strong convergence rates for cox ingersoll ross processes full parameter range abstract we study strong pathwise approximation of cox ingersoll ross processes we propose a milstein type scheme that is suitably truncated close to zero where the diffusion coefficient fails to be locally lipschitz continuous for this scheme we prove positive convergence rates for the full parameter range including the accessible boundary regime the error criterion is given by the maximal l p distance of the solution and its approximation on a compact interval in the particular case of a squared bessel process of dimension 0 the convergence rate is given by min 1 2 p

extinction time for stochastic population models the analysis of interacting population models is the subject of much interest in mathematical ecology moreover the persistence and extinction of these models is one of the most interesting and important topics because it provides insight into their behavior the mean extinction time for stochastic population models considered in this paper depends on the initial population size and satisfies a stationary partial differential equation related to the backward kolmogorov differential equation a linear second order partial differential equation with variable coefficients in this communication we review several papers where we have proposed some numerical techniques in order to estimate the mean extinction time for stochastic population models besides we will compare the theoretical predictions and numerical simulations for stochastic differential equations sdes this work can be viewed as a unified review of the contributions de la hoz and vadillo 2012 de la hoz et al 2014 and doubova and vadillo 2014

cross correlations in scaling analyses of phase transitions thermal or finite size scaling analyses of importance sampling monte carlo computer simulations in the vicinity of phase transition points often combine different estimates for the same quantity such as a critical exponent with the intent to reduce statistical fluctuations we point out that the origin of such estimates in the same time series often results in pronounced cross correlations which are usually ignored even in high precision studies generically leading to significant underestimation of statistical fluctuations we suggest to use a simple extension of the conventional analysis taking correlation effects into account which leads to improved estimators with often substantially reduced statistical fluctuations at almost no extra cost in terms of computation time

monte carlo simulation with fluctuating boundary conditions abstract i study the finite size behavior of the ratio of partition functions with periodic and antiperiodic boundary conditions in the vicinity of the critical coupling of the 3d ising model this ratio is obtained as a simple expectation value in an ensemble where i average over both kinds of boundary conditions based on such simulations i obtain c 0 221650 5 and 0 630 2 these results are compared with fourth order cumulant results obtained in the same monte carlo simulations

a computational study of residual kpp front speeds in time periodic cellular flows in the small diffusion limit abstract the minimal speeds c of the kolmogorov petrovsky piskunov kpp fronts at small diffusion 1 in a class of time periodic cellular flows with chaotic streamlines is investigated in this paper the variational principle of c reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advection diffusion operator with space time periodic coefficients and small diffusion to solve the advection dominated time dependent eigenvalue problem efficiently over large time a combination of spectral methods and finite element as well as the associated fast solvers are utilized to accelerate computation in contrast to the scaling c o 1 4 in steady cellular flows a new relation c o 1 as 1 is revealed in the time periodic cellular flows due to the presence of chaotic streamlines residual propagation speed emerges from the lagrangian chaos which is quantified as a sub diffusion process

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

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

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

high temperature series for scalar field lattice models generation and analysis an implementation of the free embedding scheme for high temperature series generation on the body centered cubic family of lattices in arbitrary dimensiond is described series to order 21 in inverse temperature are tabulated for several scalar field models both for the magnetic susceptibility and for the second moment of the spin correlation function the critical behavior of a family of 3 dimensional double gaussian models which interpolate continuously between the spin 1 2 ising model and the gaussian model is analyzed in detail away from the gaussian model limit using confluent inhomogeneous secondorder differential approximants with our best estimate of the correction to scaling exponent 0 52 0 03 the leading exponents for the susceptibility and correlation length for this family are consistent with universality and are given by 1 237 0 002 and 0 630 0 0015 respectively and 2 0 0359 0 0007

a monte carlo study of leading order scaling corrections of phi 4 theory on a three dimensional lattice we present a monte carlo study of the one component 4 model on a cubic lattice in three dimensions leading order scaling corrections are studied using the finite size scaling method we compute the corrections to scaling exponent with high precision we determine the value of the coupling at which leading order corrections to scaling vanish using this result we obtain estimates for critical exponents that are more precise than those obtained with field theoretic methods

improved high temperature expansion and critical equation of state of three dimensional ising like systems high temperature series are computed for a generalized three dimensional ising model with arbitrary potential three specific improved potentials suppressing leading scaling corrections are selected by monte carlo computation critical exponents are extracted from high temperature series specialized to improved potentials achieving high accuracy our best estimates are gamma 1 2371 4 nu 0 630 02 23 alpha 0 1099 7 eta 0 0364 4 beta 0 326 48 18 by the same technique the coefficients of the small field expansion for the effective potential helmholtz free energy are computed these results are applied to the construction of parametric representations of the critical equation of state a systematic approximation scheme based on a global stationarity condition is introduced the lowest order approximation reproduces the linear parametric model this scheme is used for an accurate determination of universal ratios of amplitudes a comparison with other theoretical and experimental determinations of universal quantities is presented

convergence and non negativity preserving of the solution of balanced method for the delay cir model with jump abstract in this work we propose the balanced implicit method bim to approximate the solution of the delay cox ingersoll ross cir model with jump which often gives rise to model an asset price and stochastic volatility dependent on past data we show that this method preserves non negativity property of the solution of this model with appropriate control functions we prove the strong convergence and investigate the p th moment boundedness of the solution of bim finally we illustrate those results in the last section

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

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

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

high order discretization schemes for the cir process application to affine term structure and heston models this paper presents weak second and third order schemes for the cox ingersoll ross cir process without any restriction on its parameters at the same time it gives a general recursive construction method to get weak second order schemes that extends the one introduced by ninomiya and victoir cite nv combining these both results this allows to propose a second order scheme for more general affine diffusions simulation examples are given to illustrate the convergence of these schemes on cir and heston models algorithms are stated in a pseudocode language

improved model for mixtures of polymers and hard spheres extensive monte carlo simulations are used to investigate how model systems of mixtures of polymers and hard spheres approach the scaling limit we represent polymers as lattice random walks of length l with an energy penalty w for each intersection domb joyce model interacting with hard spheres of radius r c via a hard core pair potential of range where r mon is identified as the monomer radius we show that the mixed polymer colloid interaction gives rise to new confluent corrections the leading ones scale as where is the usual flory exponent finally we determine optimal values of the model parameters w and r mon that guarantee the absence of the two leading confluent corrections this improved model shows a significantly faster convergence to the asymptotic limit and is amenable for extensive and accurate numerical simulations at finite density with only a limited computational effort

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

generalized ait sahalia type interest rate model with poisson jumps and convergence of the numerical approximation abstract in this paper we consider a generalized ait sahalia interest rate model with poisson jumps in finance the analytical properties including positivity boundedness and pathwise asymptotic estimations of the solution to this model are investigated moreover we prove that the euler maruyama em numerical solution converges to the true solution of the model in probability finally we apply the em solution to compute some financial quantities a numerical example is provided to demonstrate the effectiveness of our results

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

first order strong approximations of scalar sdes defined in a domain we are interested in strong approximations of one dimensional sdes which have non lipschitz coefficients and which take values in a domain under a set of general assumptions we derive an implicit scheme that preserves the domain of the sdes and is strongly convergent with rate one moreover we show that this general result can be applied to many sdes we encounter in mathematical finance and bio mathematics we will demonstrate flexibility of our approach by analyzing classical examples of sdes with sublinear coefficients cir cev models and wright fisher diffusion and also with superlinear coefficients 3 2 volatility ait sahalia model our goal is to justify an efficient multilevel monte carlo method for a rich family of sdes which relies on good strong convergence properties

the truncated milstein method for stochastic differential equations with commutative noise inspired by the truncated eulermaruyama method developed in mao 2015 we propose the truncated milstein method in this paper the strong convergence rate is proved to be close to 1 for a class of highly non linear stochastic differential equations with commutative noise numerical examples are given to illustrate the theoretical results

ising universality in three dimensions a monte carlo study we investigate three ising models on the simple cubic lattice by means of monte carlo methods and finite size scaling these models are the spin 1 2 ising model with nearest neighbour interactions a spin 1 2 model with nearest neighbour and third neighbour interactions and a spin 1 model with nearest neighbour interactions the results are in accurate agreement with the hypothesis of universality analysis of the finite size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field we find that the correction to scaling amplitudes are strongly dependent on the introduction of further neighbour interactions or a third spin state in a spin 1 ising model these corrections appear to be very small this is very helpful for the determination of the universal constants of the ising model the renormalization exponents of the ising model are determined as yt 1 587 2 yh 2 4815 15 and yi 0 82 6 the universal ratio q m2 2 m4 is equal to 0 6233 4 for periodic systems with cubic symmetry the critical point of the nearest neighbour spin 1 2 model is kc 0 2216546 10

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

two point correlation function of three dimensional o n models the critical limit and anisotropy in three dimensional o n models we investigate the low momentum behavior of the two point green s function g x in the critical region of the symmetric phase we consider physical systems whose criticality is characterized by a rotationally invariant fixed point several approaches are exploited such as strong coupling expansion of lattice n vector model and 1 n expansion field theoretical methods within the ensuremath varphi 4 continuum formulation non gaussian corrections to the universal low momentum behavior of g x are evaluated and found to be very small in nonrotationally invariant physical systems with o n invariant interactions the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent ensuremath rho which is universal and is related to the leading irrelevant operator breaking rotational invariance at n ensuremath infty one finds ensuremath rho 2 we show that for all values of ng 0 ensuremath rho ensuremath simeq 2