pricing bermudan options under local levy models with default we consider a defaultable asset whose risk neutral pricing dynamics are described by an exponential levy type martingale this class of models allows for a local volatility local default intensity and a locally dependent levy measure we present a pricing method for bermudan options based on an analytical approximation of the characteristic function combined with the cos method due to a special form of the obtained characteristic function the price can be computed using a fast fourier transform based algorithm resulting in a fast and accurate calculation the greeks can be computed at almost no additional computational cost error bounds for the approximation of the characteristic function as well as for the total option price are given

a comparison of some methods for the evaluation of highly oscillatory integrals abstract in part i the extended clenshaw curtis method for finite fourier integrals is discussed and a number of timed comparisons are made between the various implementations which appear in the literature part ii deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation their effectiveness is discussed in the light of a set of test examples

turan type inequalities for regular coulomb wave functions turan mitrinovic adamovic and wilker type inequalities are deduced for regular coulomb wave functions the proofs are based on a mittag leffler expansion for the regular coulomb wave function which may be of independent interest moreover some complete monotonicity results concerning the coulomb zeta functions and some interlacing properties of the zeros of coulomb wave functions are given

characterization of the critical submanifolds in quantum ensemble control landscapes the quantum control landscape is defined as the functional that maps the control variables to the expectation values of an observable over the ensemble of quantum systems analyzing the topology of such landscapes is important for understanding the origins of the increasing number of laboratory successes in the optimal control of quantum processes this paper proposes a simple scheme to compute the characteristics of the critical topology of the quantum ensemble control landscapes showing that the set of disjoint critical submanifolds one to one corresponds to a finite number of contingency tables that solely depend on the degeneracy structure of the eigenvalues of the initial system density matrix and the observable whose expectation value is to be maximized the landscape characteristics can be calculated as functions of the table entries including the dimensions and the numbers of positive and negative eigenvalues of the hessian quadratic form of each of the connected components of the critical submanifolds typical examples are given to illustrate the effectiveness of this method

a fast and accurate fft based method for pricing early exercise options under levy processes a fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented the method is based on a quadrature technique and relies heavily on fourier transformations the main idea is to reformulate the well known risk neutral valuation formula by recognizing that it is a convolution the resulting convolution is dealt with numerically by using the fast fourier transform this novel pricing method which we dub the convolution method is applicable to a wide variety of payoffs and requires only the knowledge of the characteristic function of the model as such the method is applicable within many regular affine models among which is the class of exponential levy models for an m times exercisable bermudan option the overall complexity is o mn log 2 n with n grid points used to discretize the price of the underlying asset american options are priced efficiently by applying richardson extrapolation to the prices of bermudan options

on contingent claim valuation in continuous time for volatility models of ornstein uhlenbeck type the paper addresses the valuation of contingent claims in stochastic volatility models of ornstein uhlenbeck type stressing the situation when volatility is driven by purely discontinuous levy processes a reduction series methodology is developed for this purpose which also provides a way for the numerical study of the value functionals the methodology is illustrated in the options case and in models based on gig distributions numerical examples are provided these examples show how the series enable computation accuracies of some three decimal places with just a single digit number of terms

pricing and hedging european style options in levy based stochastic volatility models considering the leverage effect abstract this paper develops a semi analytical exact solution for the discrete time variance optimal hedging strategy based on two dimensional fourier cosine series expansions this approach is novel and is more efficient than those previously suggested in the literature numerical studies for different models also suggest that the algorithm has better accuracy and stability for the pricing and hedging problems a levy based stochastic volatility model considering the leverage effect is also proposed we have employed data on apple stock prices and the corresponding option contracts to test this model to evaluate its ability to perform pricing and hedging in summary our model design is relatively reliable in the context of the known empirical patterns such as the existence of jumps the volatility smile and the negative correlation between asset returns and volatilities

forward starting options pricing with double stochastic volatility stochastic interest rates and double jumps abstract we present an extension of double heston stochastic volatility model by introducing cir stochastic interest rate and double exponential jumps in the stock price process we derive the characteristic function and forward characteristic function of the log asset price and thereby forward starting options are well evaluated by the cos method we also provide efficient simulation of the proposed model and monte carlo solutions to forward starting options pricing based on the qe scheme numerical results show that the cos method is fast and efficient for pricing forward starting options

linear dynamically varying lq control of nonlinear systems over compact sets linear quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed typically such systems exhibit complicated dynamics i e have nontrivial recurrence the controllers which use small perturbations of the nominal dynamics as input actuators are based on modeling the tracking error as a linear dynamically varying ldv system necessary and sufficient conditions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic riccati equation fare it is shown that despite the lack of continuity of the asymptotic trajectory relative to initial conditions the cost to stabilize about the trajectory as given by the solution to the fare is continuous an ergodic theory method for solving the fare is presented furthermore it is shown that wrapping the ldv controller around the nonlinear system secures a stable tracking dynamics finally an example of controlling the henon map to follow an aperiodic orbit is presented

transfer function approach to quantum control part i dynamics of quantum feedback systems this paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators a quantum optical closed loop including a plant and controller is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view in order to facilitate the use of control theory in microscopic world described by quantum theory in particular an input output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed this would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics this paper is divided into two parts the first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device cavity that can be regarded as a unit of quantum dynamics of bosonic modes the formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations the input output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems in the follow up paper the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory uncertainty principle e g and some applications are presented

the lie algebra structure and controllability of spin systems in this paper we study the controllability properties and the lie algebra structure of networks of particles with spin immersed in an electro magnetic field we relate the lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active for networks with different gyromagnetic ratios we provide a necessary and sufficient condition of controllability in terms of the properties of the above mentioned graph and describe the lie algebra structure in every case for these systems all the controllability notions including the possibility of driving the evolution operator and or the state are equivalent for general networks with possibly equal gyromagnetic ratios we give a sufficient condition of controllability a general form of interaction among the particles is assumed which includes both ising and heisenberg models as special cases assuming heisenberg interaction we provide an analysis of low dimensional cases number of particles less than or equal to three which includes necessary and sufficient controllability conditions as well as a study of their lie algebra structure this also provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

the hurwitz type theorem for the regular coulomb wave function via hankel determinants abstract we derive a closed formula for the determinant of the hankel matrix whose entries are given by sums of negative powers of the zeros of the regular coulomb wave function this new identity applied together with results of grommer and chebotarev allows us to prove a hurwitz type theorem about the zeros of the regular coulomb wave function as a particular case we obtain a new proof of the classical hurwitz s theorem from the theory of bessel functions that is based on algebraic arguments in addition several hankel determinants with entries given by the rayleigh function and bernoulli numbers are also evaluated

pricing early exercise and discrete barrier options by fourier cosine series expansions we present a pricing method based on fourier cosine expansions for early exercise and discretely monitored barrier options the method works well for exponential levy asset price models the error convergence is exponential for processes characterized by very smooth rm c infty a b in mathbb r transitional probability density functions the computational complexity is o m 1 n log n with n a small number of terms from the series expansion and m the number of early exercise monitoring dates this paper is the follow up of fang and oosterlee in siam j sci comput 31 2 826 848 2008 in which we presented the impressive performance of the fourier cosine series method for european options

symmetry and subspace controllability for spin networks with a single node control we consider the relation of symmetries and subspace controllability for spin 1 2 networks with xxz couplings subject to perturbation of a single node by a local potential z control the hamiltonians for such networks decompose into excitation subspaces focusing on the single excitation subspace it is shown for single node z controls that external symmetries are characterized by eigenstates of the system hamiltonian which have zero overlap with the control node and there are no internal symmetries it is further shown that there are symmetries which persist even in the presence of random perturbations for xxz chains with uniform coupling strengths a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control we then show for heisenberg and xx chains with uniform coupling strength subject to single node z control that the lack of symmetry is both necessary and sufficient for subspace controllability finally the latter approach is generalized to establish controllability results for simple branched networks

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

pricing early exercise and discrete barrier options by shannon wavelet expansions we present a pricing method based on shannon wavelet expansions for early exercise and discretely monitored barrier options under exponential levy asset dynamics shannon wavelets are smooth and thus approximate the densities that occur in finance well resulting in exponential convergence application of the fast fourier transform yields an efficient implementation and since wavelets give local approximations the domain boundary errors can be naturally resolved which is the main improvement over existing methods

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

error analysis for the computation of zeros of regular coulomb wave function and its first derivative in 1975 one of the coauthors ikebe showed that the problem of computing the zeros of the regular coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices approximation by truncation is justified but no error estimates are given there the class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by ikebe et al in 1993 where an extremely accurate asymptotic error estimate is shown in this paper we apply this error formula to the former case to obtain error formulas in a closed explicit form

complete controllability of finite level quantum systems complete controllability is a fundamental issue in the field of control of quantum systems not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables in this paper we investigate the question of complete controllability for finite level quantum systems subject to a single control field for which the interaction is of dipole form sufficient criteria for complete controllability of a wide range of finite level quantum systems are established and the question of limits of complete controllability is addressed finally the results are applied to give a classification of complete controllability for four level systems

an iterative splitting method for pricing european options under the heston model abstract in this paper we propose an iterative splitting method to solve the partial differential equation pde in option pricing problems we focus on the heston stochastic volatility model and the derived two dimensional pde we take the european option as an example and conduct numerical experiments using different boundary conditions the iterative splitting method transforms the two dimensional equation into two quasi one dimensional equations with the variable on the other dimension fixed which helps to lower the computational cost numerical results show that the iterative splitting method together with an artificial boundary condition abc based on the method by li and huang 2019 gives the most accurate option price and greeks compared to the classic finite difference method with the commonly used boundary conditions in heston 1993

accurate evaluation of european and american options under the cgmy process a finite difference method for integro differential equations arising from le vy driven asset processes in finance is discussed the equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula the discretization is shown to be second order accurate for a relevant parameter range determining the degree of the singularity in the le vy measure the singularity is dealt with by means of an integration by parts technique an application of the fast fourier transform gives the overall amount of work o n t n log n rendering the method fast

optimal control of quantum systems a projection approach this paper considers the optimal control of quantum systems the controlled quantum systems are described by the probability density matrix based liouville von neumann equation using projection operators the states of the quantum system are decomposed into two sub spaces namely the main state space and the remaining state space since the control energy is limited a solution for optimizing the external control force is proposed in which the main state is brought to the desired main state at a certain target time while the population of the remaining state is simultaneously suppressed in order to diminish its effects on the final population of the main state the optimization problem is formulated by maximizing a general cost functional of states and control force an efficient algorithm is developed to solve the optimization problem finally using the hydrogen fluoride hf molecular population transfer problem as an illustrative example the effectiveness of the proposed scheme for a quantum system initially in a mixed state or in a pure state is investigated through numerical simulations

dual control monte carlo method for tight bounds of value function under heston stochastic volatility model abstract the aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the heston stochastic volatility model with general utility functions it is well known there is a closed form solution to the hjb equation for power utility due to its homothetic property it is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the hjb equation for the heston model in this paper we propose an efficient dual control monte carlo method for computing tight lower and upper bounds of the value function we identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions including power non hara and yaari utilities finally we perform some numerical tests to see the efficiency accuracy and robustness of the method the numerical results support strongly our proposed scheme

orthogonal polynomials associated with coulomb wave functions abstract a new class of orthogonal polynomials associated with coulomb wave functions is introduced these polynomials play a role analogous to that the lommel polynomials have in the theory of bessel functions the orthogonality measure for this new class is described in detail in addition the orthogonality measure problem is discussed on a more general level apart from this various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of bessel functions a key role in these derivations is played by a jacobi tridiagonal matrix j l whose eigenvalues coincide with the reciprocal values of the zeros of the regular coulomb wave function f l the spectral zeta function corresponding to the regular coulomb wave function or more precisely to the respective tridiagonal matrix is studied as well

efficient pricing of european options on two underlying assets by frame duality abstract the proj method for pricing european options on one underlying asset was proposed by j lars kirkby and then was applied to price bermudan and asian options in this paper we extend the method to higher dimensions especially two dimensions in which some exotic options can be priced our method does not rely on a priori truncation of the integration range and exhibits excellent performance compared with other state of the art methods particularly for fatter tailed short maturity models we also discuss the errors introduced in each approximation and give corresponding error bounds numerical results on implementation of this method to price for popular two assets options under both the geometric brownian motion and variance gamma dynamics demonstrate remarkable accuracy and robustness

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

the β meixner model we propose to approximate the meixner model by a member of the b family introduced by kuznetsov 2010 in 2 the advantage of the approximation is the semi explicit formulae for the running extrema under the b family processes which enables us to produce more efficient algorithms for pricing path dependent options through the wiener hopf factors we will explore the performance of the approximation both in an equity framework and in the credit risk setting where we use the approximation to calibrate a surface of credit default swaps the paper follows the approach of the study made by schoutens and damme 2010 in 1 where the aim was to approximate the variance gamma we will contextualize the results by schoutens and damme 2010 in 1 and the ones here with respect to the approach taken by jeannin and pistorius 2010 in 15 an asymptotic expression for the rate of convergence of the approximation is derived

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

peaks and jumps reconstruction with b splines scaling functions we consider a methodology based on b splines scaling functions to numerically invert fourier or laplace transforms of functions in the space l 2 r the original function is approximated by a finite combination of jth order b splines basis functions and we provide analytical expressions for the recovered coefficients the methodology is particularly well suited when the original function or its derivatives present peaks or jumps due to discontinuities in the domain we will show in the numerical experiments the robustness and accuracy of the method

model risk and discretisation of locally risk minimising strategies we consider two models for the price process a time continuous jump diffusion and a time discretisation of it then we study the robustness of the related locally risk minimising strategy to this model choice where we focus mainly on hedging asian and spread options using the discretisation scheme and the convergence results on backward stochastic differential equations as studied in khedher and vanmaele 2016 we show that the discrete time locally risk minimising strategies converge to the corresponding continuous time strategies in an l 2 sense we present different numerical examples to illustrate our results

the evaluation of american options in a stochastic volatility model with jumps an efficient finite element approach we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the asset price is described by a jump diffusion stochastic differential equation in which the jump term consists of a levy process of compound poisson type while the volatility is modeled as a cir type process correlated with the asset price pricing american options under the bates model requires us to solve a partial integro differential equation with the final condition and boundary conditions prescribed on a free boundary in this paper a numerical method for solving such a problem is proposed in particular first of all using a richardson extrapolation technique the problem is reduced to a problem with fixed boundary then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit explicit time stepping an operator splitting technique and a non uniform mesh of right angled triangles numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast in particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing american options under the bates model

recombined multinomial tree based on saddle point approximation and its application to levy models options pricing abstract this paper studies the constructing methods of a recombined multinomial tree based on saddle point approximation and its application to levy models options pricing firstly the levy process and the european option pricing are introduced then we used the characteristic function of levy process to generate density function value at discrete point in a certain range based on the saddle point approximation method then we provide the method to construct recombined multinomial tree and give the pricing formula of european option and path dependent option pricing based on the backward iteration finally we used the cgmy process to demonstrate its application to european option american option and american barrier option pricing it proves that saddle point approximation turns the inverse fourier integral transform problem to several function value calculation comparing to ifft the speed of calculation using this method is faster and it avoid the negative probability density function value based on ifft because of the linear growth of the number of nodes we can extend the saddle point approximation method to the calculation of path dependent option

switching control of closed quantum systems via the lyapunov method in this paper we consider the state transfer problem for closed quantum systems under a degenerate case where the linearized system around the target state is not controllable it is known that the traditional lyapunov control methods may fail to guarantee the convergence to the target state under the degenerate case hence we propose to use multiple lyapunov functions and design a switching control strategy to achieve more accurate state transfer it is shown that the system can converge to the intersection of invariant sets including the target state the explicit analysis of the convergence is provided to design the switching law moreover the effectiveness of open loop lyapunov control is discussed simulation studies are presented to show the improved control performance

rbf pu method for pricing options under the jump diffusion model with local volatility abstract meshfree methods based on radial basis functions rbfs are of general interest for solving partial differential equations pdes because they can provide high order or spectral convergence for smooth solutions in complex geometries for global rbf methods one of the major disadvantages is the computational cost associated with the dense linear systems that arise therefore this paper is currently directed toward localized rbf approximations known as the rbf partition of unity rbf pu method for partial integro differential equation pide arisen in option pricing problems in jump diffusion model rbf pu method produces algebraic systems with sparse matrices which have small condition number also for comparison some stable time discretization schemes are combined with the operator splitting method to get a fully discrete problem numerical examples are presented to illustrate the convergence and stability of the proposed algorithms for pricing european and american options with merton and kou models

feedback stabilization of isospectral control systems on complex flag manifolds application to quantum ensembles the convex set of density operators of an n level quantum mechanical system foliated as a complex flag manifold where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix for an isospectral bilinear control system evolving on such an orbit the state feedback stabilization problem admits a natural lyapunov based time varying feedback design a global description of the domain of attraction of the closed loop system can be provided based on a ldquoroot spacerdquo like structure of the cone of density operators the converging conditions are time independent but depend on the topology of the flag manifold it is shown that the closed loop must have a number of equilibria at least equal to the euler characteristic of the manifold thus imposing topological obstructions to global stabilizability

fourier cosine method for ruin probabilities in theory ruin probabilities in classical insurance risk models can be expressed in terms of an infinite sum of convolutions but its inherent complexity makes efficient computation almost impossible in contrast fourier transforms of convolutions could be evaluated in a far simpler manner this feature aligns with the heuristic of the recently popular work by fang and oosterlee in particular they developed a numerical method based on fourier transform for option pricing we here promote their philosophy to ruin theory in this paper we not only introduce the fourier cosine method to ruin theory which has o n computational complexity but we also enhance the error bound for our case that are not immediate from fang and oosterlee 2009 we also suggest a robust method on estimation of ruin probabilities with respect to perturbation of the moments of both claim size and claim arrival distributions rearrangement inequality will also be adopted to amplify the fourier cosine method resulting in an effective global estimation

a highly efficient shannon wavelet inverse fourier technique for pricing european options in the search for robust accurate and highly efficient financial option valuation techniques we here present the swift method shannon wavelets inverse fourier technique based on shannon wavelets swift comes with control over approximation errors made by means of sharp quantitative error bounds the nature of the local shannon wavelets basis enables us to adaptively determine the proper size of the computational interval numerical experiments on european style options show exponential convergence and confirm the bounds robustness and efficiency

on the data driven cos method in this paper we present the data driven cos method ddcos it is a fourier based financial option valuation method which assumes the availability of asset data samples a characteristic function of the underlying asset probability density function is not required as such the presented technique represents a generalization of the well known cos method 1 the convergence of the proposed method is o 1 n in line with monte carlo methods for pricing financial derivatives the ddcos method is then particularly interesting for density recovery and also for the efficient computation of the options sensitivities delta and gamma these are often used in risk management and can be obtained at a higher accuracy with ddcos than with plain monte carlo methods

the gradient flow for control of closed quantum systems the gradient flow is commonly used to numerically solve the problem of maximizing quantum observables for finite dimensional closed quantum systems in this note we analyze the asymptotic behavior of the gradient flow we show that under a regularity assumption on the controls the flow almost always converges to a solution of the maximization problem

two dimensional fourier cosine series expansion method for pricing financial options the cos method for pricing european and bermudan options with one underlying asset was developed in f fang and c w oosterlee siam j sci comput 31 2008 pp 826 848 and f fang and c w oosterlee numer math 114 2009 pp 27 62 in this paper we extend the method to higher dimensions with a multidimensional asset price process the algorithm can be applied to for example pricing two color rainbow options but also to pricing under the popular heston stochastic volatility model for smooth density functions the resulting method converges exponentially in the number of terms in the fourier cosine series summations otherwise we achieve algebraic convergence the use of an fft algorithm for asset prices modeled by levy processes makes the algorithm highly efficient we perform extensive numerical experiments

implicit lyapunov control of finite dimensional schrodinger equations an implicit lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system the main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable the controlled lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem the convergence analysis is done using lasalle invariance principle closed loop simulations illustrate the performance of such feedback laws for the open loop control of a test case considered by chemists

robust pricing of european options with wavelets and the characteristic function we present a novel method for pricing european options based on the wavelet approximation method and the characteristic function we focus on the discounted expected payoff pricing formula and compute it by means of wavelets we approximate the density function associated to the underlying asset price process by a finite combination of jth order b splines and recover the coefficients of the approximation from the characteristic function two variants for wavelet approximation will be presented where the second variant adaptively determines the range of integration the compact support of a b splines basis enables us to price options in a robust way even in cases where fourier based pricing methods may show weaknesses the method appears to be particularly robust for pricing long maturity options fat tailed distributions as well as staircase like density functions encountered in portfolio loss computations

identification of dynamical lie algebras for finite level quantum control systems the problem of identifying the dynamical lie algebras of finite level quantum systems subject to external control is considered with special emphasis on systems that are not completely controllable in particular it is shown that the dynamical lie algebra for an n level system with symmetrically coupled transitions such as a system with equally spaced energy levels and uniform transition dipole moments is a subalgebra of so n if n 2 1 and a subalgebra of sp if n 2 general criteria for obtaining either so 2 1 or sp are established

efficient pricing of discrete arithmetic asian options under mean reversion and jumps based on fourier cosine expansions we propose an efficient pricing method for arithmetic asian options based on fourier cosine expansions in particular we allow for mean reversion and jumps in the underlying price dynamics there is an extensive body of empirical evidence in the current literature that points to the existence and prominence of such anomalies in the prices of certain asset classes such as commodities our efficient pricing method is derived for the discretely monitored versions of the european style arithmetic asian options the analytical solutions obtained from our fourier cosine expansions are compared to the benchmark fast fourier transform based pricing for the examination of its accuracy and computational efficiency

efficient numerical fourier methods for coupled forward backward sdes we develop three numerical methods to solve coupled forward backward stochastic differential equations we propose three different discretization techniques for the forward stochastic differential equation a theta discretization of the time integrands is used to arrive at schemes with conditional expectations these conditional expectations are approximated by using the cos method which relies on the availability of the conditional characteristic function of the discrete forward process the numerical methods are applied to different problems including a financial problem richardson extrapolation is used to obtain more accurate results resulting in the observation of second order convergence in the number of time steps advantages and disadvantages of each method are compared against each other

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

analysis of lyapunov method for control of quantum states the natural trajectory tracking problem is studied for generic quantum states represented by density operators a control design based on the hilbert schmidt distance as a lyapunov function is considered the control dynamics is redefined on an extended space where the lasalle invariance principle can be correctly applied even for non stationary target states lasalle s invariance principle is used to derive a general characterization of the invariant set which is shown to always contain the critical points of the lyapunov function critical point analysis of the latter is used to show that for generic states it is a morse function with n isolated critical points including one global minimum one global maximum and n 2 saddles it is also shown however that the actual dynamics of the system is not a gradient flow and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points this analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular and in fact convergence to the target state trajectory in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space on the other hand if either of these sufficient conditions is not satisfied the target state ceases to be asymptotically stable a center manifold emerges around the target state and the control design ceases to be effective

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

notes on sliding mode control of two level quantum systems in dong d and petersen i r 2012 sliding mode control of two level quantum systems automatica 48 725 735 a sliding mode control approach has been proposed for two level quantum systems to deal with bounded uncertainties in the system hamiltonian this paper further extends these results in two directions one extension is to consider the effect of uncertainties during the process of driving the system s state back to the sliding mode domain from outside and we propose two approaches to accomplish this control task the other extension generalizes the previous design approach to consider uncertainties described as perturbations in the free hamiltonian

option pricing with legendre polynomials here we develop an option pricing method based on legendre series expansion of the density function the key insight relying on the close relation of the characteristic function with the series coefficients allows to recover the density function rapidly and accurately based on this representation for the density function approximations formulas for pricing european type options are derived to obtain highly accurate result for european call option the implementation involves integrating high degree legendre polynomials against exponential function some numerical instabilities arise because of serious subtractive cancellations in its formulation 96 in proposition a 1 to overcome this difficulty we rewrite this quantity as solution of a second order linear difference equation and solve it using a robust and stable algorithm from olver derivation of the pricing method has been accompanied by an error analysis errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions but rather by the density function of the underlying stochastic models this is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions the numerical experiments on a class of models widely used in quantitative finance show exponential convergence

quantum state filter with disturbance and noise a quantum state filter qsf is proposed in this paper to estimate a low rank quantum density matrix from informationally incomplete and contaminated measurements there exist sparse disturbances on the quantum density matrix and gaussian noise in the measurements a proximal jacobian variant of the alternating direction method of multipliers pj admm is proposed to design the qsf the closed form solutions to three resulting subproblems are given and the iterative qsf is developed the proposed qsf is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state of the art methods

the kolmogorov forward fractional partial differential equation for the cgmy process with applications in option pricing abstract in this paper we derive the kolmogorov forward fractional partial differential equation fpde for the cgmy process the resulting fpde is solved numerically with a second order method in space and backward differentiation formula of order two in time we price options by integrating the resulting probability density function multiplied by the pay off function of the option hence we only have to solve one fpde to price several options this is useful in practical applications where it is common to price many options simultaneously for the same underlying diffusion model the traditional way to price options requires the solution of one fpde per option since the fpdes are of similar type and complexity the advantage of our suggested method is obvious in the case of pricing multiple european type options on the same underlying asset when only the pay off function differs the numerical experiments verify that the extra cost of pricing several options compared to only one option is very small with our suggested method

criteria for reachability of quantum states we address the question of which quantum states can be inter converted under the action of a time dependent hamiltonian in particular we consider the problem as applied to mixed states and investigate the difference between pure and mixed state controllabilities introduced in previous work we provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group we also address the problem of which states can be prepared if the dynamical lie group is not sufficiently large to allow the system to be controllable

an efficient algorithm based on eigenfunction expansions for some optimal timing problems in finance this paper considers the optimal switching problem and the optimal multiple stopping problem for one dimensional markov processes in a finite horizon discrete time framework we develop a dynamic programming procedure to solve these problems and provide easy to verify conditions to characterize connectedness of switching and exercise regions when the transition or feynman kac semigroup of the markov process has discrete spectrum we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem we also prove that the algorithm converges exponentially in the series truncation level our method is applicable to a rich family of markov processes which are widely used in financial applications including many diffusions as well as jump diffusions and pure jump processes that are constructed from diffusion through time change in particular many of these processes are often used to model mean reversion we illustrate the versatility of our method by considering three applications valuation of combination shipping carriers interest rate chooser flexible caps and commodity swing options numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical pde methods that are typically used to solve such problems

wavefunction controllability for finite dimensional bilinear quantum systems we present controllability results for quantum systems interacting with lasers exact controllability for the wavefunction in these bilinear systems is proved in the finite dimensional case under very natural hypotheses

optimal control of one qubit gates we consider the problem of carrying an initial bloch vector to a final bloch vector in a specified amount of time under the action of three control fields a vector control field we show that this control problem is solvable and therefore it is possible to optimize the control we choose the physically motivated criteria of minimum energy expended in the control minimum magnitude of the rate of change of the control and a combination of both we find exact analytical solutions determine the fields for a general one qubit gate and use the y gate as an example we argue that in the case of less than three controls only the physical intuition does not provide a straight reasonable solution and solve the problem in the case of a unique control minimizing the energy consumption

the evaluation of barrier option prices under stochastic volatility this paper considers the problem of numerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of heston 1993 7 we develop a method of lines approach to evaluate the price as well as the delta and gamma of the option the method is able to efficiently handle both continuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features in the latter case we can calculate the early exercise boundary of an american barrier option in both the continuously and discretely monitored cases

on the calibration of the 3 2 model abstract we consider the problem of calibrating the 3 2 stochastic volatility model to option data in comparison to the characteristic function of the heston model the characteristic function of the 3 2 model can be up to 50 times slower to evaluate this makes the standard least squares calibration with finite difference gradients unreasonably slow to address this problem we derive the analytic gradient of the characteristic function in compact form we then propose a computational method for the analytic gradient formula which caches intermediate results across the partial derivatives in addition to the strike dimension and the maturity dimension compared to the fastest method of calibrating the 3 2 model which we could find in the literature the method proposed in this paper is more than 10 times faster we also discuss the issue of apparent non convexity in the least squares calibration of the 3 2 model for market data to tackle it we propose a regularized calibration where the regularization point is obtained using risk neutral mcmc estimation of the model we find that this approach is particularly well suited for the calibration problem as it generates naturally a consistent damping matrix for the parameter estimates in addition to being very fast

control landscapes are almost always trap free a geometric assessment a proof is presented that almost all closed finite dimensional quantum systems have trap free i e free from local optima landscapes for a large and physically general class of circumstances which includes qubit evolutions in quantum computing this result offers an explanation for why gradient based methods succeed so frequently in quantum control the role of singular controls is analyzed using geometric tools in the case of the control of the propagator and thus in the case of observables as well singular controls have been implicated as a source of landscape traps the conditions under which singular controls can introduce traps and thus interrupt the progress of a control optimization are discussed and a geometrical characterization of the issue is presented it is shown that a control being singular is not sufficient to cause control optimization progress to halt and sufficient conditions for a trap free landscape are presented it is further shown that the local surjectivity full rank assumption of landscape analysis can be refined to the condition that the end point map is transverse to each of the level sets of the fidelity function this mild condition is shown to be sufficient for a quantum system s landscape to be trap free the control landscape is shown to be trap free for all but a null set of hamiltonians using a geometric technique based on the parametric transversality theorem numerical evidence confirming this analysis is also presented this new result is the analogue of the work of altifini wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation these collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps

controllability of open quantum systems with kraus map dynamics this paper presents a constructive proof of complete kinematic state controllability of finite dimensional open quantum systems whose dynamics are represented by kraus maps for any pair of states pure or mixed on the hilbert space of the system we explicitly show how to construct a kraus map that transforms one state into another moreover we prove by construction the existence of a kraus map that transforms all initial states into a predefined target state such a process may be used for example in quantum information dilution thus in sharp contrast to unitary control kraus map dynamics allows for the design of controls which are robust to variations in the initial state of the system the capabilities of non unitary control for population transfer between pure states illustrated for an example of a two level system by constructing a family of non unitary kraus maps to transform one pure state into another the problem of dynamic state controllability of open quantum systems i e controllability of state to state transformations given a set of available dynamical resources such as coherent controls incoherent interactions with the environment and measurements is also discussed

lyapunov control of bilinear schrodinger equations a lyapunov based approach for trajectory tracking of the schrodinger equation is proposed in the finite dimensional case convergence is precisely analyzed connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied when the linearized system is controllable such a feedback ensures almost global asymptotic convergence when the linearized system is not controllable the stability of the closed loop system is not asymptotic to overcome such lack of convergence we propose when the reference trajectory is an eigenstate a modification based on adiabatic invariance simulations illustrate the simplicity and also the interest for trajectory generation

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

application of the fast gauss transform to option pricing in many of the numerical methods for pricing american options based on the dynamic programming approach the most computationally intensive part can be formulated as the summation of gaussians though this operation usually requireso nn work when there aren summations to compute and the number of terms appearing in each summation isn we can reduce the amount of work too n n by using a technique called the fast gauss transform in this paper we apply this technique to the multinomial method and the stochastic mesh method and show by numerical experiments how it can speed up these methods dramatically both for the black scholes model and merton s lognormal jump diffusion model we also propose extensions of the fast gauss transform method to models with non gaussian densities

degrees of controllability for quantum systems and application to atomic systems precise definitions for different degrees of controllability for quantum systems are given and necessary and sufficient conditions for each type of controllability are discussed the results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies

erratum for feedback stabilization of isospectral control systems on complex flag manifolds application to quantum ensembles nov 07 2019 2028 the results in the above titled paper ibid vol 52 no 11 pp 2019 2028 nov 07 are contradicted slightly by those found in two others cited here the results are explained here