a power penalty method for a 2d fractional partial differential linear complementarity problem governing two asset american option pricing in this paper we propose a power penalty method for a linear complementarity problem lcp involving a fractional partial differential operator in two spatial dimensions arising in pricing american options on two underlying assets whose prices follow two independent geometric lvy processes we first approximate the lcp by a nonlinear 2d fractional partial differential equation fpde with a penalty term we then prove that the solution to the fpde converges to that of the lcp in a sobolev norm at an exponential rate depending on the parameters used in the penalty term the 2d fpde is discretized by a 2nd order finite difference method in space and cranknicolson method in time numerical experiments on a model basket option pricing problem were performed to demonstrate the convergent rates and the effectiveness of the penalty method

a robust and accurate finite difference method for a generalized black scholes equation in this paper we present a numerical method for a generalized black scholes equation which is used for option pricing the method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique our scheme is stable for arbitrary volatility and arbitrary interest rate and is second order convergent with respect to the spatial variable furthermore the present paper efficiently treats the singularities of the non smooth payoff function numerical results support the theoretical results

a power penalty method for solving a nonlinear parabolic complementarity problem abstract in this paper we present a penalty method for solving a complementarity problem involving a second order nonlinear parabolic differential operator in this work we first rewrite the complementarity problem as a nonlinear variational inequality then we define a nonlinear parabolic partial differential equation pde approximating the variational inequality using a power penalty term with a penalty constant 1 a power parameter k 0 and a smoothing parameter e we prove that the solution to the penalized pde converges to that of the variational inequality in an appropriate norm at an arbitrary exponential rate of the form o k e 1 e 1 k 1 2 numerical experiments performed to verify the theoretical results show that the computed rates of convergence in both and k are close to the theoretical ones

an upwind numerical approach for an american and european option pricing model the numerical solution of several mathematical models arising in financial economics for the valuation of both european and american call options on different types of assets is considered all the models are based on the black scholes partial differential equation in the case of european options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented when treating with american options an additional inequality constraint leads to a discretized linear complementarity problem in each case the numerical approximation of option values is computed by means of optimization algorithms in particular uzawa s method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

robust h control for linear discrete time systems with norm bounded time varying uncertainty the problem of robust h analysis and synthesis for linear discrete time systems with norm bounded time varying uncertainty is studied in this paper it will be shown that this problem is equivalent to the problem of h analysis and synthesis of an auxiliary system the necessary and sufficient conditions for the equivalency are proved thus the original problem can be solved by existing h control methods

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

h sub infinity analysis and synthesis of discrete time systems with time varying uncertainty the problems of h sub infinity analysis and synthesis of discrete time systems with block diagonal real time varying uncertainty are considered it is shown that these problems can be converted into scaled h sub infinity analysis and synthesis problems the problems of quadratic stability analysis and quadratic stabilization of these types of systems are dealt with as a special case the results on synthesis are established for general linear dynamic output feedback control

robust h control for uncertain descriptor systems by proportional derivative state feedback this article deals with the problem of robust h control for uncertain descriptor systems the parameter uncertainties are assumed to be norm bounded appearing not only in the state and input matrices but also in the derivative matrix a proportional plus derivative pd state feedback is designed such that the closed loop descriptor system is quadratically normal and quadratically stable and satisfies a prescribed h norm bound a necessary and sufficient condition for the solvability of this problem is obtained in terms of linear matrix inequalities the explicit expression of desired pd state feedback controller is given two illustrative examples are provided to demonstrate the applicability of the proposed approaches

optimal convergence rate of the explicit finite difference scheme for american option valuation an optimal convergence rate o dx for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition s email protected dx 2 1 using completely pde methods as a corollary a binomial tree scheme of an american put option where s email protected dx 2 1 is convergent unconditionally with the rate o dt 1 2

pole assignment for uncertain systems in a specified disk by state feedback this paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system the uncertainty being norm bounded for this the quadratic d stabilizability concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived this condition expressed as a parameter dependent discrete riccati equation enables one to design the control gain matrix by solving iteratively a discrete riccati equation

an rbf fd method for pricing american options under jump diffusion models abstract american option prices under jump diffusion models are determined as solutions to partial integro differential equations pide in this paper a new combination of a time and spatial discretization applied to a linear complementary formulation lcp of the free boundary pide is proposed first a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain an implicit explicit time discretization applied to the reformulated lcp on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system the radial basis function rbf finite difference method is a local method resulting in a sparse linear system in contrast to global rbf methods which lead to ill conditioned dense matrix systems for the corresponding european option we prove consistency stability and second order convergence in a discrete l 2 norm we derive mild conditions for the model parameters under which these results hold numerical experiments are performed with european and american options and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm

option pricing using a computational method based on reproducing kernel abstract one of the most important subject in financial mathematics is the option pricing the most famous result in this area is black scholes formula for pricing european options this paper is concerned with a method for solving a generalized black scholes equation in a reproducing kernel hilbert space subsequently the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method furthermore the error estimates for obtained approximation in reproducing kernel hilbert space are presented finally a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method

robust h spl infin control for uncertain stochastic systems with state delay this note deals with the problems of robust stochastic stabilization and robust h control for uncertain stochastic systems with a time varying delay in the state the parameter uncertainties are assumed to be time varying norm bounded appearing in both the state and input matrices the purpose of the robust stochastic stabilization problem is the design of a memoryless state feedback controller such that the closed loop system is mean square asymptotically stable for all admissible uncertainties in the robust h control problem in addition to the mean square asymptotic stability requirement a prescribed h performance is required to be achieved in terms of a linear matrix inequality sufficient conditions for the solvability of these problems are proposed respectively the expressions of desired state feedback controllers are given an example illustrating the proposed approach is provided

non fragile robust h control for uncertain spacecraft rendezvous system with pole and input constraints this article investigates the robust h control problem for a class of spacecraft rendezvous systems with parameter uncertainties and subject to input constraint pole constraint and external and controller perturbations based on the lyapunov theory a sufficient condition for the existence of the non fragile robust state feedback controller is given in terms of linear matrix inequalities lmis then proper non fragile controller design can be cast as a convex optimisation problem subject to lmi constraints with the obtained controller the spacecraft rendezvous mission can be accomplished an illustrative example is provided to show the effectiveness of the proposed control design method

an inverse problem for a parabolic variational inequality arising in volatility calibration with american options in finance the price of an american option is obtained from the price of the underlying asset by solving a parabolic variational inequality the free boundary associated with this variational inequality can be interpreted as the price for which the option should be exercised the calibration of volatility from the observations of the prices of an american option yields an inverse problem for the previously mentioned parabolic variational inequality after studying the variational inequality and the exercise price we give results concerning the sensitivity of the option price and of the exercise price with respect to the variations of the volatility the inverse problem is addressed by a least square method with suitable regularization terms we give necessary optimality conditions involving an adjoint state for a simplified inverse problem and we study the differentiability of the cost function optimality conditions are also given for the genuine calibration problem

finite element error estimates for a nonlocal problem in american option valuation based on new exact formulations of american option problems on bounded domains error estimates are established for finite element approximations of american option prices under admissible regularity some numerical results are also presented

numerical analysis of american option pricing in a jump diffusion model we discuss pricing formulae for american options in merton s jump diffusion model with the help of variational inequalities we derive some regularity properties of price functions using the finite difference method a discretization scheme is presented and a convergence theorem for the first order derivatives is proved numerical methods and results are also discussed

a super convergent unsymmetric finite volume method for convection diffusion equations abstract the design of the dual mesh of a conventional fitted finite volume method is revisited with the aim to obtain better convergence rates a new choice of dual mesh points is proposed and it is proved that at these points the approximate flux density and the solution have a second order accuracy the results of numerical experiments substantiate the theoretical findings

convergence of a fitted finite volume method for the penalized black scholes equation governing european and american option pricing in this paper we present an analysis of a numerical method for a degenerate partial differential equation called the black scholes equation governing american and european option pricing the method is based on a fitted finite volume spatial discretization and an implicit time stepping technique the analysis is performed within the framework of the vertical method of lines where the spatial discretization is formulated as a petrov galerkin finite element method with each basis function of the trial space being determined by a set of two point boundary value problems we establish the stability and an error bound for the solutions of the fully discretized system numerical results are presented to validate the theoretical results

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

finite difference scheme with a moving mesh for pricing asian options in this paper we propose a stable numerical method for pricing asian call options which is based on a central difference scheme with a moving mesh in the spatial discretization and the rannacher time stepping scheme in the time discretization at each time mesh point we make a piecewise uniform mesh to discretise the space interval which ensures that the matrix associated with the discrete operator is an m matrix hence the spatial discretization scheme is maximum norm stable for arbitrary volatility and arbitrary interest rate we show that the scheme is second order convergent with respect to both time and spatial variables numerical results support the theoretical results

a superlinearly convergent method for a class of complementarity problems with non lipschitzian functions we consider a class of complementarity problems involving functions which are not lipschitz continuous in this paper we reformulate this class of non lipschitzian complementarity problems into a lipschitzian complementarity problem then we propose an inexact smoothing newton method to solve this lipschitzian complementarity problem we prove that our proposed method converges quadratically and globally under a mild condition numerical results show that this method is promising this method can solve these kinds of complementarity problems with one million variables in reasonable time on a pc with 1 gb of ram

a fast numerical method for the black scholes equation of american options this paper introduces a fast numerical method for computing american option pricing problems governed by the black scholes equation the treatment of the free boundary is based on some properties of the solution of the black scholes equation an artificial boundary condition is also used at the other end of the domain the finite difference method is used to solve the resulting problem computational results are given for some american call option problems the results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method

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

the h functional calculus based on the s spectrum for quaternionic operators and for n tuples of noncommuting operators abstract in this paper we extend the h functional calculus to quaternionic operators and to n tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus called s functional calculus the s functional calculus has two versions one for quaternionic valued functions and one for clifford algebra valued functions and can be considered the riesz dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties the s functional calculus is based on the notion of s spectrum which in the case of quaternionic normal operators on a hilbert space is also the notion of spectrum that appears in the quaternionic spectral theorem the main purpose of this paper is to construct the h functional calculus based on the notion of s spectrum for both quaternionic operators and for n tuples of noncommuting operators we remark that the h functional calculus for n 1 tuples of operators applies in particular to the dirac operator

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

a new theory of regular functions of a quaternionic variable abstract in this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable the theory we describe follows a classical idea of cullen but we use a more geometric formulation to show that it is possible to build a rather complete theory our theory allows us to extend some important results for polynomials in the quaternionic variable to the case of power series

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

robust h control of discrete switched system with time delay this paper deals with the stabilization and robust h infinity control of discrete switched system with time delay the switching is time driven and the design is based on the average dwell time method the controller parameters and the permissible switching sequence subject to the given average dwell time can be obtained by solving a set of linear matrix inequalities lmis two examples are given to demonstrate the proposed methods

a fixed point method for a class of super large scale nonlinear complementarity problems we consider a class of complementarity problems involving functions which are nonlinear in this paper we reformulate this nonlinear complementarity problem as a system of absolute value equations which is nonsmooth then we propose a fixed point method to solve this nonsmooth system we prove that the proposed method is globally linearly convergent under a mild condition the proposed method is greatly effective not only for small and medium size problems but also for large and super large scale problems especially our method can efficiently solve super large scale problems with a million variables in a few tens of minutes on a pc

quintic b spline collocation approach for solving generalized black scholes equation governing option pricing we develop a numerical algorithm for solving generalized black scholes partial differential equation which arises in european option pricing the method comprises the horizontal method of lines for time integration and method 1 2 1 1 corresponds to the back ward euler method and 1 2 corresponds to the crank nicolson method to discretize in temporal direction and the quintic b spline collocation method in uniform spatial direction the convergence analysis and stability of proposed method are discussed in detail it is justifying that the approximate solution converges to the exact solution of orders o k h 3 for the back ward euler method and o k 2 h 3 for the crank nicolson method where k and h are mesh sizes in the time and space directions respectively the proposed method is also shown to be unconditionally stable this scheme applied on some test examples the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence results shown by this method are found to be in good agreement with the known exact solutions the produced results are also seen to be more accurate than some available results given in the literature

robust h control for uncertain discrete singular systems with pole placement in a disk abstract this paper addresses the design of robust h controllers for uncertain discrete singular systems with time invariant uncertainty in both the state and measurement matrices the singular system to be controlled is not assumed to be regular a regular dynamic output feedback controller is designed such that a prescribed h performance condition is satisfied and the closed loop poles are placed in a specified disk while the regularity causality and stability of the closed loop system can be guaranteed for all admissible uncertainties the desired controller can be obtained by solving a set of matrix inequalities a numerical example is given to demonstrate the application of the proposed method

on finite termination of an iterative method for linear complementarity problems based on a well known reformulation of the linear complementarity problem lcp as a nondifferentiable system of nonlinear equations a newton type method will be described for the solution of lcps under certain assumptions it will be shown that this method has a finite termination property i e if an iterate is sufficiently close to a solution of lcp the method finds this solution in one step this result will be applied to a recently proposed algorithm by harker and pang in order to prove that their algorithm also has the finite termination property

borel functional calculus for quaternionic normal operators in this article we give an approach to borel functional calculus for quaternionic normal operators which are not necessarily bounded first we establish the definition of functional calculus for a subclass of quaternion valued borel functions and then we extend the same to the class of quaternion valued borel functions as well as l functions we also prove spectral mapping theorem as a consequence

b spline based monotone multigrid methods for the efficient numerical solution of elliptic variational inequalities on closed convex sets multigrid methods based on piecewise linear finite elements have been investigated over the past decades essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements on the other hand there are a number of problems which profit from higher order approximations among these are the problem of pricing american options formulated as a parabolic boundary value problem involving black scholes equation with a free boundary in addition to computing the free boundary the optimal exercise price of the option of particular importance are accurate pointwise derivatives of the value of the stock option up to order two the so called greek letters in this paper we propose a monotone multigrid method for discretizations in terms of b splines of arbitrary order to solve elliptic variational inequalities on a closed convex set in order to maintain monotonicity upper bound and quasi optimality lower bound of the coarse grid corrections we propose an optimized coarse grid correction ocgc algorithm which is based on b spline expansion coefficients we prove that the ocgc algorithm is of optimal complexity of the degrees of freedom of the coarse grid and therefore the resulting monotone multigrid method is of asymptotically optimal multigrid complexity finally the method is applied to a standard model for the valuation of american options in particular it is shown that a discretization based on b splines of order four enables us to compute the second derivative of the value of the stock option to high precision

pole assignment in a specified disk the problem of assigning all poles of a closed loop system in a specified disk by state feedback is considered for both continuous and discrete systems a state feedback control law is determined by using a discrete riccati equation this kind of pole assignment problem is named d pole assignment and its relation to the optimal control problem and its robustness properties are discussed the gain and phase margins for all closed loop poles to stay inside the specified disk d are determined for the proposed control

h control with regional stability constraints abstract the problem of static state feedback h control with various regional stability constraints for the closed loop system is considered a general framework for solving this problem is established using the lagrange multiplier method in conjunction with generalised lyapunov theory and riccati equation based h optimization characterization necessary conditions are given for the existence of the desired optimal feedback gains furthermore as the simplest regional stability case the stability degree problem is studied with two different approaches one an indirect overbounding method that extends the standard h problem formulation with an shift and another that uses the direct optimization of the h performance measure with stability degree as a constraint under the general framework the conservativeness of the former approach and the generality of the latter approach are clearly demonstrated finally the theory is illustrated with examples and the computational issues are discussed it is shown that the successive approximation technique of liu and of liu and yedavalli can be used to partially solve the problem

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

robust h control for uncertain discrete time systems with circular pole constraints this paper investigates the problem of robust h control for uncertain discrete time systems with circular pole constraints the system under consideration is subject to norm bounded time invariant uncertainties in both the state and input matrices the problem we address is to design state feedback controllers such that the closed poles are located within a prespecified circular region and the h norm of the closed loop transfer function is strictly less than a given positive scalar for all admissible uncertainties by introducing the notion of quadratic d stabilizability with an h norm bound the problem is solved necessary and sufficient conditions for quadratic d stabilizability with an h norm bound are derived our results can be regarded as extensions of existing results on robust h control and robust pole assignment of uncertain systems

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

shout options a framework for pricing contracts which can be modified by the investor a shout option may be broadly defined as a financial contract which can be modified by the holder according to specified rules in a simple example the holder could have the right to set the strike of an option equal to the current value of the underlying asset in such a case the holder effectively has the right to select when to take ownership of an at the money option more generally the holder could have multiple rights along these lines in some cases with a limit placed on the number of rights which may be exercised within a given time period e g four times per year the value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems this paper describes a general framework for the valuation of complex types of shout options numerical issues related to interpolation and choice of timestepping method are considered in detail some illustrative examples are provided

an exponentially fitted finite volume method for the numerical solution of 2d unsteady incompressible flow problems abstract in this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the navier stokes equations describing 2d incompressible flows the method is based on an unstructured delaunay mesh and its dual dirichlet tessellation combined with a locally constant approximation to the flux this yields a piecewise exponential approximation to the exact solution numerical tests are presented for a linear advection diffusion problem with boundary layers the method is then applied to the driven cavity problem with reynolds numbers up to 10 4 the numerical results indicate that the method is robust for a wide range of values of the reynolds number in the case re 10 4 unsteady solutions are captured if the mesh is sufficiently fine

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

robust h control for linear time invariant systems with norm bounded uncertainty in the input matrix abstract this paper focuses on the problem of robust h control design for a class of linear time invariant systems with uncertainty in the state space model we consider uncertain systems with norm bounded parameter uncertainty in the input matrix the paper presents a state feedback control design which stabilizes the plant for all admissible uncertainties and also guarantees an h norm bound constraint on disturbance attenuation paralleling to the theory of robust control the robust h control problem is solved via the notion of quadratic stabilization with an h norm bound constraint necessary and sufficient conditions for quadratic stabilization with an h norm bound are derived it is shown that the solution to this problem involves solving a parameter dependent algebraic riccati equation the results can be regarded as extensions of existing results on robust stabilization of linear uncertain systems and h optimal control

h control design for systems with structured parameter uncertainty in a recent paper a unification of the h 2 lqg and h control design problems was obtained in terms of modified algebraic riccati equations in the present paper these results are extended to guarantee robust h 2 and hoo performance in the presence of structured real valued parameter variations aa ab ac in the state space model for design flexibility the paper considers two distinct types of uncertainty bounds for both full and reduced order dynamic compensation an important special case of these results generates h2 hoo controller designs with guaranteed gain margins

pricing options under jump diffusion processes with fitted finite volume method this paper develops a numerical method for a partial integro differential equation and a partial integro differential complementarity problem arising from european and american options valuations respectively when the underlying assets are driven by a jump diffusion process the method is based on a fitted finite volume scheme for the spatial discretization and the crank nicolson scheme for the time discretization the fully discretized system is solved by an iterative method coupled with an fft for the evaluation of the discretized integral term while the constraint in the american option model is imposed by adding a penalty term to the original partial integro differential complementarity problem we show that the system matrix of the discretized system is an m matrix and propose an algorithm for solving the discretized system numerical experiments are implemented to show the efficiency and robustness of this method

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

an iterative two step algorithm for linear complementarity problems we propose an algorithm for the numerical solution of large scale symmetric positive definite linear complementarity problems each step of the algorithm combines an application of the successive overrelaxation method with projection to determine an approximation of the optimal active set with the preconditioned conjugate gradient method to solve the reduced residual systems of linear equations convergence of the iterates to the solution is proved in the experimental part we compare the efficiency of the algorithm with several other methods as test example we consider the obstacle problem with different obstacles for problems of dimension up to 24 000 variables the algorithm finds the solution in less then 7 iterations where each iteration requires about 10 matrix vector multiplications

the spectral theorem for quaternionic unbounded normal operators based on the s spectrum in this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of s spectrum the proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator with this paper we complete the foundation of spectral analysis of quaternionic operators the s spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators the lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators a prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics