continuous time red and black how to control a diffusion to a goal a player starts at x in 0 1 and tries to reach 1 the process xt t 0 of his positions moves according to a diffusion process or more generally an ito process whose infinitesimal parameters are chosen by the player at each instant of time from a set depending on his current position to maximize the probability of reaching 1 the player should choose the parameters so as to maximize 2 at least when the maximum is achieved by bounded measurable functions this implies that bold timid play is optimal for subfair superfair continuous time red and black furthermore in superfair red and black the strategy which maximizes the drift coefficient of log xt minimizes the expected time to reach 1

continuous time casino problems a gambler or an investor seeks to increase his fortune by a given amount before going bankrupt the problem can be formulated as a stochastic control problem on an interval and is called a continuous time casino if certain natural conditions are satisfied an explicit formula is given for the value function of such a problem and the collection of optimal strategies is determined fair casinos are characterized in a result analogous to one of l e dubins and l j savage for discrete time problems

time periodic solutions of quasilinear parabolic differential equations i dirichlet boundary conditions we study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable our approach is to relate the theory of such problems to the classical theory for initial boundary value problems in the process we generalize many previously known results

minimizing or maximizing the expected time to reach zero we treat the following control problems the process x 1 t with values in the interval infty 0 or 0 infty is given by the stochastic differential equation dx 1 t mu t dt sigma t dw t quad x 1 0 x 1 where the nonanticipative controls mu and sigma are to be chosen so that mu t sigma t remains in a given set mathcal s and the object is to minimize or maximize the expected time to reach the origin the minimization problem had been dicussed earlier by heath pestien and sudderth under various restrictions on the set mathcal s here an improved verification lemma is established which is used to solve the minimization and maximization problems for any mathcal s an application to a portfolio problem is discussed

spectral stability of dirichlet second order uniformly elliptic operators we prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous dirichlet boundary conditions upon domain perturbation

the nonlinear oblique derivative problem for quasilinear elliptic equations on considere des problemes aux valeurs limites pour une equation elliptique quasilineaire d ordre 2 qu 0 dans ou est un domaine borne de r n n 2 et q est donne sur c 2 par qu x a ij x u x du x d ij u x a x u x du x

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

ito s calculus in financial decision making this paper presents an introduction to ito s stochastic calculus by stating some basic definitions theorems and mathematical examples afterwards the use of ito s calculus in modern financial theory is illustrated by expositing a few representative applications the main observation of this paper is that ito s calculus which was developed from purely mathematical questions originating in wiener s work has found unexpectedly important applicability in the theory of finance from the perspective of continuous time

domain dependence of eigenvalues of elliptic type operators the dependence on the domain for the dirichlet eigenvalues of elliptic operators considered in bounded domains is studied the proximity of domains is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one this allows to compare eigenvalues corresponding to domains that have non smooth boundaries and different topology the main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity applications of this result are given the results are new for the laplace operator

survival and growth with a liability optimal portfolio strategies in continuous time we study the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time e g to pay for a liability to consume or to pay dividends the investor is allowed to invest in any or all of a given number of risky stocks whose prices follow geometric brownian motion as well as in a riskless asset which has a constant rate of return the fact that the withdrawal is continuously enforced regardless of the wealth level ensures that there is a region where there is a positive probability of ruin in the complementary region ruin can be avoided with certainty call the former region the danger zone and the latter region the safe region we first consider the problem of maximizing the probability that the safe region is reached before bankruptcy which we call the survival problem while we show among other results that an optimal policy does not exist for this problem we are able to construct explicit e optimal policies for any e 0 in the safe region where ultimate survival is assured we turn our attention to growth among other results we find the optimal growth policy for the investor i e the policy which reaches another higher valued goal as quickly as possible other variants of both the survival problem as well as the growth problem are also discussed our results for the latter are intimately related to the theory of constant proportions portfolio insurance

solvability of quasilinear elliptic equations with nonlinear boundary conditions ii abstract it is shown that solvability of the second order quasilinear elliptic equation qu 0 in with first order nonlinear boundary condition nu 0 on can be inferred from appropriate estimates on solutions of the problem qu in nu 0 on as varies over a suitable function class this result improves previous work of the author where estimates were required on solutions of qu in nu on as varies over some function space the value of this improvement is demonstrated by some examples

random dispersal versus fitness dependent dispersal abstract this work extends previous work cantrell et al 2008 9 on fitness dependent dispersal for a single species to a two species competition model both species have the same population dynamics but one species adopts a combination of random and fitness dependent dispersal and the other adopts random dispersal global existence of smooth solutions to the time dependent quasilinear parabolic system is studied when a single species has a strong tendency to move up its fitness gradient it results in a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution cantrell et al 2008 9 for the two species competition model if one species has strong tendency to move up its fitness gradient such approximately ideal free dispersal is advantageous relative to random dispersal bifurcation analysis shows that two competing species can coexist when one species has only an intermediate tendency to move up its fitness gradient and the other species has a smaller random dispersal rate

laplace eigenvalues on regular polygons a series in 1 n abstract for regular polygons p n inscribed in a circle the eigenvalues of the laplacian converge as n to the known eigenvalues on a circle we compute the leading terms of n in a series in powers of 1 n by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon the o 1 n 2 term comes from hadamard s formula and reflects the change in area this term disappears if we transcribe the polygon scaling it to have the same area as the circle

the laplacian eigenvalues of a polygon the difficulties are almost always at the boundary that statement applies to the solution of partial differential equations with a given boundary and also to shape optimization with an unknown boundary these problems require two decisions closely related but not identical o1 how to discretize the boundary conditions 2 how to discretize the boundary itself that second problem is the one we discuss here the region w is frequently replaced by a polygon or polyhedron the approximate boundary email protected n may be only a linear interpolation of the true boundary email protected a perturbation theory that applies to smooth changes of domain is often less successful for a polygon this paper concentrates on a model problem the simplest we could find and we look at eigenvalues of the laplacian the boundary email protected will be the unit circle the approximate boundary email protected n is the regular inscribed polygon with n equal sides it seems impossible that the eigenvalues of regular polygons have not been intensively studied but we have not yet located an authoritative reference the problem will be approached numerically from three directions without attempting a general theory those directions are o1 finite element discretizations of the polygons w n 2 a taylor series based on piecewise smooth perturbations of the circle 3 a series expansion of the eigenvalues in powers of 1 n the second author particularly wishes that we could have consulted george fix about this problem his harvard thesis demonstrated the tremendous improvement that singular elements can bring to the finite element method particularly when w has a reentrant corner or even a crack his numerical experiments in 1 came at the beginning of a long and successful career in applied mathematics we only wish it had been longer

on differentiability of eigenvalues of second order elliptic operators on non smooth domains abstract the regularity of eigenvalues of elliptic operators upon deformations of a given bounded domain is a classical problem in elliptic pdes which has been focused by many authors we establish a theorem on c r dependence of algebraically simple eigenvalues and eigenfunctions with respect to perturbations of c 1 class of non smooth domains and of c r class of coefficients of elliptic operators moreover we also compute the first variation of these eigenvalues in relation to both parameters for non smooth domains as a byproduct we extend hadamard s formula to second order elliptic operators for domains of c 2 class and other non smooth ones

controlling a process to a goal in finite time a player starts at x in 0 1 and seeks to reach 1 by time t0 the process xt 0 t t0 of the player s positions is a diffusion process or an ito process whose infinitesimal parameters are chosen by the player at each instant of time from a set depending on the current position the probability of reaching 1 by time t0 is maximized if the player can and does choose the parameters so that and 2 are maximized at least when these maxima are sufficiently regular this result implies that bold play is optimal for subfair continuous time red and black and roulette when there is a limit on playing time

semilinear periodic parabolic equations with nonlinear boundary conditions recently much work has been devoted to periodic parabolic equations with linear homogeneous boundary conditions however very little has been accomplished in the literature for periodic parabolic problems with nonlinear boundary conditions it is the purpose of this paper to prove existence and regularity results for classical periodic solutions to semilinear second order parabolic partial differential equations with nonlinear boundary conditions provided ordered upper and lower solutions are given fractional order function spaces ehrling gagliardo nirenberg and lions peetre calderon type interpolation inequalities for functions in anisotropic sobolev slobodeckii spaces play an important role in the obtainment ofa prioriboundary and interior estimates in proving our existence results we make use of topological degree techniques and regularity results for linear parabolic partial differential equations under linear nonhomogeneous boundary conditions we also indicate how one can obtain minimal and maximal time periodic solutions to parabolic problems with nonlinear boundary conditions

on the hadamard formula for nonsmooth domains we consider the first eigenvalue of the dirichlet laplacian in three cases c1 1 domains lipschitz domains and bounded domains without any smoothness assumptions asymptotic formula for this eig

a robust finite element solver for a multiharmonic parabolic optimal control problem this paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting in particular the desired state is assumed to be multiharmonic after eliminating the control from the optimality system we arrive at the reduced optimality system for the state and the co state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation due to the linearity the state and the co state are multiharmonic as well we discretize the fourier coefficients by the finite element method this leads to a large system of algebraic equations which fortunately decouples into smaller systems each of them defining the cosine and sine fourier coefficients for the state and co state with respect to a single frequency for these smaller systems we construct preconditioners resulting in a fast converging minimal residual solver with a parameter independent convergence rate all these systems can be solved totally in parallel