fuzzy differential games for nonlinear stochastic systems suboptimal approach a fuzzy differential game theory is proposed to solve the n person or n player nonlinear differential noncooperative game and cooperative game team problems which are not easily tackled by the conventional methods in the paper both noncooperative and cooperative quadratic differential games are considered first the nonlinear stochastic system is approximated by a fuzzy model based on the fuzzy model a fuzzy controller is proposed to deal with the noncooperative differential game in the sense of nash equilibrium strategies or with the cooperative game in the sense of pareto optimal strategies using a suboptimal approach the outcomes of the fuzzy differential games for both the noncooperative and the cooperative cases are parameterized in terms of an eigenvalue problem since the state variables are usually unavailable a suboptimal fuzzy observer is also proposed in this study to estimate the states for these differential game problems finally simulation examples are given to illustrate the design procedures and to indicate the performance of the proposed methods

uncertain statistical inference models with imprecise observations the observations of some samples from the population of a probability density function with unknown parameters are usually imprecise due to various reasons by employing uncertain variables to model these imprecise observations this paper proposes an interdiscipline called the uncertain statistical inference which is composed of statistical inference and uncertainty theory it presents three types of statistic inference problems with imprecise observations that are the point estimation the hypothesis test and the interval estimation then it proposes a method of moments and a method of likelihood function maximum likelihood estimation for the first problem and a method of likelihood ratio function for the other two problems

linear quadratic differential games saddle point and riccati differential equation zhang siam j control optim 43 2005 pp 2157 2165 recently established the equivalence between the finiteness of the open loop value of a two player zero sum linear quadratic lq game and the finiteness of its open loop lower and upper values in this paper we complete and sharpen the results of zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition 0 0 is a solution of the open loop lower value of the game for the zero initial state then we show that under the assumption of an open loop saddle point in the time horizon 0 t for all initial states there is an open loop saddle point in the time horizon s t for all initial times s 0 le s t and all initial states at time s from this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function p s and show that it is an h 1 0 t solution of the matrix riccati differential equation thence an open loop saddle point in 0 t yields closed loop optimal strategies for both players furthermore a necessary and sufficient set of conditions for the existence of an open loop saddle point in 0 t for all initial states is the convexity concavity of the utility function and the existence of an h 1 0 t symmetrical solution to the matrix riccati differential equation as an illustration of the cases where the open loop lower upper value of the game is infty infty we work out two informative examples of solutions to the riccati differential equation with and without blow up time

the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

large time asymptotic and numerical solution of a nonlinear diffusion model with memory large time behavior of solutions and finite difference approximation of a nonlinear system of integro differential equations associated with the penetration of a magnetic field into a substance is studied two initial boundary value problems are investigated the first with homogeneous conditions on whole boundary and the second with nonhomogeneous boundary data on one side of lateral boundary the rates of convergence are also given mathematical results presented show that there is a difference between stabilization rates of solutions with homogeneous and nonhomogeneous boundary conditions the convergence of the corresponding finite difference scheme is also proved the decay of the numerical solution is compared with the analytical results

numerical approach for solution to an uncertain fractional differential equation abstract uncertain fractional differential equation ufde is of importance tool for the description of uncertain dynamic systems generally we may not obtain its analytic solutions in most cases this paper focuses on proposing a numerical method for solving ufde involving caputo derivative first the concept of path to an ufde with initial value conditions is introduced which is a solution of the corresponding fractional differential equation fde involving with the same initial value conditions then the relations between its solution and associate path are investigated besides a formula is derived for calculating expected value of a monotonic function with respect to solutions of ufdes based on the established relations numerical algorithms are designed finally some numerical experiments of nonlinear ufdes are given to demonstrate the effectiveness of the numerical algorithms

unconditionally stable numerical method for a nonlinear partial integro differential equation the paper presents an unconditionally stable numerical scheme to solve a nonlinear integro differential equation which arises in mathematical modeling of the penetration of a magnetic field into a substance if the temperature is kept constant throughout the material numerical scheme comprises of the galerkin finite element method jangveladze and kiguradze 2011 for the spatial discretization followed by an implicit finite difference scheme for the time stepping we extended the results for stability estimates to a non homogeneous problem and derived optimal order error estimates for the semidiscretized and fully discretized equations using h 0 1 projection further to show the efficiency the proposed numerical method is demonstrated via numerical example

a banzhaf function for a fuzzy game in this paper another important solution for fuzzy games called the banzhaf function is investigated first a sim plified expression for the banzhaf function for games with fuzzy coalition is proposed the relationship between this simplified ex pression and two kinds of particular banzhaf functions are dis cussed in detail furthermore in the framework of games with fuzzy coalitions we give a definition of the potential function of fuzzy games and prove that the marginal values of the potential function are the components of the banzhaf function further more a new axiomatic characterization of the banzhaf function for games with fuzzy coalitions is provided

solving high order uncertain differential equations via runge kutta method high order uncertain differential equations are used to model differentiable uncertain systems with high order differentials and how to solve the high order uncertain differential equations is a core issue in practice this paper aims at proposing a numerical method to solve the high order uncertain differential equations based on the runge kutta recursion formula which is of high precision degree a procedure is designed and some numerical experiments are performed to illustrate the effectiveness and efficiency of the runge kutta method in addition this paper also presents a numerical method to calculate the expected value of a function of the solution

optimal control of stochastic hybrid system with jumps a numerical approximation the generalized class of stochastic hybrid systems consists of models with regime changes including the occurrence of impulsive behavior in this paper the stochastic hybrid processes with jumps are approximated by locally consistent markov decision processes that preserve local mean and covariance we further apply a randomized switching policy for approximating the dynamics on the switching boundaries to investigate the validity of the approximation we study a stochastic optimal control problem on the basis of the discrete approximation we solve the optimal control problem and show that the solution obtained from the discretized problem converges to the solution of the original optimal control problem

finite difference approximation of a nonlinear integro differential system finite difference approximation of the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied the convergence of the finite difference scheme is proved the rate of convergence of the discrete scheme is given the decay of the numerical solution is compared with the analytical results proven earlier

uncertain alternating renewal process and its application uncertain process is a sequence of uncertain variables indexed by time and space first this paper presents a kind of uncertain process known as the uncertain alternating renewal process whose alternating interarrival times are uncertain variables then it proves an uncertain alternating renewal theorem on the limit value of average working rate finally an application of the alternating renewal theorem is discussed

on the length of simplex paths the assignment case this paper reports on an experimental study of the distribution of the length of simplex paths for the optimal assignment problem we study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis in this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients provided these are sufficiently spread out further the mean and standard deviation grow only moderately with the size of the problem namely asd1 8 andd1 5 respectively for ad d problem implying in particular that the pivot counts concentrate around the mean with growingd the usual simplex method on the other hand gives a growth ofd1 6 hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average at least for assignment problems

linearly localized difference schemes for the nonlinear maxwell model of a magnetic field into a substance abstract a linearly localized difference scheme with the first order time approximation is proposed for solving a nonlinear maxwell model associated with the penetration of a magnetic field into a substance the new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast thomas algorithm without any newton type inner iterations it is also local in time that is only numerical solutions in one previous time level are necessary to update the current solutions such that it requires much less storage compared with the fully implicit method furthermore the exponential decaying behavior of difference solution which is analogous to that of the continuous solution is obtained to improve the time accuracy we apply the crank nicolson type time discretization to construct a second order linearly localized method numerical examples are presented to support our theoretical results

shapley mappings and the cumulative value for n person games with fuzzy coalitions abstract in this paper we prove existence and uniqueness of the so called shapley mapping which is a solution concept for a class of n person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions the shapley mapping when it exists associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency the symmetry and the null player conditions it determines a cumulative value that is the sum of all coalitional allocations for whose computation we provide an explicit formula

global smooth solutions for a class of parabolic integrodifferential equations the existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved if the integral kernel behaves like t near t 0 o with a 2 3 an existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for o

some results on two person zero sum linear quadratic differential games considered in this paper are the two person zero sum linear quadratic differential games it is shown that the value of the game exists if and only if both the upper and lower values exist as a consequence we prove that another necessary and sufficient condition for the existence of the values of the game is the existence of an open loop open loop saddle point an example is also given in which the lower value exists but the upper value does not exist

uncertain calculus with yao process uncertain calculus is a branch of mathematics that deals with the differentiation and integration of functions of uncertain processes this paper investigates the yao process defined by the yao integral and extends the uncertain calculus on the yao process some important results are developed for finite variation and linearity of integral moreover this paper proposes a concept of the multifactor yao process and also gives a fundamental theorem for it

adaptive dynamic programming for discrete time zero sum games in this paper a novel adaptive dynamic programming adp algorithm called iterative zero sum adp algorithm is developed to solve infinite horizon discrete time two player zero sum games of nonlinear systems the present iterative zero sum adp algorithm permits arbitrary positive semidefinite functions to initialize the upper and lower iterations a novel convergence analysis is developed to guarantee the upper and lower iterative value functions to converge to the upper and lower optimums respectively when the saddle point equilibrium exists it is emphasized that both the upper and lower iterative value functions are proved to converge to the optimal solution of the zero sum game where the existence criteria of the saddle point equilibrium are not required if the saddle point equilibrium does not exist the upper and lower optimal performance index functions are obtained respectively where the upper and lower performance index functions are proved to be not equivalent finally simulation results and comparisons are shown to illustrate the performance of the present method

adams simpson method for solving uncertain differential equation uncertain differential equation is a type of differential equation driven by canonical liu process how to obtain the analytic solution of uncertain differential equation has always been a thorny problem in order to solve uncertain differential equation early researchers have proposed two numerical algorithms based on euler method and runge kutta method this paper will design another numerical algorithm for solving uncertain differential equations via adams simpson method meanwhile some numerical experiments are given to illustrate the efficiency of the proposed numerical algorithm furthermore this paper gives how to calculate the expected value the inverse uncertainty distributions of the extreme value and the integral of the solution of uncertain differential equation with the aid of adams simpson method

on the diameter of partition polytopes and vertex disjoint cycle cover we study the combinatorial diameter of partition polytopes a special class of transportation polytopes they are associated to partitions of a set x x1 xn of items into clusters c1 ck of prescribed sizes 1 k we derive upper bounds on the diameter in the form of 1 2 n 1 and lfloor frac n 2 rfloor this is a direct generalization of the diameter 2 result for the birkhoff polytope the bounds are established using a constructive graph theoretical approach where we show that special sets of vertices in graphs that decompose into cycles can be covered by a set of vertex disjoint cycles further we give exact diameters for partition polytopes with k 2 or k 3 and prove that for all k 4 and all 1 2 there are cluster sizes 3 k such that the diameter of the corresponding partition polytope is at least lceil frac 4 3 kappa 2 rceil finally we provide an o n kappa 1 kappa 2 sqrt k 1 algorithm for an edge walk connecting two given vertices of a partition polytope that also adheres to our diameter bounds

stability in distribution for uncertain delay differential equation abstract as a type of differential equation uncertain delay differential equation is driven by liu process stability in measure stability in mean and stability in moment for uncertain delay differential equation have been proposed this paper mainly gives a concept of stability in distribution and proves a sufficient condition for uncertain delay differential equation being stable in distribution as a supplement moreover this paper further discusses their relationships among stability in distribution stability in measure stability in mean and stability in moment

transportation problems and simplicial polytopes that are not weakly vertex decomposable provan and billera defined the notion of weak k decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes they showed the diameter of a weakly k decomposable simplicial complex is bounded above by a polynomial function of the number of k faces in and its dimension for weakly 0 decomposable complexes this bound is linear in the number of vertices and the dimension in this paper we exhibit the first examples of non weakly 0 decomposable simplicial polytopes our examples are in fact polar to certain transportation polytopes

a strongly polynomial algorithm for a concave production transportation problem with a fixed number of nonlinear variables we show that the production transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time the algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem monotonicity of the objective function along certain directions small proportion of nonlinear variables and combinatorial properties implied by transportation constraints

uncertain wave equation with infinite half boundary wave equation is a type of second order and hyperbolic partial differential equation it is a commonly used tool to model many kinds of wave propagations such as sound wave electromagnetic wave water wave and string vibration propagations similarly uncertain wave equation is a type of uncertain partial equation driven by liu process which is widely used to model the wave propagation with uncertain noise such as vibrating string in uncertain environment the existing literature has studied uncertain wave equation with infinite boundary since infinite boundary is a much ideal condition this paper aims at studying the uncertain wave equation with infinite half boundary

some results about uncertain differential equations with time dependent delay abstract uncertain differential equations with time dependent delay are a type of differential equations driven by liu process this paper mainly proves that this type of uncertain differential equations have unique solutions in the infinite domain under some conditions in addition we also focus on the stability of uncertain differential equations with time dependent delay and some correspondence results are shown in the form of theorems

large time behavior of solutions to a nonlinear integro differential system abstract asymptotic behavior of solutions as t to the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied initial boundary value problems with two kinds of boundary data are considered the first with homogeneous conditions on whole boundary and the second with non homogeneous boundary data on one side of lateral boundary the rates of convergence are given too

galerkin finite element method for one nonlinear integro differential model galerkin finite element method for the approximation of a nonlinear integro differential equation associated with the penetration of a magnetic field into a substance is studied first type initial boundary value problem is investigated the convergence of the finite element scheme is proved the rate of convergence is given too the decay of the numerical solution is compared with the analytical results

finite difference scheme for one system of nonlinear partial integro differential equations abstract system of maxwell equations is considered reduction to the integro differential form is given existence uniqueness and large time behavior of solutions of the initial boundary value problem for integro differential model with two component and one dimensional case are studied finite difference scheme is investigated wider class of nonlinearity is studied than one has been investigated before freefem realization code and results of numerical experiments are given

quadratic diameter bounds for dual network flow polyhedra both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best case performance of linear programming algorithms we study the diameters of dual network flow polyhedra associated to b flows on directed graphs g v e g v e and prove quadratic upper bounds for both of them the minimum of v 1 cdot e v 1 e and frac 1 6 v 3 16 v 3 for the combinatorial diameter and frac v cdot v 1 2 v v 1 2 for the circuit diameter previously bounds on these diameters have only been known for bipartite graphs the situation is much more involved for general graphs in particular we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant

on a class of volterra nonlinear equations of parabolic type in this paper we show the backward uniqueness in time of solution for a class of volterra nonlinear equations of parabolic type we also give reasonable physical interpretation for our conclusion

reliability analysis of uncertain weighted k out of n systems the uncertain variable is used to model a quantity under human uncertainty and the weighted k out of n system is used to model a system of n components which functions if and only if the total weights of functioning components is greater than k considering the human uncertainty in operating the system this paper introduces the uncertain variable to the weighted k out of n system and proposes a concept of uncertain weighted k out of n system some formulas are derived to calculate the reliability index of such a system as a generalization this paper also studies an uncertain weighted k out of n system whose weights are estimated by experts and modeled by uncertain variables instead of crisp numbers in addition this paper analyzes the importance measure of the components in the uncertain weighted k out of n systems

an infeasible exterior point simplex algorithm for assignment problems the so called modified hung rom algorithm based upon theoretical considerations of hirsch paths seems to be one of the most efficient algorithms for assignment problems since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest this paper presents an algorithm of this kind for assignment problems

numerical method for solving uncertain spring vibration equation abstract as a type of uncertain differential equations uncertain spring vibration equation is driven by liu process this paper proposes a concept of path and shows that the solution of an uncertain spring vibration equation can be expressed by a family of solutions of second order ordinary differential equations this paper also proves that the inverse uncertainty distribution of solution of uncertain spring vibration equation is just the path of uncertain spring vibration equation and a numerical algorithm is designed moreover a formula to calculate the expected value of solution of uncertain spring vibration equation is derived finally several numerical examples are provided to illustrate the efficiency of the numerical method

the diameters of network flow polytopes satisfy the hirsch conjecture we solve a problem in the combinatorics of polyhedra motivated by the network simplex method we show that the hirsch conjecture holds for the diameter of the graphs of all network flow polytopes in particular the diameter of a network flow polytope for a network with n nodes and m arcs is never more than m n 1 a key step to prove this is to show the same result for classical transportation polytopes

global existence of solutions of volterra integrodifferential equations of parabolic type abstract sufficient conditions for the global existence of a strong solution of the equation u t t x t 0 k t s u x s x x ds f t x are given the kernel k satisfies r k z i k z and is increasing with sup p

finite differences versus finite elements for solving nonlinear integro differential equations consider the nonlinear integro differential equation ut x t 0t a t x ux x d f x t 0 x 0 t t with appropriate initial and boundary conditions this problem serves as a model for one dimensional heat flow in materials with memory the numerical solution via finite elements was discussed in b neta j math anal appl 89 1982 598 611 in this paper we compare the results obtained there with finite difference approximation from the point of view of accuracy and computer storage it turns out that the finite difference method yields comparable results for the same mesh spacing using less computer storage

solving uncertain heat equation via numerical method abstract uncertain heat equation is a type of uncertain partial differential equations driven by liu processes this paper proposes a concept of path for uncertain heat equation and shows that the solution of an uncertain heat equation can be represented by a family of solutions of ordinary heat equations and a formula is derived to calculate expected value of solution of uncertain heat equation moreover a numerical method is designed to solve uncertain heat equation several examples are given to illustrate the efficiency of the numerical method

cooperative fuzzy games we derive versions of known results dealing with core equilibria and shapley values of cooperative games in the case of cooperative fuzzy games i e games defined on fuzzy subsets of the set of n players a fuzzy coalition is an n vector i associating with each player i his rate of participation i 0 1 in the fuzzy coalition and the real number v is the coalition worth assumed to be positively homogeneous if also v is superadditive or equivalently concave the fuzzy game with side payments has a nonempty core associated with the coalition worth function for any ordinary game with side payments is a fuzzy extension thereof viz the fuzzy game whose coalition worth function is the least positively homogeneous superadditive function majorizing the coalition worth function of the original game the fuzzy extension always has a nonempty core moreover if the original game has a nonempty core it coincides with that of its fuzzy extension analogous results are established for games without side payments an axiomatization of values of fuzzy games with side payments is also given the results are applied to show that the set of walras equilibria coincides with the fuzzy core of an economy

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

signature methods for the assignment problem the signature of a dual feasible basis of the assignment problem is an n vector whose ith component is the number of nonbasic activities of type i j this paper uses signatures to describe a method for finding optimal assignments that terminates in at most n 1n 2 2 pivot steps and takes at most on3 work

a counterexample to the hirsch conjecture the hirsch conjecture 1957 stated that the graph of a d dimensional polytope with n facets cannot have combinatorial diameter greater than n d that is any two vertices of the polytope can be connected by a path of at most n d edges this paper presents the rst counterexample to the conjecture our polytope has dimension 43 and 86 facets it is obtained from a 5 dimensional polytope with 48 facets that violates a certain generalization of the d step conjecture of klee and walkup

hurwicz model of uncertain linear quadratic optimal control with jump abstractbased on hurwicz model of uncertain optimal control model with jump in this paper a hurwicz model of uncertain linear quadratic optimal control with jump is proposed then the necessary

an uncertain optimal control model with hurwicz criterion a new uncertain optimal control model based on the hurwicz criterion is introduced to design dynamic optimization problems applying the method of dynamic programming the principle of optimality is presented and then the equation of optimality is given to solve the proposed model as an application a manufacturing technology diffusion problem is discussed by using the equation obtained

fenchel type duality for matroid valuations the weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem given a pair of valuated matroidsmi v i i i 1 2 defined on a common ground setv find a common baseb 1 2 that maximizes 1 b 2 b this paper develops a fenchel type duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming a fenchel type min max theorem and a discrete separation theorem are given furthermore the subdifferentials of matroid valuations are investigated 1998 the mathematical programming society inc published by elsevier science b v

numerical solution of a nonlinear integro differential equation abstract galerkin s method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro differential equation u t x t 0 t a t x u x x d f x t 0 an error estimate in a suitable norm will be derived for the difference u uh between the exact solution u and the approximant uh it turns out that the rate of convergence of uh to u as h 0 is optimal this result was confirmed by the numerical experiments

least squares estimation in uncertain differential equations uncertain differential equations are a type of differential equations driven by liu processes how to estimate the parameters in an uncertain differential equation based on the observed data is a crucial problem in the real applications of these equations by means of the least squares estimation this article proposes a principle of minimum noise as an approach to the problem following this principle the estimates of the parameters in some special types of uncertain differential equations are derived which are represented as functions of the observed data in addition some numerical experiments are performed to illustrate the principle

dynamic programming for a markov switching jump diffusion abstract we consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a markov switching jump diffusion stochastic differential equation our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems more specifically we provide a detailed proof of bellman s optimality principle or dynamic programming principle and obtain the corresponding hamilton jacobi belman equation which turns out to be a partial integro differential equation due to the extra terms arising from the levy process and the markov process as an application of our results we study a finite horizon consumption investment problem for a jump diffusion financial market consisting of one risk free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state markov process we provide a detailed study of the optimal strategies for this problem for the economically relevant families of power utilities and logarithmic utilities

linear quadratic uncertain differential game with application to resource extraction problem uncertain differential game investigates interactive decision making of players over time and the system dynamics is described by an uncertain differential equation this paper goes further to study the two player zero sum uncertain differential game in order to guarantee the saddle point nash equilibrium a max min theorem is provided furthermore when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic the existence of saddle point nash equilibrium is obtained via the solvability of a corresponding riccati equation finally a resource extraction problem is analyzed by using the theory proposed in this paper