a diffusion model for optimal portfolio selection in the presence of brokerage fees we consider a financial market model with two assets one has deterministic rate of growth while the rate of growth of the second asset is governed by a brownian motion with drift we can shift money from one asset to another however there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset we want to maximize the expected rate of growth of funds it is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort

explicit solution of a general consumption investment problem this paper solves a general consumption and investment decision problem in closed form an investor seeks to maximize total expected discounted utility of consumption there are n distinct risky investments modelled by dependent geometric brownian motion processes and one riskless deterministic investment the analysis allows for a general utility function and general rates of return the model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy so this paper generalizes many of the results of lehoczky sethi and shreve lehoczky j s sethi s shreva 1983 optimal consumption and investment policies allowing consumption constraints and bankruptcy math oper res 8 613 636 the value function is determined explicitly as are the optimal consumption and investment policies the analysis is extended to consider more general risky investments under certain conditions the value functions derived for geometric brownian motion are shown to provide upper and lower bounds on the value functions in the more general context

table erratum table of integrals series and products academic press new york 1980 mr0582453 81g 33001 by i s gradshteĭn and i m ryzhik

symmetric periodic solutions of the anisotropic manev problem we consider the manev potential in an anisotropic space i e such that the force acts differently in each direction using a generalization of the poincare continuation method we study the existence of periodic solutions for weak anisotropy in particular we find that the symmetric periodic orbits of the manev system are perturbed to periodic orbits in the anisotropic problem

infinitely many periodic orbits for the rhomboidal five body problem we prove the existence of infinitely many symmetric periodic orbits for a regularized rhomboidal five body problem with four small masses placed at the vertices of a rhombus centered in the fifth mass the main tool for proving the existence of such periodic orbits is the analytic continuation method of poincare together with the symmetries of the problem

multi asset investment consumption model with transaction costs abstract in this paper we consider the multi asset optimal investment consumption model a riskless asset and d risky assets when the initial time is t 0 for a proportional transaction costs and discount factors we proof that the value function of the model is a unique viscosity solution of a hamilton jacobi bellman hjb equations

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

on an integral related to biaxially anisotropic media an integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail the integral has three integer two real and one complex parameter the integrand involves a product of bessel functions of different argument and order several generalizations are discussed

numerical analysis of american option pricing in a jump diffusion model we discuss pricing formulae for american options in merton s jump diffusion model with the help of variational inequalities we derive some regularity properties of price functions using the finite difference method a discretization scheme is presented and a convergence theorem for the first order derivatives is proved numerical methods and results are also discussed

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the dynamic programming equation for the problem of optimal investment under capital gains taxes this paper considers an extension of the merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes we derive the dynamic programming equation in the sense of constrained viscosity solutions we next introduce a family of functions v varepsilon varepsilon 0 which converges to our value function uniformly on compact subsets and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation in particular this result justifies the numerical results reported in the accompanying paper i ben tahar h m soner and n touzi 2005 modeling continuous time financial markets with capital gains taxes preprint http www cmap polytechnique fr sim touzi bst06 pdf

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

additive control of stochastic linear systems with finite horizon we consider a dynamic system whose state is governed by a linear stochastic differential equation with time dependent coefficients the control acts additively on the state of the system our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process it is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space time domain by using some decomposition arguments the problems of a two sided control i e optimal corrections and the case with constraints on the resources i e finite fuel can be reduced to a simpler case of only one sided control i e a monotone follower these results are applied to solving some examples by the so called method of similarity solutions

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

finite horizon optimal investment and consumption with transaction costs this paper concerns continuous time optimal investment and the consumption decision of a constant relative risk aversion crra investor who faces proportional transaction costs and a finite time horizon in the no consumption case it has been studied by liu and loewenstein review of financial studies 15 2002 pp 805 835 and dai and yi j differential equations 246 2009 pp 1445 1469 mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints the problem gives rise to two free boundaries which stand for the optimal buying and selling strategies respectively we present an analytical approach to analyze the behaviors of free boundaries the regularity of the value function is studied as well our approach is essentially based on the connection between singular control and optimal stopping which is first revealed in the present problem

finite horizon optimal investment with transaction costs a parabolic double obstacle problem abstract this paper concerns optimal investment problem of a crra investor who faces proportional transaction costs and finite time horizon from the angle of stochastic control it is a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies this enables us to make use of the well developed theory of obstacle problem to attack the problem the c 2 1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized

stability in a rhomboidal 5 body problem with generalized central forces we consider a system of five mass points r1 r2 r3 and r4 with masses m1 m2 m and m3 m4 m moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system we assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a manev s type potential i e characterized by a potential of the form 1r er2 on the other hand is assumed that the attraction between the bodies r1 r2 r3 and r4 is of the newtonian type this model represents several cases for instance when the central body is a spheroid or a radiating source first we prove the existence of three different relative rhomboidal solutions and its central configuration is as follows 1 the rhombus is a square and all primaries have equal masses 2 the rhombus is not a square but all masses are equal and 3 the rhombus is not a square and the pairs of primaries have different masses the first two cases present two pa

on some cheap control problems for diffusion processes on considere plusieurs cas de problemes de commande de processus de diffusion dans lesquels la fonctionnelle de cout ne depend pas explicitement de la commande on demontre la continuite de la fonction cout optimale et on en donne une caracterisation a l aide d une inegalite quasivariationnelle interpretant le probleme comme la limite d un probleme de commande impulsionnelle dont le cout tend vers zero

table errata table of integrals series and products corrected and enlarged edition academic press new york 1980 mr 81g 33001 by i s gradshteyn i s gradshteĭn and i m ryzhik

table errata table of integrals series and products by i s gradshteĭn and i m ryzhik academic press boston 1994 mr1243179 94g 00008

investment consumption models with transaction fees and markov chain parameters this paper considers an infinite horizon investment consumption model in which a single agent consumes and distributes his wealth in two assets a bond and a stock the problem of maximization of the total utility from consumption is treated state amount allocated in assets and control consumption rates of trading constraints are present it is shown that the value function is the unique viscosity solution of a system of variational inequalities with gradient constraints

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given