convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay this paper can be regarded as the continuation of the work contained in papers milosevic 2011 2013 at the same time it represents the extension of the paper wu et al 2010 in this paper the one sided lipschitz condition is employed in the context of the backward euler method for a class of neutral stochastic differential equations with constant delay sufficient conditions for this method to be well defined are revealed under certain nonlinear growth conditions the convergence in probability is established for the continuous forward backward euler method as well as for the discrete backward euler method additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without requiring for the drift coefficient to satisfy the linear growth condition this manuscript represents the continuation of the previous work of the author related to a class of highly nonlinear neutral stochastic differential equations with time dependent delay main results of this manuscript are convergence in probability and almost sure exponential stability of the backward euler approximate solution for a class of stochastic differential equations with constant delay conditions under which both the exact and approximate solutions share the property of the almost sure exponential stability are revealed this paper also illustrates that in some cases stochastic differential equations with constant delay should be studied separately from those with time dependent delay in that sense it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time dependent delay by the constant delay

on the equilibrium of shell membranes under normal loading hidden integrability it is established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact integrable the membranes are shown to have geometries within the integrable class of so called o surfaces the membrane o surfaces include inter alia minimal constant mean curvature constant gaussian curvature and more generally linear weingarten surfaces as well as canal surfaces and dupin cyclides

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

strong convergence rates of modified truncated em methods for neutral stochastic differential delay equations abstract the aim of this paper is to investigate strong convergence of modified truncated euler maruyama method for neutral stochastic differential delay equations introduced in lan 2018 strong convergence rates of the given numerical scheme to the exact solutions at fixed time t are obtained under local lipschitz and khasminskii type conditions moreover convergence rates over a time interval 0 t are also obtained under additional polynomial growth condition on g without the weak monotonicity condition which is usually the standard assumption to obtain the convergence rate two examples are presented to interpret our conclusions

strong convergence rates of modified truncated em method for stochastic differential equations abstract motivated by truncated euler maruyama em method introduced by mao 2015 a new explicit numerical method named modified truncated euler maruyama method is developed in this paper strong convergence rates of the given numerical scheme to the exact solutions to stochastic differential equations are investigated under given conditions in this paper compared with truncated em method the given numerical simulation strongly converges to the exact solution at fixed time t and over a time interval 0 t under weaker sufficient conditions meanwhile the convergence rates are also obtained for both cases two examples are provided to support our conclusions

a distributional version of functional equations and their stabilities abstract generalizing the results in j math anal appl 286 2003 177 186 j math anal appl 295 2004 107 114 arch math to appear j math anal appl 299 2004 578 586 that consider the hyers ulam stability problems of several functional equations in the spaces of the schwartz tempered distributions and the fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

stability of exponential equations in schwartz distributions abstract we reformulate the superstability of exponential equation and cosine functional equation j a baker the stability of cosine equation proc amer math soc 80 1980 411 416 in some spaces of generalized functions such as the schwartz distributions sato hyperfunctions and gelfand generalized functions which completes the previous results of partial generalizations of the stability problems j chung a distributional version of functional equations and their stabilities nonlinear anal 62 2005 1037 1051 j chung s y chung d kim the stability of cauchy equations in the space of schwartz distributions j math anal appl 295 2004 107 114

implicit numerical methods for highly nonlinear neutral stochastic differential equations with time dependent delay abstract this paper represents the continuation of the analysis from papers milosevic 2011 10 and milosevic 2013 11 the main aim of this paper is to establish certain results for the backward euler method for a class of neutral stochastic differential equations with time dependent delay for that purpose the split step backward euler method which represents an extension of the backward euler method is introduced for this class of equations conditions under which the split step backward euler method and thus the backward euler method is well defined are revealed moreover the convergence in probability of the backward euler method is proved under certain nonlinear growth conditions including the one sided lipschitz condition this result is proved using the technique which is based on the application of the continuous time approximation for this reason the discrete forward backward euler method is involved since it allows its continuous version to be well defined from the aspect of measurability the convergence in probability is established for the continuous forward backward euler solution which is essential for proving the same result for both discrete forward backward and backward euler methods additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient of the equation as usual the whole consideration is affected by the presence and properties of the delay function

action functional for diffusions in discontinuous media the action functional i e the rate function governing the large deviations is obtained for a family of stochastic processes with discontinuous drift and small diffusion a well known method of continuous mapping is developed which proves to be efficient in a so called stable case

strong convergence of split step backward euler method for stochastic age dependent capital system with markovian switching abstract in this paper we present and analyze the split step backward euler method for the stochastic capital system with markovian switching under the one sided local lipschitz condition on the drift and local lipschitz condition on the diffusion we prove the split step backward euler method converges with strong order of one half to the true solution a numerical example is provided to illustrate the theoretical results

a backlund transformation for l isothermic surfaces it is established that a backlund transformation for l isothermic surfaces is associated with a darboux transformation for a non homogeneous linear schrodinger equation a lax pair for l isothermic surfaces is presented and it is shown that a quartet of eigenfunctions contained therein may be explicitly represented in terms of linearly independent solutions of a linear schrodinger equation with a potential involving the backlund parameter a permutability theorem is presented whereby l isothermic surfaces may be constructed and the action of the backlund transformation on a class of generalized dupin cyclides is considered

convergence and stability of two classes of theta milstein schemes for stochastic differential equations abstract this paper examines convergence and stability of the two classes of theta milstein schemes for stochastic differential equations sdes with non global lipschitz continuous coefficients the split step theta milstein sstm scheme and the stochastic theta milstein stm scheme for 1 2 1 this paper concludes that the two classes of theta milstein schemes converge strongly to the exact solution with the order 1 for 0 1 2 under the additional linear growth condition for the drift coefficient these two classes of the theta milstein schemes are also strongly convergent with the standard order this paper also investigates exponential mean square stability of these two classes of the theta milstein schemes for 1 2 1 these two theta milstein schemes can share the exponential mean square stability of the exact solution for 0 1 2 similar to the convergence under the additional linear growth condition these two theta milstein schemes can also reproduce the exponential mean square stability of the exact solution

strong convergence of split step backward euler method for stochastic differential equations with non smooth drift in this paper we are concerned with the numerical approximation of stochastic differential equations with discontinuous nondifferentiable drifts we show that under one sided lipschitz and general growth conditions on the drift and global lipschitz condition on the diffusion a variant of the implicit euler method known as the split step backward euler ssbe method converges with strong order of one half to the true solution our analysis relies on the framework developed in d j higham x mao and a m stuart strong convergence of euler type methods for nonlinear stochastic differential equations siam journal on numerical analysis 40 2002 1041 1063 and exploits the relationship which exists between explicit and implicit euler methods to establish the convergence rate results

stability of functional equations in the spaces of distributions and hyperfunctions making use of the fundamental solution of the heat equation we prove the stability theorems of quadratic functional equation and d alembert equation in the spaces of schwartz distributions and sato hyperfunctions

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

l isothermic and l minimal surfaces it has recently been shown that a second order linear nonhomogeneous differential equation is associated with a surface with an isothermic representation of their lines of curvature l isothermic surface schief et al 2007 j math phys 48 073510 the 6 parameter group acting on linearly independent solutions of the homogeneous version of the latter equation generates a laguerre transformation of the surface the weierstrass representation of the surfaces which are both l isothermic and l minimal is presented

stresses in lipid membranes the stresses in a closed lipid membrane described by the helfrich hamiltonian quadratic in th ee xtrinsic curvature are identified using noether s theorem three equations describe the conservation of the stress tensor the norma lp rojection is identified as the shape equation describing equilibrium configurations the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes the corresponding torque tensor is also identified the use of the stress tensor as a basis for perturbation theory is discussed the conservation laws are cast in terms of the forces and torques on closed curves as an application the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along the circles of constant latitude

the stability of cauchy equations in the space of schwartz distributions abstract we reformulate the stability theorem of d h hyers in the schwartz tempered distributions and prove that every additive tempered distribution can be approximated by a linear function we also consider the superstability of the cauchy equation f x y f x f y

on shell membranes of enneper type generalized dupin cyclides it is demonstrated that a class of generalized dupin cyclides arises naturally out of a classical system of equilibrium equations for shell membranes this class consists of all families of parallel canal surfaces on which the lines of curvature are planar various examples of viable membrane geometries such as particular l minimal surfaces are presented

the lame equation in shell membrane theory a classical nonlinear shell membrane system has recently been demonstrated to have underlying integrable structure here a wide class of corresponding parallel membranes is shown to be generated via a schrodinger equation of lame type this class is characterized by the existence of a multiplicity of stress distributions for a given membrane geometry a variety of viable membrane geometries such as generalized dupin cyclides are constructed explicitly together with the associated one parameter families of stress resultants a lax pair which encapsulates both the gauss mainardi codazzi and constrained equilibrium equations is also recorded