lack of strong completeness for stochastic flows it is well known that a stochastic differential equation sde on a euclidean space driven by a brownian motion with lipschitz coefficients generates a stochastic flow of homeomorphisms when the coefficients are only locally lipschitz then a maximal continuous flow still exists but explosion in finite time may occur if in addition the coefficients grow at most linearly then this flow has the property that for each fixed initial condition x the solution exists for all times almost surely if the exceptional set of measure zero can be chosen independently of x then the maximal flow is called strongly complete the question whether an sde with locally lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years in this paper we construct a two dimensional sde with coefficients which are even bounded and smooth and which is not strongly complete thus answering the question in the negative

strong convergence of a tamed theta scheme for nsddes with one sided lipschitz drift abstract this paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one sided lipschitz drift strong convergence rate is revealed under a global one sided lipschitz condition while for a local one sided lipschitz condition the tamed theta scheme is modified to ensure the well posedness of implicit numerical schemes then we show the convergence of the numerical solutions

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

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

an euler type method for the strong approximation of the cox ingersoll ross process we analyse the strong approximation of the cox ingersoll ross cir process in the regime where the process does not hit zero by a positivity preserving drift implicit euler type method as an error criterion we use the p th mean of the maximum distance between the cir process and its approximation on a finite time interval we show that under mild assumptions on the parameters of the cir process the proposed method attains up to a logarithmic term the convergence of order 1 2 this agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with lipschitz coefficients despite the fact that the cir process has a non lipschitz diffusion coefficient

optimal boundary control of hyperbolic equations with pointwise state constraints in this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints in contrast to parabolic dynamics such systems have not been sufficiently studied in the literature the reason is the lack of regularity in the case of hyperbolic dynamics we present necessary optimality conditions for both neumann and dirichlet boundary control problems and discuss differences and relationships between them

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

global flows for stochastic differential equations without global lipschitz conditions we consider stochastic differential equations driven by wiener processes the vector fields are supposed to satisfy only local lipschitz conditions the lipschitz constants of the drift vector field valid on balls of radius r are supposed to grow not faster than log r while those of the diffusion vector fields are supposed to grow not faster than sqrt log r we regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the wiener process on small intervals by showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation we simultaneously establish the existence of a global flow for the stochastic equation under local lipschitz conditions

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when

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

boundary optimal control of the westervelt and the kuznetsov equations this paper is concerned with optimal neumann boundary control for the westervelt and the kuznetsov equations which are equations of nonlinear acoustics specifically functionals of tracking type with applications in noninvasive ultrasonic medical treatments are considered existence of optimal controls is established and first order necessary optimality conditions are derived stability of the minimizer with respect to perturbations in the data as well as convergence of the controls when the regularization parameter tends to zero is shown

numerical integration of stochastic differential equations with nonglobally lipschitz coefficients we propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations sdes with nonglobally lipschitz coefficients following this concept we discard the approximate trajectories which leave a sufficiently large sphere we prove that accuracy of any method of weak order p is estimated by varepsilon o h p where varepsilon can be made arbitrarily small with increasing radius of the sphere the results obtained are supported by numerical experiments

loss of regularity for kolmogorov equations the celebrated hormander condition is a sufficient and nearly necessary condition for a second order linear kolmogorov partial differential equation pde with smooth coefficients to be hypoelliptic as a consequence the solutions of kolmogorov pdes are smooth at all positive times if the coefficients of the pde are smooth and satisfy hormander s condition even if the initial function is only continuous but not differentiable first order linear kolmogorov pdes with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth in this article we consider the intermediate regime of nonhypoelliptic second order kolmogorov pdes with smooth coefficients the main observation of this article is that there exist counterexamples to regularity preservation in that case more precisely we give an example of a second order linear kolmogorov pde with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the pde is not even locally holder continuous from the perspective of probability theory the existence of this example pde has the consequence that there exists a stochastic differential equation sde with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally holder continuous in other words degenerate noise can have a roughening effect a further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the sde more precisely we prove for an example sde with globally bounded and smooth coefficients that the standard euler approximations converge to the exact solution of the sde in the strong and numerically weak sense but at a rate that is slower then any power law

a monotone method for constructing extremal solutions to fourth order periodic boundary value problems in this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problemu 4 t f t u t u t email protected 0 email protected u 0 u email protected u 0 u email protected u 0 u email protected u 0 u email protected in the presence of an upper solution b and a lower solution a with b a

existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4 abstract this article shows the existence of positive periodic solutions for retarded advanced neutral and ordinary differential equations whose order is a multiple of 4 first we find a positive green s function explicitly then assuming that the coefficient of the unknown in the linear part of the equation is bounded above and below by positive constants we find one and then three solutions by applying the krasnoselskii and legget william fixed point theorems

the global control of a nonlinear wave equation we consider in this paper the existence and numerical estimation of the distributed control of a nonlinear wave equation with an integral performance criterion and fixed final states the problem is modified into one consisting of the minimization of a linear form over a set of pairs of positive measures satisfying linear constrains the minimization in the new problem is global and can be approximated by a finite dimensional linear programming problem the solution of this problem is used to construct a nearly optimal control an example is given to illustrate the procedure

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

semigroup splitting and cubature approximations for the stochastic navier stokes equations approximation of the marginal distribution of the solution of the stochastic navier stokes equations on the two dimensional torus by high order numerical methods is considered the corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on wiener space applied to a spectral galerkin discretization of degree n while the estimates exhibit a strong n dependence convergence is obtained for appropriately chosen time step sizes results of numerical simulations are provided and confirm the applicability of the methods

approximation for non smooth functionals of stochastic differential equations with irregular drift abstract this paper aims at developing a systematic study for the weak rate of convergence of the euler maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients we apply our method to obtain the rates of approximation for the expectation of various non smooth functionals of both stochastic differential equations and killed diffusion we also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is holder continuous

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

convergence of numerical schemes for the solution of parabolic stochastic partial differential equations we consider the numerical solution of the stochastic partial differential equation u t 2 u x 2 u w x t where w is space time white noise using finite differences for this equation gyongy has obtained an estimate of the rate of convergence for a simple scheme based on integrals of w over a rectangular grid we investigate the extent to which this order of convergence can be improved and find that better approximations are possible for the case of additive noise u 1 if we wish to estimate space averages of the solution rather than pointwise estimates or if we are permitted to generate other functionals of the noise but for multiplicative noise u u we show that no such improvements are possible

on the use of state constraints in optimal control of singular pdes abstract we consider optimal control of nonlinear partial differential equations involving potentially singular solution dependent terms singularity can be prevented by either restricting controls to a closed admissible set for which well posedness of the equation can be guaranteed or by explicitly enforcing pointwise bounds on the state by means of an elliptic model problem we contrast the requirements for deriving the existence of solutions and first order optimality conditions for both the control constrained and the state constrained formulation our analysis as well as numerical tests illustrate that control constraints lead to severe restrictions on the attainable states which is not the case for state constraints

periodic boundary value problems for nonlinear higher order ordinary differential equations we study a class of boundary value problems for linear and nonlinear ordinary differential equations of higher order we study the solution set and present a maximum principle as a consequence we show the validity of the monotone method and obtain the extremal solutions between the upper and lower solutions

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

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

optimal control of a singular pde modeling transient mems with control or state constraints abstract a particular feature of certain microelectromechanical systems mems is the appearance of a so called pull in instability corresponding to a singularity in the underlying pde model we here consider a transient mems model and its optimal control via the dielectric properties of the membrane and or the applied voltage in contrast to the static case the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions for this purpose we here use a relaxation approach combined with a localization technique

a note on strong approximation of sdes with smooth coefficients that have at most linearly growing derivatives abstract in the literature on strong approximation of sdes polynomial error rate results for numerical schemes are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition in the present article we show that the latter condition is far from being sufficient for polynomial error rates we construct an sde with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can at best be approximated with a logarithmic rate whatever numerical method based on observations of the driving brownian motion at finitely many fixed times is used most interestingly it furthermore turns out that using a method that properly adjusts the number of evaluations of the driving brownian motion to its actual path the latter sde can in fact be approximated with polynomial rate 1 in terms of the average number of evaluations that are used to the best of our knowledge this is only the second example in the literature of an sde for which there exist adaptive methods that perform superior to non adaptive ones with respect to the convergence rate

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

stochastic initial boundary value problems subject to distributed and boundary noise and their optimal control abstract in this paper we consider a class of stochastic evolution equations arising from initial boundary value problems with both boundary and distributed noise we prove existence and regularity of mild solutions then we consider a controlled version of the model and prove the existence of optimal controls and develop the necessary conditions of optimality for partially observed problems using relaxed controls

optimal approximation of stochastic differential equations by adaptive step size control we study the pathwise strong approximation of scalar stochastic differential equations with respect to the global error in the l2 norm for equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving brownian motion as a consequence higher order methods do not exist if the global error is analyzed we introduce an adaptive step size control for the euler scheme which performs asymptotically optimally in particular the new method is more efficient than an equidistant discretization this superiority is confirmed in simulation experiments for equations with additive noise as well as for general scalar equations

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

on stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions in a recent article jentzen et al 2016 commun math sci 14 1477 1500 doi 10 4310 cms 2016 v14 n6 a1 it has been established that for every arbitrarily slow convergence speed and every nat

strong convergence of the partially truncated euler maruyama scheme for a stochastic age structured sir epidemic model abstract we study in this paper strong convergence of the partially truncated euler maruyama scheme for an age structured susceptible infected removed sir epidemic model with environmental noise using the semigroup theory the existence and uniqueness of global positive solution for the model is first proved we then define a truncated function and develop a partially truncated em numerical solutions to the stochastic age structured sir epidemic model we present the pth moment boundedness of the partially truncated em numerical approximate solutions under appropriate conditions furthermore the strong lq convergence is established for the condition of 2 q