions provides a viscosity solution to a system of variational inequalities with interconnected obstacles

bayesian switching multiple disorder problems the switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders or change points at which the observable process changes its probability characteristics we study a bayesian formulation of this problem for an observable brownian motion with switching constant drift rates the method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic type free boundary problem we derive analytic form estimates for the bayesian risk function and the optimal switching boundaries for the components of the posterior probability process

optimal switching in an economic activity under uncertainty this paper considers the problem of finding the optimal sequence of opening starting and closing stopping times of a multi activity production process given the costs of opening running and closing the activities and assuming that the state of the economic system is a stochastic process the problem is formulated as an extended impulse control problem and solved using stochastic calculus as an application the optimal starting and stopping strategy are explicitly found for a resource extraction when the price of the resource is following a geometric brownian motion

laplacian spectral moment and laplacian estrada index of random graphs abstract let g be a simple graph with n vertices and let 1 2 n 0 be the laplacian eigenvalues of g the k th laplacian spectral moment of g is defined to be l m k g i 1 n i k g where k is a non negative integer and the laplacian estrada index of g is defined as l e e g i 1 n e i in this paper we first estimate these two indices for almost all graphs and then we give lower and upper bounds to these two indices for almost all multipartite graphs

using the bootstrap to quantify the authority of an empirical ranking the bootstrap is a popular and convenient method for quantifying the authority of an empirical ordering of attributes for example of a ranking of the performance of institutions or of the influence of genes on a response variable in the first of these examples the number p of quantities being ordered is sometimes only moderate in size in the second it can be very large often much greater than sample size however we show that in both types of problem the conventional bootstrap can produce inconsistency moreover the standard n out of n bootstrap estimator of the distribution of an empirical rank may not converge in the usual sense the estimator may converge in distribution but not in probability nevertheless in many cases the bootstrap correctly identifies the support of the asymptotic distribution of ranks in some contemporary problems bootstrap prediction intervals for ranks are particularly long and in this context we also quantify the accuracy of bootstrap methods showing that the standard bootstrap gets the order of magnitude of the interval right but not the constant multiplier of interval length the m out of n bootstrap can improve performance and produce statistical consistency but it requires empirical choice of m we suggest a tuning solution to this problem we show that in genomic examples where it might be expected that the standard synchronous bootstrap will successfully accommodate non independence of vector components that approach can produce misleading results an independent component bootstrap can overcome these difficulties even in cases where components are not strictly independent

3 regular digraphs with optimum skew energy abstract the skew energy of a digraph d is defined as the sum of the singular values of its skew adjacency matrix s d in this paper we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by adiga et al c adiga r balakrishnan wasin so the skew energy of a graph linear algebra appl 432 2010 1825 1835 of determining all 3 regular connected digraphs with optimum skew energy

solution examples of an impulse control problem we study an impulse control problem with switching technology in infinite horizon we solve the impulse control problem in case of deterministic impulse times on specific transition kernel examples our objective is to exhibit an optimal strategy which maximizes the value function in these cases

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

modeling the variability of rankings for better or for worse rankings of institutions such as universities schools and hospitals play an important role today in conveying information about relative performance they inform policy decisions and budgets and are often reported in the media while overall rankings can vary markedly over relatively short time periods it is not unusual to find that the ranks of a small number of highly performing institutions remain fixed even when the data on which the rankings are based are extensively revised and even when a large number of new institutions are added to the competition in the present paper we endeavor to model this phenomenon in particular we interpret as a random variable the value of the attribute on which the ranking should ideally be based more precisely if p items are to be ranked then the true but unobserved attributes are taken to be values of p independent and identically distributed variates however each attribute value is observed only with noise and via a sample of size roughly equal to n say these noisy approximations to the true attributes are the quantities that are actually ranked we show that if the distribution of the true attributes is light tailed e g normal or exponential then the number of institutions whose ranking is correct even after recalculation using new data and even after many new institutions are added is essentially fixed formally p is taken to be of order n c for any fixed c 0 and the number of institutions whose ranking is reliable depends very little on p on the other hand cases where the number of reliable rankings increases significantly when new institutions are added are those for which the distribution of the true attributes is relatively heavy tailed for example with tails that decay like x for some 0 these properties and others are explored analytically under general conditions a numerical study links the results to outcomes for real data problems

connections between singular control and optimal switching this paper builds a new theoretical connection between singular control of finite variation and optimal switching problems this correspondence provides a novel method for solving high dimensional singular control problems and enables us to extend the theory of reversible investment sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions consequently our regularity result links singular controls and dynkin games through sequential optimal stopping problems

a problem of sequential entry and exit decisions combined with discretionary stopping we consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty this problem combines features of both stochastic impulse control and optimal stopping the aim is to discover the form of the optimal strategy it turns out that this has a priori rather unexpected features the results that we establish are of an explicit nature we also construct an example whose value function does not possess c1 regularity

a class of singular control problems and the smooth fit principle this paper analyzes a class of singular control problems for which value functions are not necessarily smooth necessary and sufficient conditions for the well known smooth fit principle along with the regularity of the value functions are given explicit solutions for the optimal policy and for the value functions are provided in particular when payoff functions satisfy the usual inada conditions the boundaries between action and continuation regions are smooth and strictly monotonic as postulated and exploited in the existing literature see a k dixit and r s pindyck investment under uncertainty princeton university press princeton nj 1994 m h a davis et al adv in appl probab 19 1987 pp 156 176 t o kobila stochastics stochastics rep 43 1993 pp 29 63 a b abel and j c eberly j econom dynam control 21 1997 pp 831 852 a oksendal finance stoch 4 2000 pp 223 250 j a scheinkman and t zariphopoulou j econom theory 96 2001 pp 180 207 a merhi and m zervos siam j control optim 46 2007 pp 839 876 and l h alvarez a general theory of optimal capacity accumulation under price uncertainty and costly reversibility working paper helsinki center of economic research helsinski finland 2006 illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions

optimal switching strategy of a mean reverting asset over multiple regimes we solve optimal iterative three regime switching problems with transaction costs with investment in a mean reverting asset that follows an ornstein uhlenbeck process and find the explicit solutions the investor can take either a long short or square position and can switch positions during the period modeling the short sales position is necessary to study optimal trading strategies such as the pair trading few studies provide explicit solutions to problems with multiple more than two regimes states the value function is proved to be a unique viscosity solution of a hamilton jacobi bellman variational inequality hjb vi multiple regime switching problems are more difficult to solve than conventional two regime switching problems because they need to identify not only when to switch but also where to switch therefore multiple regime switching problems need to identify the structure of the continuation switching regions in the free boundary problem for each regime if the number of the states n is two only two regions have to be identified but if n 3 n p 2 6 regions have to be detected we identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach

optimal switching over multiple regimes this paper studies the optimal switching problem for a general one dimensional diffusion with multiple more than two regimes this is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties a viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities leading to sharp qualitative characterizations of the switching regions these characterizations in turn reduce the switching problem into one of finding a finite number of threshold values in a state that would trigger switchings the results of our analysis take several qualitatively different forms depending on model parameters and the issue of when and where it is optimal to switch is addressed the general results are then demonstrated by the three regime case where a quasi explicit solution is obtained and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one dimensional diffusions

remarks on the energy of regular graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix this note is about the energy of regular graphs it is shown that graphs that are close to regular can be made regular with a negligible change of the energy also a k regular graph can be extended to a k regular graph of a slightly larger order with almost the same energy as an application it is shown that for every sufficiently large n there exists a regular graph g of order n whose energy g satisfies g 1 2 n 3 2 n 13 10 several infinite families of graphs with maximal or submaximal energy are given and the energy of almost all regular graphs is determined

optimal dynamic risk taking we analyze a continuous time stochastic control problem that arises in the study of several important issues in financial economics an agent controls the drift and volatility of a diffusion output process by dynamically selecting one of an arbitrary but finite number of projects and the termination time the optimal policy depends on the projects risk adjusted drifts that are determined by their drifts volatilities and the curvature or relative risk aversion of the agent s payoff function we prove that the optimal policy only selects projects in the spanning subset furthermore if the projects risk adjusted drifts are consistently ordered for all output values then the optimal policy is characterized by at most k 1 switching triggers where k is the number of projects in the spanning subset we also characterize the optimal policy when the consistent ordering condition does not hold and we outline a general and tractable computational algorithm to derive the optimal policies

explicit solution to an optimal switching problem in the two regime case this paper considers the problem of determining the optimal sequence of stopping times for a diffusion process subject to regime switching decisions this is motivated in the economics literature by the investment problem under uncertainty for a multi activity firm involving opening and closing decisions we use a viscosity solutions approach combined with the smooth fit property and explicitly solve the problem in the two regime case when the state process is of geometric brownian nature the results of our analysis take several qualitatively different forms depending on model parameter values