markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

valuation of stock loans with regime switching this paper is concerned with stock loan valuation in which the underlying stock price is dictated by geometric brownian motion with regime switching the stock loan pricing is quite different from that for standard american options because the associated variational inequalities may have infinitely many solutions in addition the optimal stopping time equals infinity with positive probability variational inequalities are used to establish values of stock loans and reasonable values of critical parameters such as loan sizes loan rates and service fees in terms of certain algebraic equations numerical examples are included to illustrate the results

a homotopy method for getting a local minimum of constrained nonconvex programming abstract in this paper a combined interior point homotopy method is used to solve constrained nonconvex programming problems some results for the homotopy pathway are obtained it is known that a k k t point can be obtained from this homotopy method and it proves that when the homotopy map is a regular map the k k t point must be a local minimum point on choosing a proper homotopy equation

optimal control of the risk process in a regime switching environment this paper is concerned with cost optimization of an insurance company the surplus of the insurance company is modeled by a controlled regime switching diffusion in which the regime switching mechanism provides the fluctuations of the random environment the goal is to find an optimal control that minimizes the total cost up to a stochastic exit time a weaker sufficient condition than that of fleming and soner 2006 section v 2 for the continuity of the value function is obtained further the value function is shown to be a viscosity solution of a hamilton jacobi bellman equation

portfolio selection in a two regime world abstract standard mean variance analysis is based on the assumption of normal return distributions however a growing body of literature suggests that the market oscillates between two different regimes one with low volatility and the other with high volatility in such a case even if the return distributions are normal in both regimes the overall distribution is not it is a mixture of normals mean variance analysis is inappropriate in this framework and one must either assume a specific utility function or alternatively employ the more general and distribution free second degree stochastic dominance ssd criterion this paper develops the ssd rule for the case of mixed normals the ssdmn rule this rule is a generalization the mean variance rule the cost of ignoring regimes and assuming normality when the distributions are actually mixed normal can be quite substantial it is typically equivalent to an annual rate of return of 2 3 percent

theory of globally convergent probability one homotopies for nonlinear programming for many years globally convergent probability one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective convergence theory has been derived for a few particular problems and considerable fixed point theory exists but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking this paper derives some probability one homotopy convergence theorems for unconstrained and inequality constrained optimization for linear and nonlinear inequality constraints and with and without convexity some insight is provided into why the homotopies used in engineering practice are so successful and why this success is more than dumb luck by presenting the theory as variations on a prototype probability one homotopy convergence theorem the essence of such convergence theory is elucidated

an m ary detection approach for asset allocation abstract we develop a continuous time asset allocation model which incorporates both model uncertainty and structural changes in economic conditions a dynamic m ary detection framework for a continuous time hidden markov chain partially observed in a gaussian process is used to model the price dynamics of the risky asset and the hidden states of an economy the goal of an investor is to select an optimal asset portfolio mix so as to maximize the expected utility of terminal wealth filtering theory is used first to turn the problem into one with complete observations and then to derive m ary detection filters for the hidden system the hamilton jacobi bellman dynamic programming approach is used to solve the asset allocation problem with complete observations an explicit solution is obtained for the power utility case

stock trading an optimal selling rule trading in stock markets consists of three major steps select a stock purchase a number of shares and eventually sell them to make a profit the timing to buy and sell is extremely crucial a selling rule can be specified by two preselected levels a target price and a stop loss limit this paper is concerned with an optimal selling rule based on the model characterized by a number of geometric brownian motions coupled by a finite state markov chain such a policy can be obtained by solving a set of two point boundary value differential equations moreover the corresponding expected target period and probability of making money and that of losing money are derived analytic solutions are obtained in one and two dimensional cases finally a numerical example is considered to demonstrate the effectiveness of our method

optimal investment consumption strategy with liability and regime switching model under value at risk constraint abstract this paper considers the optimal investment consumption problem with liability subject to a maximum value at risk denoted by mvar constraint the model also contains regime switching market modes whose states are interpreted as the states of the economy in each state of economy state we constrain a var value for the portfolio in a short time duration and mvar is defined as the maximum value of the vars in all economy states we suppose that both the price dynamics of the risky asset and the liability value process are governed by a markov modulated geometric brownian motion with the objective of maximizing the discounted utility of consumption we obtain a system of hjb equations corresponding to the economy states by using the dynamic programming principle then by adopting the techniques of chen et al 2010 we get explicit expressions of value functions moreover with the help of lagrange multiplier method we derive the optimal investment and the optimal consumption finally a numerical example is investigated and the effects of many parameters on the optimal investment on the optimal consumption and on var value are studied furthermore we also explore how var value affects the optimal investment and the optimal consumption

optimality conditions for a nondifferentiable minimax programming in complex spaces abstract in this paper we study a nondifferentiable minimax programming in complex spaces we establish the kuhn tucker type necessary optimality conditions and prove the existence theorem for optimality in complex programming under generalized convexity

mean variance portfolio selection under a non markovian regime switching model abstract in this paper we investigate the mean variance portfolio selection problem in a continuous time setting we assume that the coefficients in the model are random and adapted to the filtration generated by a markov chain instead of using the embedding approach which is widely adopted in the existing literature we study the problem from the viewpoint of mean field formulation and provide a distinctive and straightforward approach by introducing and discussing a new system of mean field backward stochastic differential equations driven by a markov chain we obtain both the optimal strategy and the efficient frontier in explicit forms in particular we revisit the markovian regime switching model in which the coefficients are deterministic functions of the markov chain

recursive algorithms for stock liquidation a stochastic optimization approach this work develops stochastic optimization algorithms for a class of stock liquidation problems the stock liquidation rules are based on hybrid geometric brownian motion models allowing regime changes that are modulated by a continuous time finite state markov chain the optimal selling policy is of threshold type and can be obtained by solving a set of two point boundary value problems the total number of equations to be solved is the same as the number of states of the underlying markov chain to reduce the computational burden using a stochastic optimization approach recursive algorithms are constructed to approximate the optimal threshold values convergence and rates of convergence of the algorithm are studied simulation examples are presented and the computation results are compared with the analytic solutions finally the algorithms are tested using real market data

dual method for continuous time markowitz s problems with nonlinear wealth equations continuous time mean variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method a necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique it is also shown that the optimal wealth and portfolio is the solution of a forward backward stochastic differential equation with constraints

numerical solutions of quantile hedging for guaranteed minimum death benefits under a regime switching jump diffusion formulation this work develops numerical approximation methods for quantile hedging involving mortality components for contingent claims in incomplete markets in which guaranteed minimum death benefits gmdbs could not be perfectly hedged a regime switching jump diffusion model is used to delineate the dynamic system and the hedging function for gmdbs where the switching is represented by a continuous time markov chain using markov chain approximation techniques a discrete time controlled markov chain with two component is constructed under simple conditions the convergence of the approximation to the value function is established examples of quantile hedging model for guaranteed minimum death benefits under linear jumps and general jumps are also presented

solving the karush kuhn tucker system of a nonconvex programming problem on an unbounded set abstract in the papers g c feng b yu combined homotopy interior point method for nonlinear programming problems in h fujita m yamaguti eds advances in numerical mathematics proceedings of the second japan china seminar on numerical mathematics in lecture notes in numerical and applied analysis vol 14 kinokuniya tokyo 1995 pp 9 16 g c feng z h lin b yu existence of an interior pathway to a karush kuhn tucker point of a nonconvex programming problem nonlinear analysis 32 1998 761 768 z h lin b yu g c feng a combined homotopy interior point method for convex programming problem applied mathematics and computation 84 1997 193 211 a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition however for when the feasible set is not bounded no result has so far been obtained in this paper a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered under suitable additional assumptions boundedness of the homotopy path and hence global convergence is proven

superreplication under gamma constraints in a financial market consisting of a nonrisky asset and a risky one we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint this is a constraint on the unbounded variation part of the hedging portfolio we first consider the case in which the prices are given as general markov diffusion processes and prove a verification theorem which characterizes the superreplication cost as the unique solution of a quasi variational inequality in the context of the black scholes model i e when volatility is constant this theorem allows us to derive an explicit solution of the problem these results are based on a new dynamic programming principle for general stochastic target problems

spectral method for analysis of switching diffusions the analysis problem of switching diffusions is considered this paper presents a new approach based on the spectral method formalism for solving generalized fokker planck equations the proposed method allows to transform partial differential equations into the linear algebraic equations and to arrive at a solution in an explicit form the aspects of applications are discussed a numerical example is given to illustrate the efficiency of the proposed method

approximation pricing and the variance optimal martingale measure let x be a semimartingale and let be the space of all predictable x integrable processes v such that so dx is in the space s 2 of semi martingales we consider the problem of approximating a given random variable h l 2 p by the sum of a constant c and a stochastic integral s 0 t v s dx s with respect to the l 2 p norm this problem comes from financial mathematics where the optimal constant v 0 can be interpreted as an approximation price for the contingent claim h an elementary computation yields v 0 as the expectation of h under the variance optimal signed martingale measure p and this leads us to study p in more detail in the case of finite discrete time we explicitly construct p by backward recursion and we show that p is typically not a probability but only a signed measure in a continuous time framework the situation becomes rather different we prove that p is nonnegative if x has continuous paths and satisfies a very mild no arbitrage condition as an application we show how to obtain the optimal integrand in feedback form with the help of a backward stochastic differential equation

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

a mean variance optimization problem for discounted markov decision processes in this paper we consider a mean variance optimization problem for markov decision processes mdps over the set of deterministic stationary policies different from the usual formulation in mdps we aim to obtain the mean variance optimal policy that minimizes the variance over a set of all policies with a given expected reward for continuous time mdps with the discounted criterion and finite state and action spaces we prove that the mean variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and markov properties then we show that a mean variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations we also address related issues such as a mutual fund theorem and illustrate our results with an example

optimal portfolio selection with liability management and markov switching under constrained variance in this paper we mainly discuss an optimal portfolio selection model with liability management and markov switching which maximize the expected final surplus under constrained variance because linear quadratic control is a basic method for the m v problem in this paper we begin with the general stochastic linear quadratic model and obtain the optimal solution of the problem exactly the analytical optimal portfolio strategy is derived in this paper furthermore we demonstrate that a special case is consistent with those results of chiu and li 2006 3

discrete time mean variance optimal control of linear systems with markovian jumps and multiplicative noise in this article we consider the stochastic optimal control problem of discrete time linear systems subject to markov jumps and multiplicative noise under three kinds of performance criterions related to the final value of the expectation and variance of the output in the first problem it is desired to minimise the final variance of the output subject to a restriction on its final expectation in the second one it is desired to maximise the final expectation of the output subject to a restriction on its final variance and in the third one it is considered a performance criterion composed by a linear combination of the final variance and expectation of the output of the system we present explicit sufficient conditions for the existence of an optimal control strategy for these problems generalising previous results in the literature we conclude this article presenting a numerical example of an asset liabilities management model for pension funds with regime switching

parametric duality on minimax programming involving generalized convexity in complex space in this paper we employ generalized convexity of complex functions to establish several sufficient optimality conditions for minimax programming in complex spaces using such criteria we constitute a parametrical dual and establish the weak strong and strict converse duality theorems in the framework

a combined homotopy interior point method for convex nonlinear programming abstract in this paper we present a new interior point method combined homotopy interior point method chip method for convex nonlinear programming without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set we prove that for any 0 an solution of the problem can be obtained by the chip method to our knowledge strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well known center path following method therefore the chip method essentially reduces the assumptions of the center path following method and can be applied to more general problems

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

optimal portfolios with regime switching and value at risk constraint we consider the optimal portfolio selection problem subject to a maximum value at risk mvar constraint when the price dynamics of the risky asset are governed by a markov modulated geometric brownian motion gbm here the market parameters including the market interest rate of a bank account the appreciation rate and the volatility of the risky asset switch over time according to a continuous time markov chain whose states are interpreted as the states of an economy the mvar is defined as the maximum value of the vars of the portfolio in a short time duration over different states of the chain we formulate the problem as a constrained utility maximization problem over a finite time horizon by utilizing the dynamic programming principle we shall first derive a regime switching hamilton jacobi bellman hjb equation and then a system of coupled hjb equations we shall employ an efficient numerical method to solve the system of coupled hjb equations for the optimal constrained portfolio we shall provide numerical results for the sensitivity analysis of the optimal portfolio the optimal consumption and the var level with respect to model parameters these results are also used to investigating the effect of the switching regimes

optimal mean variance control for discrete time linear systems with markovian jumps and multiplicative noises in this paper we consider the stochastic optimal control problem of discrete time linear systems subject to markov jumps and multiplicative noises under two criteria the first one is an unconstrained mean variance trade off performance criterion along the time and the second one is a minimum variance criterion along the time with constraints on the expected output we present explicit conditions for the existence of an optimal control strategy for the problems generalizing previous results in the literature we conclude the paper by presenting a numerical example of a multi period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor

optimality and duality for nonsmooth multiobjective fractional programming problems involving exponential v r invexity abstract this paper is to study a subdifferentiable multiobjective fractional programming problem under exponential v r invexity we establish sufficient optimality conditions for multiobjective fractional programming problems involving exponential v r invex lipschitz functions by optimality conditions the parametric dual model is formulated consequently the duality theorems are proved so that the optimal values of the duality problems are equal to the primary problem under the framework of exponential v r invexity for the lipschitz function

almost sure stabilization for feedback controls of regime switching linear systems with a hidden markov chain this work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown markov chain the control system displays continuous dynamics represented by differential equations and discrete events given by a hidden markov chain in the previous investigation on this class of problems averaging criteria were used which provides only the system behavior in some expectation sense a closer scrutiny of the system behavior necessarily requires the consideration of sample path properties different from previous work on stabilization of adaptive controlled systems with a hidden markov chain where average criteria were considered this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes under simple conditions it is shown that as long as the feedback controls have linear growth in the continuous component the resulting process is regular moreover by appropriate choice of the lyapunov functions it is shown that the adaptive system is stabilizable almost surely as a by product it is also established that the controlled process is positive recurrent

optimality conditions for nondifferentiable minimax fractional programming with complex variables we show that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space we establish the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities

duality for nondifferentiable minimax programming in complex spaces abstract employing the optimality necessary and sufficient conditions of a nondifferentiable minimax programming problem in complex spaces we formulate a one parametric dual and a parameter free dual problems on both dual problems we establish three duality theorems weak strong and strict converse duality theorem and prove that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem

duality for a class of nondifferentiable mathematical programming problems in complex space craven and mond 1 have given fritz john type necessary conditions for a class of nonlinear programming problems in complex space over polyhedral cones here we derive the necessary and sufficient conditions for the static minimax problems in complex space which are extensions of the corresponding real space conditions of 4 these conditions are then used to extend some duality results of 2 for a general class of nondifferentiable programming problems to complex space over arbitrary polyhedral cones the complex minimax problem that we consider seeks to choose e so e c g c e s which minimizes

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

a lattice method for option pricing with two underlying assets in the regime switching model abstract in this work we develop an efficient lattice approach for option pricing with two underlying assets whose prices are governed by regime switching models jump amplitudes are specified in a way such that the lattice achieves complete node recombination along each asset variable and grows quadratically as the number of time steps increases jump probabilities are obtained by solving a related quadratic programming problem the weak convergence of the discrete lattice approximations to the continuous time regime switching diffusion processes is established the lattice is employed to price both european and american options written on the maximum and minimum of two assets in different regimes numerical results are provided and compared for the european options with a monte carlo simulation approach

robust m ary detection filters and smoothers for continuous time jump markov systems in this paper we consider a dynamic m ary detection problem when markov chains are observed through a wiener process these systems are fully specified by a candidate set of parameters whose elements are a rate matrix for the markov chain and a parameter for the observation model further we suppose these parameter sets can switch according to the state of an unobserved markov chain and thereby produce an observation process generated by time varying jump stochastic parameter sets given such an observation process and a specified collection of models we estimate the probabilities of each model parameter set explaining the observation by defining a new augmented state process then applying the method of reference probability we compute matrix valued dynamics whose solutions estimate joint probabilities for all combinations of candidate model parameter sets and values taken by the indirectly observed state process these matrix valued dynamics satisfy a stochastic integral equation with a wiener process integrator using the gauge transformation techniques introduced by clark and a pointwise matrix product we compute robust matrix valued dynamics for the joint probabilities on the augmented state space in these new dynamics the observation wiener process appears as a parameter matrix in a linear ordinary differential equation rather than an integrator in a stochastic integral equation it is shown that these robust dynamics when discretised enjoy a deterministic upper bound which ensures nonnegative probabilities for any observation sample path in contrast no such upper bounds can be computed for taylor expansion approximations such as the euler maryauana and milstein schemes finally by exploiting a duality between causal and anticausal robust detector dynamics we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations a computer simulation demonstrating performance is included

indefinite quadratic with linear costs optimal control of markov jump with multiplicative noise systems in this paper we consider the stochastic optimal control problem of discrete time markov jump with multiplicative noise linear systems the performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables the weighting matrices of the state and control for the quadratic part are allowed to be indefinite we present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized riccati difference equations interconnected with a set of coupled linear recursive equations for the case in which the quadratic term matrices are non negative this necessary and sufficient condition can be written in a more explicit way the results are applied to a problem of portfolio optimization