a positivity preserving numerical method for stochastic r d model abstract the stochastic research and development r d model plays an important role in the growth rates of technological progress and capital accumulation however we can not obtain the explicit solution of stochastic r d model to explore a numerical approximate method preserving positivity for the r d model with uncertainty from the population growth we first construct an explicit euler maruyama em method and then verify it converges to the true solution with strong order 1 2 1 1 p but the em method may lead to a negative approximation which is unrealistic so we establish the balanced explicit method bem to overcome the defect of em method and give sufficient conditions to preserve the positivity of bem then the convergence order of bem is obtained finally numerical simulations are carried out to verify our theoretical work

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

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

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

the diffusion kernel filter a particle filter method is presented for the discrete time filtering problem with nonlinear ito stochastic ordinary differential equations sode with additive noise supposed to be analytically integrable as a function of the underlying vector wiener process and time the diffusion kernel filter is arrived at by a parametrization of small noise driven state fluctuations within branches of prediction and a local use of this parametrization in the bootstrap filter the method applies for small noise and short prediction steps with explicit numerical integrators the operations count in the diffusion kernel filter is shown to be smaller than in the bootstrap filter whenever the initial state for the prediction step has sufficiently few moments the established parametrization is a dual formula for the analysis of sensitivity to gaussian initial perturbations and the analysis of sensitivity to noise perturbations in deterministic models showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an sode should be modeled i e defined for a gaussian initial deterministic problem to be cast into an sode problem from it a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches tests are made with the lorenz 63 equations showing good results both for the filter and the definition of prediction

quasi isometry invariance of group splittings we show that a finitely presented one ended group which is not commensurable to a surface group splits over a two ended group if and only if its cayley graph is separated by a quasi line this shows in particular that splittings over two ended groups are preserved by quasi isometries

the projected explicit ito taylor methods for stochastic differential equations under locally lipschitz conditions and polynomial growth conditions abstract although the numerical methods to stochastic differential equations with the coefficients of locally lipschitz and polynomial growth have been discussed commonly by some authors there are few works on the high strong order numerical methods in this paper the mean square convergence of the general projected explicit ito taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition suitable local lipschitz conditions and polynomial growth conditions our analysis follows the idea of stochastic c stability and stochastic b consistency after giving the selection strategy of optimal parameters we prove that the projected explicit ito taylor methods owning optimal parameters share the same mean square convergence orders with non projected ones finally two numerical experiments are presented to show the effectiveness of theoretical results

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

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

sampling the posterior an approach to non gaussian data assimilation the viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a bayesian framework in the absence of model error the correct solution to the data assimilation problem is to find the posterior distribution implied by this bayesian setting methods for dealing with data assimilation should then be judged by their ability to probe this distribution in this paper we propose a range of techniques for probing the posterior distribution based around the langevin equation and we compare these new techniques with existing methods when the underlying dynamics is deterministic the posterior distribution is on the space of initial conditions leading to a sampling problem over this space when the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths by writing down a density and conditioning on observations it is possible to define a range of markov chain monte carlo mcmc methods which sample from the desired posterior distribution and thereby solve the data assimilation problem the basic building blocks for the mcmc methods that we concentrate on in this paper are langevin equations which are ergodic and whose invariant measures give the desired distribution in the case of path space sampling these are stochastic partial differential equations spdes two examples are given to show how data assimilation can be formulated in a bayesian fashion the first is weather prediction and the second is lagrangian data assimilation for oceanic velocity fields furthermore the relationship between the bayesian approach outlined here and the commonly used kalman filter based techniques prevalent in practice is discussed two simple pedagogical examples are studied to illustrate the application of bayesian sampling to data assimilation concretely finally a range of open mathematical and computational issues arising from the bayesian approach are outlined

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

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

existence and uniqueness of the solutions and convergence of semi implicit euler methods for stochastic pantograph equations abstract in this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given i e the local lipschitz condition and the linear growth condition under the lipschitz condition and the linear growth condition it is proved that the semi implicit euler method is convergence with strong order 1 2

numerical solutions of sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper the numerical solutions to stochastic differential equations with markovian switching and jumps under non lipschitz conditions are considered firstly we study the existence and uniqueness of solutions to this class of equations then we present the moment bounds of the exact and numerical solutions it is also proved that the strong rate of convergence of the euler scheme is equal to 1 2 with the drift coefficient satisfying one sided lipschitz condition while the diffusion and jump coefficients satisfy global lipschitz conditions some numerical examples are given to confirm the theoretical results

a class of contractive second derivatives multistep methods in this paper we study some properties of contractivity regions of second derivative multistep methods these properties lead to the design of classes of a o and a contractive methods

exponential stability of the split step θ method for neutral stochastic delay differential equations with jumps abstract the exponential mean square stability of the split step method for neutral stochastic delay differential equations nsddes with jumps is considered new conditions for jumps are proposed to ensure the exponential mean square stability of the trivial solution if the drift coefficient satisfies the linear growth condition it is shown that the split step method can reproduce the exponential mean square stability of the trivial solution for the constrained stepsize then by applying the chebyshev inequality and the borel cantelli lemma the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained since split step method covers euler maruyama em method and split step backward euler ssbe method the conclusions are valid for these two methods moreover they can adapt to the nsddes and the sddes with jumps finally a numerical example illustrates the effectiveness of the theoretical 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

the a contractive second order multistep formulas with variable steps the k 1 parameter family of all second order k step formulas which are a contractive in the max norm is derived for arbitrary variable integration steps a contractive methods are suitable for generating stable numerical solutions of stiff problems which lack smoothness for example a contractive one leg methods are guaranteed to be stable with respect to dot x lambda t x for any lambda t satisfying operatorname re lambda t leqslant 0 and arbitrary step sequences h n there also is evidence that a contractive one leg methods give very accurate amplitude responses when applied with variable steps to certain stiff problems with oscillatory nonstiff modes and little or no dissipation

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

semi implicit split step numerical methods for a class of nonlinear stochastic differential equations with non lipschitz drift terms abstract in this paper we discuss numerical solutions of a class of nonlinear stochastic differential equations using semi implicit split step methods under some monotonicity conditions on the drift term we study moment estimates and strong convergence properties of the numerical solutions with a focus on stochastic ginzburg landau equations moreover we compare the performance of various numerical methods including the tamed euler truncated euler implicit euler and split step procedures in particular we discuss the empirical rate of convergence and the computational cost of these methods for certain parameter values of the models used

matrix valued versions of a result of von neumann with an application to time discretization abstract we study the size of the operator c a where c z is an n m matrix with rational elements such that c z 1 in some norm for all complex z in the unit disc and a is a linear operator in a hilbert space with norm at most 1 an application to variable step integration of initial value problems using one leg methods is given

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