analysis of an optimal control problem for the three dimensional coupled modified navier stokes and maxwell equations the mathematical formulation and analysis of an optimal control problem associated with a viscous incompressible electrically conducting fluid in a bounded three dimensional domain with fixed perfectly conducting boundaries is considered the objective of control is the matching of the velocity and magnetic fields to given target fields control is effected through distributed mechanical force and current controls the existence of optimal solutions is shown the g teaux differentiability for the magnetohydrodynamic system with respect to controls is proved and the optimality system is obtained

numerical approximation of equations involving minimal maximal operators by successive solution of obstacle problems abstract let r 2 be a polygonal domain and let l i i 1 2 be two elliptic operators of the form l i u x div a i x u x c i x u x f i x motivated by the results in blanc et al 2016 we propose a numerical iterative method to compute the numerical approximation to the solution of the minimal problem min l 1 u l 2 u 0 in u 0 on the convergence of the method is proved and numerical examples illustrating our results are included

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions abstract let x be a banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict opial condition let c be a bounded closed convex subset of x kc x the family of all compact convex subsets of x and t a nonexpansive mapping from c into kc x with bounded range we prove that t has a fixed point the nonstrict opial condition can be removed if in addition t is an 1 contractive mapping

methods for pricing american options under regime switching we analyze a number of techniques for pricing american options under a regime switching stochastic process the techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable in the case of implicit methods we also compare a number of iterative procedures for solving the associated nonlinear algebraic equations numerical tests indicate that a fixed point policy iteration coupled with a direct control formulation is a reliable general purpose method finally we remark that we formulate the american problem as an abstract optimal control problem hence our results are applicable to more general problems as well

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

boundary effects in a phase field approach to topology optimization a phase field based topology optimization approach is considered for the maximum stiffness or minimum compliance problem the objective functional to be minimized consists in addition to the compliance a cost for gray solutions and a cost for interfaces between void and full material since the interfaces between void and full material are penalized via a volume integral in the original phase field formulation there is no penalty associated with interfaces along the external boundaries in the present contribution an additional term representing the cost of interfaces at external boundaries is added to the functional subject to minimization it is shown that the new boundary term enters the optimization as a robin boundary condition the method is implemented in a finite element setting and numerical simulations of typical structures are considered the results indicate that the optimal designs are influenced by the cost of interfaces to a large less

tame functions are semismooth superlinear convergence of the newton method for nonsmooth equations requires a semismoothness assumption in this work we prove that locally lipschitz functions definable in an o minimal structure in particular semialgebraic or globally subanalytic functions are semismooth semialgebraic or more generally globally subanalytic mappings present the special interest of being order semismooth where is a positive parameter as an application of this new estimate we prove that the error at the kth step of the newton method behaves like o 2 1 gamma k

optimal control of evolutionary quasi newtonian fluids we are interested in the necessary optimality conditions for optimal control problems of systems describing the unsteady flow of incompressible shear thickening fluids to overcome the difficulties related to the differentiability of the control to state mapping consequence of the lack of regularity of the state variable we first establish some intermediate optimality conditions and then pass to the limit

a deterministic control based approach motion by curvature the level set formulation of motion by mean curvature is a degenerate parabolic equation we show that its solution can be interpreted as the value function of a deterministic two person game more precisely we give a family of discrete time two person games whose value functions converge in the continuous time limit to the solution of the motion by curvature pde for a convex domain the boundary s first arrival time solves a degenerate elliptic equation this corresponds in our game theoretic setting to a minimum exit time problem for a nonconvex domain the two person game still makes sense we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the positive part of the curvature these results are unexpected because the value function of a deterministic control problem is normally the solution of a first order hamilton jacobi equation our situation is different because the usual first order calculation is singular 2005 wiley periodicals inc

existence theorems for perturbed abstract measure differential equations abstract in this paper an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of dhage b c dhage on some nonlinear alternatives of leray schauder type and functional integral equations arch math brno 42 2006 11 23 under the mixed generalized lipschitz and caratheodory conditions the existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of dhage given in the above mentioned reference on ordered banach spaces our existence results include the existence results of sharma r r sharma an abstract measure differential equation proc amer math soc 32 1972 503 510 joshi s r joshi a system of abstract measure delay differential equations j math phy sci 13 1979 497 506 and shendge and joshi g r shendge s r joshi abstract measure differential inequalities and applications acta math hung 41 1983 53 54 as special cases under weaker continuity condition

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

state constrained optimal control of the three dimensional stationary navier stokes equations abstract in this paper an optimal control problem for the stationary navier stokes equations in the presence of state constraints is investigated existence of optimal solutions is proved and first order necessary conditions are derived the regularity of the adjoint state and the state constraint multiplier is also studied lipschitz stability of the optimal control state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

optimal control of the stationary navier stokes equations with mixed control state constraints in this paper we consider the distributed optimal control of the navier stokes equations in the presence of pointwise mixed control state constraints after deriving a first order necessary condition the regularity of the mixed constraint multiplier is investigated second order sufficient optimality conditions are studied as well in the last part of the paper a semismooth newton method is applied for the numerical solution of the control problem the convergence of the method is proved and numerical experiments are carried out

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

distributed control for a class of non newtonian fluids we consider control problems with a general cost functional where the state equations are the stationary incompressible navier stokes equations with shear dependent viscosity the equations are quasi linear the control function is given as the inhomogeneity of the momentum equation in this paper we study a general class of viscosity functions which correspond to shear thinning or shear thickening behavior the basic results concerning existence uniqueness boundedness and regularity of the solutions of the state equations are reviewed the main topic of the paper is the proof of g teaux differentiability which extends known results it is shown that the derivative is the unique solution to a linearized equation moreover necessary first order optimality conditions are stated and the existence of a solution of a class of control problems is shown

large strain phase field based multi material topology optimization summary a multi material topology optimization scheme is presented the formulation includes an arbitrary number of phases with different mechanical properties to ensure that the sum of the volume fractions is unity and in order to avoid negative phase fractions an obstacle potential function which introduces infinity penalty for negative densities is utilized the problem is formulated for nonlinear deformations and the objective of the optimization is the end displacement the boundary value problems associated with the optimization problem and the equilibrium equation are solved using the finite element method to illustrate the possibilities of the method it is applied to a simple boundary value problem where optimal designs using multiple phases are considered copyright 2015 john wiley sons ltd

discontinuous galerkin finite element approximation of hamilton jacobi bellman equations with cordes coefficients we propose an hp version discontinuous galerkin finite element method for fully nonlinear second order elliptic hamilton jacobi bellman equations with cordes coefficients the method is proved to be consistent and stable with convergence rates that are optimal with respect to mesh size and suboptimal in the polynomial degree by only half an order numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme an existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

regularity of flows and optimal control of shear thinning fluids abstract we study optimal control problems of systems describing the flow of incompressible shear thinning fluids taking advantage of regularity properties of the flows we derive necessary optimality conditions under a restriction on the optimal control

hybrid fixed points of multivalued operators in metric spaces with applications abstract this paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space our claim is also illustrated with applications to a class of integral inclusions for proving the existence results

hybrid fixed point theory for strictly monotone increasing multi valued mappings with applications in this paper a general hybrid fixed point theorem for the strict monotone increasing multi valued mappings in ordered banach spaces is proved via measure of noncompactness and it is further applied to perturbed functional nonconvex differential inclusions for proving the existence results for the extremal solutions under mixed lipschitz compactness and strict monotonic conditions

optimal control of shear thickening flows we study optimal control problems of systems describing the flow of incompressible shear thickening fluids we prove existence of solutions and derive necessary optimality conditions under precise restrictions on the optimal control

fixed points of multivalued operators in ordered metric spaces with applications abstract the purpose of this paper is to present some new hybrid fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space an example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results

convergent semi lagrangian methods for the monge ampere equation on unstructured grids this paper is concerned with developing and analyzing convergent semi lagrangian methods for the fully nonlinear elliptic monge ampere equation on general triangular grids this is done by establishing an equivalent in the viscosity sense hamilton jacobi bellman formulation of the monge ampere equation a significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built in property of the new formulation moreover this new approach allows one to tap the wealthy numerical methods such as semi lagrangian schemes for hamilton jacobi bellman equations to solve monge ampere type equations it is proved that the considered numerical methods are monotone pointwise consistent and uniformly stable consequently its solutions converge uniformly to the unique convex viscosity solution of the monge ampere dirichlet problem a superlinearly convergent howard s algorithm which is a newton type method is utilized as the nonlinear

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

discrete miranda talenti estimates and applications to linear and nonlinear pdes abstract in this article we construct simple and convergent finite element methods for linear and nonlinear elliptic differential equations in non divergence form with discontinuous coefficients the methods are motivated by discrete miranda talenti estimates which relate the h 2 semi norm of piecewise polynomials with the l 2 norm of its laplacian on convex domains we develop a stability and convergence theory of the proposed methods and back up the theory with numerical experiments

a general multi valued hybrid fixed point theorem and perturbed differential inclusions abstract combining three basic multi valued versions of banach schauder and tarski fixed point theorems a general hybrid fixed point theorem for multi valued mappings in banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed lipschitz compactness and monotone conditions

optimal control of shear thinning fluids the aim of this paper is to establish necessary optimality conditions for optimal control problems governed by steady incompressible navier stokes equations with shear dependent viscosity the main difficulty is related to the differentiability of the control to state mapping and is overcome by introducing a family of smooth approximate control problems and by passing to the limit in the corresponding optimality conditions

combined fixed point and policy iteration for hamilton jacobi bellman equations in finance implicit methods for hamilton jacobi bellman hjb partial differential equations give rise to highly nonlinear discretized algebraic equations the classic policy iteration approach may not be efficient in many circumstances in this article we derive sufficient conditions to ensure convergence of a combined fixed point policy iteration scheme for the solution of discretized equations numerical examples are included for a singular stochastic control problem arising in insurance a guaranteed minimum withdrawal benefit where the underlying risky asset follows a jump diffusion and an american option assuming a regime switching process

howard s algorithm in a phase field topology optimization approach abstract in undetermined the topology optimization problem is formulated in a phase field approach the solution procedure is based on the allan cahn diffusion model where the conservation of volume is enforced by a global constraint the functional defining the minimization problem is selected such that no penalization is imposed for full and void materials upper and lower bounds of the density function are enforced by infinite penalty at the bounds a gradient term that introduces cost for boundaries and thereby regularizing the problem is also included in the objective functional conditions for stationarity of the functional are derived and it is shown that the problem can be stated as a variational inequality or a max min problem both defining a double obstacle problem the numerical examples used to demonstrate the method are solved using the fem whereas the double obstacle problem is solved using howard s algorithm less

exponential convergence and stability of howard s policy improvement algorithm for controlled diffusions optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system starting from an initial guess howard s policy impro

a data assimilation approach for non newtonian blood flow simulations in 3d geometries abstract blood flow simulations can play an important role in medical training and diagnostic predictions associated to several pathologies of the cardiovascular system the main challenge at the present stage is to obtain reliable numerical simulations in the particular districts of the cardiovascular system that we are interested in here we propose a data assimilation procedure in the form of a non linear optimal control problem of dirichlet type to reconstruct the blood flow profile from known data available in certain parts of the computational domain this method will allow us to obtain the boundary conditions not fully determined by the physics of the model in order to recover more accurate simulations to solve the control problem we propose a discretize then optimize do approach based on a stabilized finite element method numerical simulations on 3d geometries are performed to validate this procedure in particular we consider some idealized geometries of interest and real geometries such as a saccular aneurysm and a bypass we assume blood as an homogeneous fluid with non newtonian inelastic shear thinning behavior the results show that even in the presence of noisy data accuracy can be improved using the optimal control approach

some regularity and convergence results for parabolic hamilton jacobi bellman equations in bounded domains abstract we study parabolic hamilton jacobi bellman hjb equations in bounded domains with strong dirichlet boundary conditions we work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well posedness and regularity of a related switching system and the convergence of its components to the hjb equation in particular we show existence of a viscosity solution to the switching system by a novel construction of sub and supersolutions and application of perron s method error bounds for monotone schemes for the hjb equation are then derived from estimates near the boundary where the standard regularisation procedure for viscosity solutions is not applicable and are found to be of the same order as known results for the whole space we deduce error bounds for some common finite difference and truncated semi lagrangian schemes

convergence properties of policy iteration this paper analyzes asymptotic convergence properties of policy iteration in a class of stationary infinite horizon markovian decision problems that arise in optimal growth theory these problems have continuous state and control variables and must therefore be discretized in order to compute an approximate solution the discretization may render inapplicable known convergence results for policy iteration such as those of puterman and brumelle math oper res 4 1979 pp 60 69 under certain regularity conditions we prove that for piecewise linear interpolation policy iteration converges quadratically also under more general conditions we establish that convergence is superlinear we show how the constants involved in these convergence orders depend on the grid size of the discretization these theoretical results are illustrated with numerical experiments that compare the performance of policy iteration and the method of successive approximations

a convergent difference scheme for a class of partial integro differential equations modeling pricing under uncertainty in this paper we present a finite difference scheme to approximate viscosity solutions of a class of partial integro differential equations describing pricing under model uncertainty we establish

iterative methods for the solution of a singular control formulation of a gmwb pricing problem discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep we consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit gmwb where the underlying asset is assumed to follow a jump diffusion process we use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges our methods take advantage of the special structure of the gmwb problem in order to obtain a rapidly convergent iteration the direct control method has a scaling parameter which must be set by the user we give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round off error example computations verify that these estimates are of the correct order finally we compare the scaled direct control formulation to a formulation based on a block version of the penalty method huang and forsyth in ima j numer anal 32 320 351 2012 we show that the scaled direct control method has some advantages over the penalty method

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

accuracy of numerical solutions using the euler equation residuals in this paper we derive sorne asymptotic properties on the accuracy of numerical solutions we silow tilat the approximation error of the policy function is of the same order of magnitude as the size of the euler equation residuals moreover for bounding this approximation error tile most relevant parameters are the discount factor and the curvature of the return function these findings provide theoretical foundations for the construction of tests that can assess the performance of alternative computational methods

distributed control of systems governed by a general class of quasilinear elliptic equations abstract in this paper we are concerned with optimal control problems governed by a general class of quasilinear elliptic equations our aim is to prove existence of solution and derive optimality conditions in order to deal with this last question we begin investigating differentiability of state solution of equation with respect to the control second term of equation obtaining it only in certain cases a perturbation method is used to treat other cases as a consequence of the optimality system regularity of optimal control and state is establised

local minimization algorithms for dynamic programming equations the numerical realization of the dynamic programming principle for continuous time optimal control leads to nonlinear hamilton jacobi bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls this minimization is often performed by comparison over a finite number of elements of the control set in this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively the considered class of equations includes nonsmooth control problems with ell 1 penalization which lead to sparse controls

an efficient policy iteration algorithm for dynamic programming equations we present an accelerated algorithm for the solution of static hamilton jacobi bellman equations related to optimal control problems our scheme is based on a classic policy iteration procedure which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution this limitation often degenerates into a behavior similar to a value iteration method with an increased computation time the new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling the method starts with a coarse mesh value iteration phase and then switches to a fine mesh policy iteration procedure when a certain error threshold is reached a delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution we analyze the methods and efficient coupling in a number of examples

on the convergence of finite element methods for hamilton jacobi bellman equations we study the convergence of monotone p1 finite element methods on unstructured meshes for fully nonlinear hamilton jacobi bellman equations arising from stochastic optimal control problems with possibly degenerate isotropic diffusions using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by barles and souganidis we obtain strong uniform convergence of the numerical solutions and under nondegeneracy assumptions strong l 2 convergence of the gradients

weakly chained matrices policy iteration and impulse control this work is motivated by numerical solutions to hamilton jacobi bellman quasi variational inequalities hjbqvis associated with combined stochastic and impulse control problems in particular we consider i direct control ii penalized and iii semi lagrangian discretization schemes applied to the hjbqvi problem scheme i takes the form of a bellman problem involving an operator which is not necessarily contractive we consider the well posedness of the bellman problem and give sufficient conditions for convergence of the corresponding policy iteration to do so we use weakly chained diagonally dominant matrices which give a graph theoretic characterization of weakly diagonally dominant m matrices we compare schemes i iii under the following examples a optimal control of the exchange rate b optimal consumption with fixed and proportional transaction costs and c pricing guaranteed minimum withdrawal benefits in variable annuities we find that one should abstain from using sche