boundary behavior of solutions to some singular elliptic boundary value problems abstract let be a bounded domain with smooth boundary in r n n 1 for the more general weight b some nonlinearities f and singularities g by two kinds of nonlinear transformations a new perturbation method which was introduced by garcia melian in j garcia melian boundary behavior of large solutions to elliptic equations with singular weights nonlinear anal 67 2007 818 826 and comparison principles we show that the boundary behavior of solutions to a boundary blow up elliptic problem w b x f w w 0 x w and a singular dirichlet problem u b x g u u 0 x u 0 has the same form under the nonlinear transformations which can be determined in terms of the inverses of the transformations

on a class of singular elliptic systems we study the semilinear elliptic system u u p v q u 0in v u r v s v 0in u v 0on where rn n 1 is a smooth and bounded domain p q r s 0 under suitable ranges of exponents we obtain various results regarding the well posedness of our system

option pricing stochastic volatility singular dynamics and constrained path integrals abstract stochastic volatility models have been widely studied and used in the financial world the heston model heston 1993 7 is one of the best known models to deal with this issue these stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter which relates the two brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset solutions to the heston model in the context of option pricing using a path integral approach are found in lemmens et al 2008 21 while in baaquie 2007 1997 12 13 propagators for different stochastic volatility models are constructed in all previous cases the propagator is not defined for extreme cases 1 it is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator in this paper we study in detail a general class of stochastic volatility models for extreme values 1 and show that in these two cases the associated classical dynamics corresponds to a system with second class constraints which must be dealt with using dirac s method for constrained systems dirac 1958 1967 22 23 in order to properly obtain the propagator in the form of a euclidean hamiltonian path integral henneaux and teitelboim 1992 25 after integrating over momenta one gets an euclidean lagrangian path integral without constraints which in the case of the heston model corresponds to a path integral of a repulsive radial harmonic oscillator in all the cases studied the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral

lane emden systems with negative exponents we study the elliptic system u u pv qin v u rv sin u v 0on in a bounded domain rn n 1 with a smooth boundary p s 0 and q r 0 we investigate the existence non existence and uniqueness of c2 c solutions in terms of p q r and s a necessary and sufficient condition for the c1 regularity of solutions up to the boundary is also obtained

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

boundary behaviour of the unique solution to a singular dirichlet problem with a convection term by karamata regular variation theory and constructing comparison functions we derive that the boundary behaviour of the unique solution to a singular dirichlet problem u b x g u u q u 0 x u 0 which is independent of u q w here is a bounded domain with smooth boundary in r n r q 0 2 lims 0 g s a ndb is nonnegative on which may be vanishing on the boundary

multi parameter bifurcation and asymptotics for the singular lane emden fowler equation with a convection term to the second variable and g is unbounded around the origin the asymptotic behaviour of the solution around the bifurcation point is also established provided g u behaves like u around the origin for some 0 1 our approach relies on finding explicit sub and supersolutions combined with various techniques related to the maximum principle for elliptic equations the analysis we develop in this paper shows the key role played by the convection term u p

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

on a singular elliptic system with quadratic growth in the gradient abstract in this paper we deal with the following singular elliptic system u u 2 u p p q a x v q u p 2 u f x r n v v 2 v q p q a x u p v q 2 v g x r n where n 3 n 2 4 p q 1 and p q n 2 n 2 we show through the sub and supersolutions method the existence of a nonnegative solution for an approximated system the limit of the approximated solution is a positive solution in the case 0 p q and f g we prove the uniqueness of a solution among others we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle a minimization method and with the help of nehari manifold the proofs rely on the concentration compactness principle

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

the exact boundary behavior of the unique solution to a singular dirichlet problem with a nonlinear convection term abstract in this paper we analyze the exact boundary behavior of the unique solution to the singular nonlinear dirichlet problem u b x g u u q u 0 x u 0 where is a bounded domain with smooth boundary in r n q 0 2 0 g c 1 0 0 lim s 0 g s g is decreasing on 0 and b c loc is positive in may be vanishing or singular on the boundary we reveal that the nonlinear convection term u q does not affect the first expansion of the unique solution near the boundary to the problem

singular elliptic problems existence non existence and boundary behavior abstract we deal with the existence uniqueness and boundary behavior of positive solutions of the semilinear elliptic equation u a x g u b x f u in under dirichlet boundary conditions where r n is a bounded domain with smooth boundary a b g f are continuous non negative real valued functions the main feature here is that either g or f or both of them are singular at 0 in the sense that g t f t t 0 and 0 are parameters our results require no symmetry from either a or b and no monotonicity on f or g penalty arguments as well as variational principles are exploited

boundary behavior for the solutions to dirichlet problems involving the infinity laplacian abstract in this paper by constructing suitable comparison functions we mainly analyze the exact boundary behavior for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s g is normalized regularly varying at zero with index 1 and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary

steady state solutions for a general activator inhibitor model abstract we discuss existence results for a singular gierer meinhardt elliptic system with zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology and has interesting and challenging mathematical properties the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the general activator inhibitor model

symmetry analysis of the bidimensional lane emden systems abstract we carry out a complete group classification of the nonlinear lane emden systems in dimension two the noether symmetries are found and their corresponding conservation laws are established

fluctuations of casimir forces on finite objects i spheres and hemispheres the mean square forces that result from the zero point fluctuations of quantized fields are calculated when acting on spheres and hemispheres of variable sizes for the maxwell field the boundary conditions of a perfectly conducting surface are imposed the scalar field is investigated for neumann and dirichlet boundary conditions the force is averaged over a finite time t small and large objects are distinguished on the scale of ct the results for the sphere and the hemisphere are compared with those for a piston that is embedded in an infinite plane a small hemisphere and a small piston are found to have fluctuations of the same order of magnitude while on a small sphere the fluctuations are by two orders of magnitude smaller because of correlations of fluctuations on the two sides of the sphere large spheres are shown to fit into the picture of large objects being composed of many patches each with the fluctuations impinging as on a large piston

positive solutions of lane emden systems with negative exponents existence boundary behavior and uniqueness abstract we study the existence boundary behavior and uniqueness of solutions for the singular elliptic system u u p v q v u r v s u 0 v 0 x u v 0 where is a bounded domain with smooth boundary in r n p s 0 and q r 0 our results are obtained in a range of p q r s different from those in m ghergu lane emden systems with negative exponents j funct anal 258 2010 3295 3318

vector fields on a disk with mixed boundary conditions vector fields on a disk satisfying two types of boundary conditions are studied these boundary conditions are selected by brst invariance in electrodynamics they also appear in the de rham complex the construction of the harmonic expansion is presented the eigenfunctions of the vector laplace operator are expressed in terms of fields satisfying pure dirichlet or robin boundary conditions for the case of four dimensional disk several first coefficients of the heat kernel expansion are computed an error in the analytical expression by branson and gilkey is corrected

positive versus compact support solutions to a singular elliptic problem abstract the semilinear elliptic problem u k x u q u p u 0 in u 0 on is considered in this paper where is a bounded domain of r n n 2 with c 2 boundary 0 q p 1 and k x as x we mainly study the effect of the blow up rate of k x near to the existence of positive and compact support solutions furthermore an optimal compact support principle is given for a class of elliptic differential inequalities

path integral approach to asian options in the black scholes model abstract we derive a closed form solution for the price of an average strike as well as an average price geometric asian option by making use of the path integral formulation our results are compared to a numerical monte carlo simulation we also develop a pricing formula for an asian option with a barrier on a control process combining the method of images with a partitioning of the set of paths according to the average along the path this formula is exact when the correlation is zero and is approximate when the correlation increases

modelling stock market volatility bridging the gap to continuous time understanding and specifying the discrete time model d b nelson modelling stock market volatility changes d b nelson stationarity and persistence in the garch i i model d b nelson conditional heteroskedasticity in asset returns a new approach p a braun d b nelson and a m sunier good news bad news volatility and betas continuous time limits and optimal filtering for arch models d b nelson arch models as diffusion approximations d b nelson filtering and forecasting with misspecified arch models i getting the right variance with the wrong model d b nelson and d p foster filtering and forecasting with misspecified arch models ii making the right forecast with the wrong model d b nelson and d p foster asymptotic filtering theory for univariate arch models d b nelson asymptotic filtering theory for multivariate arch models d b nelson and d b nelson continuous record asymptotics for rolling sample variance estimators specification and estimation of continuous time processes r f engle and g g j lee estimating diffusion models of stochastic volatility a r gallant and g tauchen specification analysis of continuous time models in finance l p hansen and j a scheinkman back to the future generating moment implications for continuous time markov processes y ait sahalia nonparametric pricing of interest rate derivative securities index

a three solution theorem for singular nonlinear elliptic boundary value problems abstract we establish a three solution theorem for singular elliptic boundary value problems of the form u f u u in u 0 on where is a bounded domain in r n n 1 with a smooth boundary here f 0 0 is a c 1 function in 0 with f 0 0 and 0 1 in particular if there exist two pairs of sub supersolutions 1 1 and 2 2 where 1 2 1 1 2 1 with 2 2 and 2 2 are strict sub and supersolutions then we establish existence of three solutions u 1 u 2 and u 3 for the above boundary value problem such that u 1 1 2 u 2 2 1 and u 3 1 1 1 2 2 1 our results extend the work in 1 and also in 17 where such multiplicity results were discussed in the non singular case 0 further our results strengthen the multiplicity results in 12 and 13 for such singular problems 0 namely establish an additional third solution we also establish a sub super solution theorem for infinite semipositone problems i e the case when f 0 0

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

existence and optimal estimates of solutions for singular nonlinear dirichlet problems abstract by applying the properties of the unique classical solution to the singular boundary value problem on half line p s g p s p s 0 s 0 p 0 0 lims p s b 0 and constructing the new comparison functions they show the existence and the optimal global estimates of solutions to singular nonlinear dirichlet problems u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in rn g s is nonincreasing and positive in 0 1 g t dt and lim s 0 g s k c is positive in and may be singular or zero on the boundary

pricing currency options in the mixed fractional brownian motion abstract this paper deals with the problem of pricing european currency options in the mixed fractional brownian environment both the pricing formula and the mixed fractional partial differential equation for european call currency options are obtained some greeks and the estimator of volatility are also provided empirical studies and simulation results confirm the theoretical findings and show that the mixed fractional brownian pricing model is a reasonable one

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

existence and uniqueness of solutions for a singular semilinear elliptic system abstract in this paper we study the existence and the uniqueness of solutions for the singular semilinear elliptic system u f x u 2 g x u 1 p 2 p q a x u p v 1 q x r n v f x v 2 g x v 1 q 2 p q a x u 1 p v q x r n where 0 p 1 and 0 q 1 we show the existence of a solution for an approximated system by using a perturbation method which eliminates the singularity and via variational methods the limit of the approximated solutions is a positive solution in the case g 0 p q and 1 2 p 1 we prove the uniqueness of solution the proofs rely on the concentration compactness principle

a priori estimates and existence for elliptic systems via bootstrap in weighted lebesgue spaces we present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains this method is based on a bootstrap procedure used alternatively on each component in the scale of weighted lebesgue spaces lp lp x dx where x is the distance to the boundary using this method we significantly improve the known existence results for various classes of elliptic systems

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

global existence for a singular gierer meinhardt system abstract this paper is concerned with existence results for a singular gierer meinhardt system subject to zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology the mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure we use a functional method to obtain both upper and lower bounds for the perturbed system and then use sobolev embedding theorem to prove the existence of a pair of positive solutions under suitable conditions this method is first used in a singular parabolic system and is completely different than the traditional methods of sub and super solutions

positive solutions of the p laplace equation with singular nonlinearity abstract for a given bounded domain in r n with c 1 boundary for some 0 1 and a possibly singular nonlinearity f on 0 we give sufficient conditions on f so that the p laplace equation p u f x u admits a solution u w 0 1 p on the basis of a comparison principle we will give a sufficient condition under which such a problem admits a unique solution

the existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents a b s t r a c t in this paper we consider a class of singular elliptic boundary problems u u p x u u 0 in u 0 on with 1 0 1 and p c 0 0 1 we emphasize that there is no restriction on the boundedness of from above by means of the approximating variational arguments and truncation two classical positive solutions are obtained 2014 the authors published by elsevier ltd this is an open access article under the cc by nc nd license http creativecommons org licenses by nc nd 3 0

finite element approximation of a semilinear elliptic problem with a singular nonlinearity given nu in bbb r we consider the following problem find u 0 such that where omega subset bbb r d d 1 2 or 3 and c 0 in bar omega we prove h 1 and l infty error bounds for the standard continuous piecewise linear galerkin finite element approximation with a weakly acute triangulation our bounds are nearly optimal in addition for d 1 and 2 and c in bbb r we analyze a more practical scheme involving numerical integration on the nonlinear term we obtain nearly optimal h 1 and l infty error bounds for d 1 for this case we also present some numerical results

bounded solutions to a singular parabolic system abstract in this paper we are concerned with the singular parabolic system u t u f x v p v t v g x u q in a smooth bounded domain r n subject to zero dirichlet conditions with initial conditions u 0 x v 0 x 0 this problem is of interest as it is related to some problems in biology and physics under suitable assumptions on p q and f x g x some existence results of weak and classical solutions are obtained using a functional method this method is motivated by such results found in 4 and 5 when dealing with singular parabolic systems and the related references within

the pricing of european options on two underlying assets with delays abstract in the paper the pricing of european options on two underlying assets with delays is discussed by using the approach of equivalent martingale measure transformation the market is proved to be complete with exchange option as a particular example we obtain the explicit pricing formula in a subinterval of option period the robust euler maruyama method is combined with the monte carlo simulation to compute exchange option prices within the whole option period numerical experiments indicate that there is an increasing possibility of the difference between the delayed and black scholes option prices with the increase of delay

existence and regularity results for fully nonlinear equations with singularities we consider the dirichlet boundary value problem for a singular elliptic pde like f u p x u where p 0 in a bounded smooth domain of mathbb r n the nondivergence form operator f is assumed to be of hamilton jacobi bellman or isaacs type we establish existence and regularity results for such equations

an exact estimate result for a class of singular equations with critical exponents abstract we consider the singular boundary value problem u h x u u p 1 in u 0 on with p 2 n n 2 0 1 it is well known that there exists 0 such that the problem has a solution for all 0 and no solution for we obtain an exact result for p h

estimating option greeks under the stochastic volatility using simulation abstract as the black scholes bs equation being widely used to price options which is based on a hypothesis that the underlying bonds stocks volatility is constant many scholars proposed the extended version of this formula to predict the behavior of the volatility so stochastic volatility model is the improved version in which fixed volatility is replaced the purpose of this study is to adopt one of the famous stochastic volatility models heston model 1993 to price european call options put option values can easily obtained by call put parity if it is needed simulation has proved to be a valuable tool for estimating options price derivatives i e greeks this paper proposes the method for the simulation of stock prices and variance under the heston stochastic volatility model we consider three different models based on the heston model we present two direct methods a path wise method and likelihood ratio method for estimating the derivatives of options then we compare it with black scholes equation and make a sensitivity analysis for its parameters by using estimator s approaches

liouville type theorems monotonicity results and a priori bounds for positive solutions of elliptic systems this paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations

the functional integral for fields in a cavity the functional integral for a scalar field confined in a cavity and subjected to linear boundary conditions is discussed herein it is shown how the functional measure can be conveniently dealt with by modifying the classical action with boundary corrections the nonuniqueness of the boundary actions is described with a three parameter family of them giving identical boundary conditions in some cases the corresponding green s function will define a kind of generalized gaussian measure on function space the vacuum energy is discussed paying due attention to its anomalous scale dependence and the physical issues involved are considered

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

uniformly bounded weak and classical solutions to a singular parabolic system and applications abstract equations of both elliptic and parabolic type featuring singular nonlinearities have appeared in numerous works throughout the years in this work we consider a time dependent problem featuring nonlinearities of the form u p v q u r v s subject to homogeneous dirichlet boundary conditions and prove the existence of uniformly bounded weak and classical solutions under appropriate conditions on p q r s these results can in some ways be seen as a generalization of the results presented in 9 these results are obtained using a functional method motivated by works found in 8 22 6 etc and the boundary behaviour of a fundamental singular elliptic equation described in 16