utility maximization with habit formation dynamic programming and stochastic pdes this paper studies the habit forming preference problem of maximizing total expected utility from consumption net of the standard of living a weighted average of past consumption we describe the effective state space of the corresponding optimal wealth and standard of living processes identify the associated value function as a generalized utility function and exploit the interplay between dynamic programming and feynman kac results via the theory of random fields and stochastic partial differential equations spdes the resulting value random field of the optimization problem satisfies a nonlinear backward spde of parabolic type widely referred to as the stochastic hamilton jacobi bellman equation the dual value random field is characterized further in terms of a backward parabolic spde which is linear progressively measurable versions of stochastic feedback formulae for the optimal portfolio and consumption choices are obtained as well

the obstacle problem for quasilinear stochastic pdes analytical approach we prove existence and uniqueness of the solution of quasilinear stochas tic pdes with obstacle our method is based on analytical technics coming from the parabolic potential theory the solution is expressed as pair u where u is a pre dictable continuous process which takes values in a proper sobolev space andis a random regular measure satisfying minimal skohorod condition

reflected solutions of backward sde s and related obstacle problems for pde s we study reflected solutions of one dimensional backward stochastic differential equations the reflection keeps the solution above a given stochastic process we prove uniqueness and existence both by a fixed point argument and by approximation via penalization we show that when the coefficient has a special form then the solution of our problem is the value function of a mixed optimal stopping optimal stochastic control problem we finally show that when put in a markovian framework the solution of our reflected bsde provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

convergence of consistent and inconsistent finite difference schemes and an acceleration technique this paper states and generalizes in part some recent results on finite difference methods for dirichlet problems in a bounded domain which the author has obtained by himself or with coworkers after stating a superconvergence property of finite difference solution for the case where the exact solution u belongs to c4 it is remarked that such a property does not hold in general if u c4 next a convergence theorem is given for inconsistent schemes under some assumptions furthermore it is shown that the accuracy of the approximate solution can be improved by a coordinate transformation numerical examples are also given

on the ito wentzell formula for distribution valued processes and related topics we prove the ito wentzell formula for processes with values in the space of generalized functions by using the stochastic fubini theorem and the ito wentzell formula for real valued processes appropriate versions of which are also proved

on semi linear degenerate backward stochastic partial differential equations in this paper we study a class of one dimensional degenerate semilinear backward stochastic partial differential equations bspdes for short of parabolic type by establishing some new a priori estimates for both linear and semilinear bspdes we show that the regularity and uniform boundedness of the adapted solution to the semilinear bspde can be determined by those of the coefficients a special feature that one usually does not expect from a stochastic differential equation the proof follows the idea of the so called bootstrap method which enables us to analyze each of the derivatives of the solution under consideration some related results including some comparison theorems of the adapted solutions for semilinear bspdes as well as a nonlinear stochastic feynman kac formula are also given

a self adaptive method for solving general mixed variational inequalities abstract the general mixed variational inequality containing a nonlinear term is a useful and an important generalization of variational inequalities the projection method cannot be applied to solve this problem due to the presence of the nonlinear term to overcome this disadvantage noor m a noor pseudomonotone general mixed variational inequalities appl math comput 141 2003 529 540 used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities in this paper we present a new self adaptive iterative method which can be viewed as a refinement and improvement of the method of noor global convergence of the new method is proved under the same assumptions as noor s method some preliminary computational results are given

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

a non markovian liquidation problem and backward spdes with singular terminal conditions we establish the existence uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition the equation describes the value function of non markovian stochastic optimal control problem in which the terminal state of the controlled process is prespecified the analysis of such control problems is motivated by models of optimal portfolio liquidation

the primal douglas rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem we apply the douglas rachford splitting algorithm to a class of multi valued equations consisting of the sum of two monotone mappings compared with the dual application of the same algorithm which is known as the alternating direction method of multipliers the primal application yields algorithms that seem somewhat involved however the resulting algorithms may be applied effectively to problems with certain special structure in particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

optimal stopping problem for stochastic differential equations with random coefficients an optimal stopping problem for stochastic differential equations with random coefficients is considered the dynamic programming principle leads to a hamiltion jacobi bellman equation which for the current case is a backward stochastic partial differential variational inequality bspdvi for short for the value function well posedness of such a bspdvi is established and a verification theorem is proved

wm p solution p 2 of linear degenerate backward stochastic partial differential equations in the whole space abstract in this paper we consider the backward cauchy problem of linear degenerate stochastic partial differential equations we obtain the existence and uniqueness results in sobolev space l p c 0 t w m p with both m 1 and p 2 being arbitrary without imposing the symmetry condition for the coefficient of the gradient of the second unknown which was introduced by ma and yong 1999 21 in the case of p 2 to illustrate the application we give a maximum principle for optimal control of degenerate stochastic partial differential equations

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

a combined relaxation method for variational inequalities with nonlinear constraints a simple iterative method for solving variational inequalities with a set valued nonmonotone mapping and a convex feasible set is proposed this set can be defined by nonlinear functions the method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization also a modification of the averaging method for the problem under consideration is proposed 1998 the mathematical programming society inc published by elsevier science b v

backward stochastic differential equations with reflection and dynkin games we establish existence and uniqueness results for adapted solutions of backward stochastic differential equations bsde s with two reflecting barriers generalizing the work of el karoui kapoudjian pardoux peng and quenez existence is proved first by solving a related pair of coupled optimal stopping problems and then under different conditions via a penalization method it is also shown that the solution coincides with the value of a certain dynkin game a stochastic game of optimal stopping moreover the connection with the backward sde enables us to provide a pathwise deterministic approach to the game

stochastic hamilton jacobi bellman equations this paper studies the following form of nonlinear stochastic partial differential equation begin gathered d phi t mathop inf v in u left frac 1 2 sum i j left sigma sigma right ij x v t partial x i x j phi t x sum i b i x v t partial x i phi t x l x v t right hfill qquad qquad left sum i j sigma ij x v t partial x i psi j t x right dt psi t x dw t quad phi t x h x hfill end gathered where the coefficients sigma ij b i l and the final datum h may be random the problem is to find an adapted pair phi psi x t uniquely solving the equation the classical hamilton jacobi bellman hjb equation can be regarded as a special case of the above problem an existence and uniqueness theorem is obtained for the case where sigma does not contain the control variable v an optimal control interpretation is given the linear quadratic case is discussed as well

a double projection algorithm for multi valued variational inequalities and a unified framework of the method in this paper we propose a double projection algorithm for a generalized variational inequality with a multi valued mapping under standard conditions our method is proved to be globally convergent to a solution of the variational inequality problem moreover we present a unified framework of projection type methods for multi valued variational inequalities preliminary computational experience is also reported

dynkin game of stochastic differential equations with random coefficients and associated backward stochastic partial differential variational inequality a dynkin game is considered for stochastic differential equations with random coefficients we first apply qiu and tang s maximum principle for backward stochastic partial differential equations to generalize a krylov estimate for the distribution of a markov process to that of a non markov process and establish a generalized ito kunita wentzell formula which allows the test function as an ito process to take values in a sobolev space we then prove the verification theorem that the nash equilibrium point and the value of the dynkin game are characterized by the strong solution of the associated hamilton jacobi bellman isaacs equation which is currently a backward stochastic partial differential variational inequality bspdvi with two obstacles finally we prove the existence and uniqueness result and a comparison theorem for strong solution of the bspdvi

a new projection method for variational inequality problems we propose a new projection algorithm for solving the variational inequality problem where the underlying function is continuous and satisfies a certain generalized monotonicity assumption e g it can be pseudomonotone the method is simple and admits a nice geometric interpretation it consists of two steps first we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem this procedure requires a single projection onto the feasible set and employs an armijo type linesearch along a feasible direction then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set thus in contrast with most other projection type methods only two projection operations per iteration are needed the method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions preliminary computational experience is also reported

on the quasi linear reflected backward stochastic partial differential equations abstract this paper is concerned with the quasi linear reflected backward stochastic partial differential equation rbspde for short basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential we first associate rbspde to a variational problem and via the penalization method we prove the existence and uniqueness of the solution for linear rbspde with laplacian leading coefficients with the continuity approach we further obtain the well posedness of general quasi linear rbspdes related results including ito formulas for backward stochastic partial differential equations with stochastic regular measures the comparison principle for solutions of rbspdes and the connections with reflected backward stochastic differential equations and optimal stopping problems are addressed as well

a subgradient extragradient algorithm for solving multi valued variational inequality in this paper we propose a subgradient extragradient method for solving multi valued variational inequality it is showed that the method converges globally to a solution of multi valued variational inequality provided the multi valued mapping is pseudomonotone with nonempty compact convex values convergence rate is analyzed preliminary computational experience is also reported

a duality analysis on stochastic partial differential equations abstract the duality equations of stochastic partial differential equations are solved in the sobolev space h m w 2 m r d and the h m norm estimates of the solutions are obtained as an application the h m norm estimates with negative m for the solutions of stochastic partial differential equations are derived

maximum principle for quasi linear reflected backward spdes abstract this paper establishes a maximum principle for quasi linear reflected backward stochastic partial differential equations rbspdes for short we prove the existence and uniqueness of the weak solution to rbspdes allowing for non zero dirichlet boundary conditions and using a stochastic version of de giorgi s iteration establish the maximum principle for rbspdes on a general domain the maximum principle for rbspdes on a bounded domain and the maximum principle for backward stochastic partial differential equations bspdes for short on a general domain can be obtained as byproducts finally the local behavior of the weak solutions is considered

l p estimates for the uniform norm of solutions of quasilinear spde s in this paper we prove l p estimates p 2 for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations spde of parabolic type our method is based on a version of moser s iteration scheme developed by aronson and serrin in the context of non linear parabolic pde

maximum principle for quasi linear backward stochastic partial differential equations abstract in this paper we are concerned with the maximum principle for quasi linear backward stochastic partial differential equations bspdes for short of parabolic type we first prove the existence and uniqueness of the weak solution to quasi linear bspdes with the null dirichlet condition on the lateral boundary then using the de giorgi iteration scheme we establish the maximum estimates and the global maximum principle for quasi linear bspdes to study the local regularity of weak solutions we also prove a local maximum principle for the backward stochastic parabolic de giorgi class

the maximum principle for partially observed optimal control of stochastic differential equations this paper concerns partially observed optimal control of possibly degenerate stochastic differential equations with correlated noises between the system and the observation the control is allowed to enter into all the coefficients a general maximum principle is proved for the partially observed optimal control and the relations among the adjoint processes are established adjoint vector fields which are adapted to the past and present observations are introduced as the solutions to some backward stochastic partial differential equations bspdes and their relations are established under suitable conditions the adjoint processes are characterized in terms of the adjoint vector fields their differentials and hessians along the optimal state process some other formulations of the partially observed stochastic maximum principle are then derived

an algorithm for generalized variational inequality with pseudomonotone mapping we suggest an algorithm for a variational inequality with multi valued mapping the iteration sequence generated by the algorithm is proven to converge to a solution provided the multi valued mapping is continuous and pseudomonotone with nonempty compact convex values

a bundle method for solving variational inequalities in this paper we present a bundle method for solving a generalized variational inequality problem this problem consists of finding a zero of the sum of two multivalued operators defined on a real hilbert space the first one f is monotone and the second is the subdifferential of a lower semicontinuous proper convex function our method is based on the auxiliary problem principle due to cohen and our strategy is to approximate in the subproblems the nonsmooth convex function by a sequence of convex piecewise linear functions as in the bundle method for nonsmooth optimization this makes the subproblems more tractable first we explain how to build step by step suitable piecewise linear approximations by means of a bundle strategy and we present a new stopping criterion to determine whether the current approximation is good enough this criterion is the same as that commonly used in the special case of nonsmooth optimization second we study the convergence of the algorithm for the case when the stepsizes are chosen going to zero and for the case bounded away from zero in the first case the convergence can be proved under rather mild assumptions the operator f is paramonotone and possibly multivalued in the second case the convergence needs a stronger assumption f is single valued and satisfies a dunn property finally we illustrate the behavior of the proposed algorithm by some numerical tests