kalman bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems abstract this paper is concerned with kalman bucy filtering problems of a forward and backward stochastic system which is a hamiltonian system arising from a stochastic optimal control problem there are two main contributions worthy pointing out one is that we obtain the kalman bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation the other is that we develop a backward separation technique which is different to wonham s separation theorem to study a partially observed recursive optimal control problem this new technique can also cover some more general situation such as a partially observed linear quadratic non zero sum differential game problem is solved by it we also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases

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

a filtering problem with uncertainty in observation abstract this paper is concerned with a generalized kalman bucy filtering model and corresponding robust problem under model uncertainty we find that this robust problem is equivalent to considering an estimate problem under some sublinear operator therefore we turn to obtaining the minimum mean square estimator under a sublinear operator by girsanov theorem and minimax theorem we obtain the optimal estimator x t of the signal process x t for given time t 0 t

solitary shock waves and other travelling waves in a general compressible hyperelastic rod in the literature it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods recently it has been shown that this conjecture is true one might guess that when compressibility is taken into account such a wave which is both a solitary wave and a shock wave can still arise one of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description it is difficult to directly tackle the fully nonlinear rod equations here by using a non dimensionalization process and the reductive perturbation technique we derive a new type of nonlinear dispersive equation as the model equation we then focus on the travelling wave solutions of this new equation as a result we obtain a system of ordinary differential equations an important feature of this system is that there is a vertical singular line in the phase plane which leads to the appearance of shock waves by considering the equilibrium points and their relative positions to the singular line we are able to determine all qualitatively different phase planes those paths in phase planes which represent physically acceptable solutions are discussed one by one it turns out that there is a variety of travelling waves including solitary shock waves solitary waves periodic shock waves etc analytical expressions for all these waves are obtained a new phenomenon is also found a solitary wave can suddenly change into a periodic wave with finite period in dynamical systems this represents a homoclinic orbit suddenly changing into a closed orbit to the authors9 knowledge such a bifurcation has not been found in any other dynamical systems

generalized reflected bsdes driven by a levy process and an obstacle problem for pdies with a nonlinear neumann boundary condition in this paper we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equations grbsdes in short driven by a levy process which involve the integral with respect to a continuous process by means of the snell envelope the penalization method and the fixed point theorem in addition we obtain the comparison theorem for the solutions of the grbsdes as an application we give a probabilistic formula for the viscosity solution of an obstacle problem for a class of partial differential integral equations pdies in short with a nonlinear neumann boundary condition

the maximum principle for partially observed optimal control problems of mean field fbsdes abstractthis paper is concerned with a partially observed optimal control problem described by mean field forward and backward stochastic differential equations moreover the control variable ente

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

a constrained control problem with degenerate coefficients and degenerate backward spdes with singular terminal condition we study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact the coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation bspde with singular terminal condition for this degenerate bspde we prove existence and uniqueness of a nonnegative solution our existence result requires a novel gradient estimate for degenerate bspdes

bifurcation of travelling wave solutions in generalized phi four equation by using the method of planar dynamical systems to a nonlinear variant of the phi four equation the existence of smooth and nonsmooth solitary wave so called peakon and valleyon and infinite many periodic wave solutions is shown under different regions of parametric spaces various sufficient conditions to guarantee the existence of above solutions are given the formulas to compute the travelling waves are also educed

numerical approximation of doubly reflected bsdes with jumps and rcll obstacles we study a discrete time approximation scheme for the solution of a doubly reflected backward stochastic differential equation dbbsde in short with jumps driven by a brownian motion and an independent compensated poisson process moreover we suppose that the obstacles are right continuous and left limited rcll processes with predictable and totally inaccessible jumps and satisfy mokobodski s condition our main contribution consists in the construction of an implementable numerical sheme based on two random binomial trees and the penalization method which is shown to converge to the solution of the dbbsde finally we illustrate the theoretical results with some numerical examples in the case of general jumps

bifurcations of travelling wave solutions in a model of the hydrogen bonded systems by using the theory of bifucations of dynamical systems to a model of the hydrogen bonded systems the existence of solitary wave kink and anti kink wave solutions and uncountably infinite many smooth and non smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given in some simple parametric conditions exact explicit and implicit solution formulas are listed

a robust kalman bucy filtering problem abstract a generalized kalman bucy model under model uncertainty and a corresponding robust problem are studied in this paper we find that this robust problem is equivalent to an estimated problem under a sublinear operator by girsanov transformation and the minimax theorem we prove that this problem can be reformulated as a classical kalman bucy filtering problem under a new probability measure the equation which governs the optimal estimator is obtained moreover the optimal estimator can be decomposed into the classical optimal estimator and a term related to the model uncertainty parameter under some condition

bifurcations of travelling wave solutions for modified nonlinear dispersive phi four equation abstract by using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi four equation we analysis all bifurcations and phase portraits in the parametric space the existence of solitary wave solutions and uncountably infinite many smooth and non smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some explicit exact solution formulas are acquired for some special cases

an approximation of the analytic solution of the shock wave equation in this article we discuss the analytic solution of the fully developed shock waves the adomian decomposition method is used to solve the shock wave equation which describes the flow of gases unlike the various numerical techniques which are usually valid for short period of time the solution of the presented equation is analytic for 0 t also the results presented here indicate that the method is reliable accurate and converges very rapidly

the maximum principles for stochastic recursive optimal control problems under partial information a maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal agent problems by virtue of a classical spike variational approach and a filtering technique the maximum principle is obtained and the related adjoint processes are characterized by the solutions of forward backward stochastic differential equations in finite dimensional spaces then our theoretical results are applied to study a partially observed linear quadratic recursive optimal control problem in addition for the case with initial and terminal state constraints the corresponding maximum principle is also obtained by using ekeland s variational principle

galerkin method for the numerical solution of the rlw equation using quintic b splines the regularized long wave equation rlw is solved numerically by using the quintic b spline galerkin finite element method the same method is applied to the time split rlw equation comparison is made with both analytical solutions and some previous results propagation of solitary waves interaction of two solitons are studied

on the equality of solutions of max min and min max systems of variational inequalities with interconnected bilateral obstacles abstract in this paper we deal with the solutions of systems of pdes with bilateral interconnected obstacles of min max and max min types these systems arise naturally in stochastic switching zero sum game problems we show that when the switching costs of one side are regular the solutions of the min max and max min systems coincide then this common viscosity solution is related to a multi dimensional doubly reflected bsde with bilateral interconnected obstacles finally its relationship with the values of a zero sum switching game is studied

generalized bsde with 2 reflecting barriers and stochastic quadratic growth abstract we study the existence of a solution for a one dimensional generalized backward stochastic differential equation with two reflecting barriers grbsde for short under assumptions on the input data which are weaker than that on the current literature in particular we construct a maximal solution for such a grbsde when the terminal condition is only f t measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any p integrability conditions the work is suggested by the interest the results might have in dynkin game problem and american game option

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

on the necessary conditions of optimal controls for stochastic partial differential equations this paper concerns optimal control of systems governed by stochastic partial differential equations in which drift and diffusion terms are second and first order differential operators respectively necessary conditions for an optimal control are derived for both nondegenerate and degenerate systems and all the coefficients appearing in the equations are allowed to depend on the control variables furthermore the results obtained in the paper can also be used to derive necessary conditions of optimality for partially observed diffusions with correlation between the signals and the observation noises

a linear quadratic optimal control problem of forward backward stochastic differential equations with partial information this paper studies a linear quadratic optimal control problem derived by forward backward stochastic differential equations where the drift coefficient of the observation equation is linear with respect to the state x and the observation noise is correlated with the state noise in the sense that the cross variation of the state and the observation is nonzero a backward separation approach is introduced combining it with variational method and stochastic filtering two optimality conditions and a feedback representation of optimal control are derived closed form optimal solutions are obtained in some particular cases as an application of the optimality conditions a generalized recursive utility problem from financial markets is solved explicitly

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

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

on solutions to backward stochastic partial differential equations for levy processes in this paper we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations bspdes for short driven by the teugels martingales associated with a levy process satisfying some moment conditions and by an independent brownian motion an example is given to illustrate the theory

stochastic optimal control for backward stochastic partial differential systems abstract this paper studies optimal controls for a class of backward stochastic partial differential systems in the abstract evolution form under the assumption of a convex control domain necessary and sufficient conditions for an admissible control to be optimal are derived in the form of stochastic maximum principles by means of a convex variation method and a duality technique as an application the optimal control for a linear backward stochastic evolution equation bsee with quadratic cost criteria called bseelq problem is discussed and the corresponding optimal control is characterized via the stochastic hamilton system which is a linear full coupled forward backward stochastic evolution equation fbsee and consists of the state equation the adjoint equation and the dual presentation of the optimal control

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

on the cauchy problem for backward stochastic partial differential equations in holder spaces this paper is concerned with solution in holder spaces of the cauchy problem for linear and semi linear backward stochastic partial differential equations bspdes of super parabolic type the pair of unknown variables are viewed as deterministic spatial functionals which take values in banach spaces of random vector processes we define suitable functional holder spaces for them and give some inequalities among these holder norms the existence uniqueness as well as the regularity of solutions are proved for bspdes which contain new assertions even on deterministic pdes

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

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 stochastic maximum principle for partially observed stochastic control systems with delay abstract this paper deals with partially observed optimal control problems for the state governed by stochastic differential equation with delay we develop a stochastic maximum principle for this kind of optimal control problems using a variational method and a filtering technique also we establish a sufficient condition without assumption of the concavity two examples that shed light on the theoretical results are established in the paper in particular in the example of an optimal investment problem with delay its numerical simulation shows the effect of delay via a discretization technique for forward backward stochastic differential equations fbsdes with delay and anticipate terms

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

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

backward stochastic partial differential equations with quadratic growth abstract this paper is concerned with the weak solution in analytic sense to the cauchy dirichlet problem of a backward stochastic partial differential equation when the nonhomogeneous term has a quadratic growth in both the gradient of the first unknown and the second unknown existence and uniqueness results are obtained under separate conditions

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

trigonometric bases for matrix weighted lp spaces abstract we give a complete characterization of 2 periodic matrix weights w for which the vector valued trigonometric system forms a schauder basis for the matrix weighted space l p t w then trigonometric quasi greedy bases for l p t w are considered quasi greedy bases are systems for which the simple thresholding approximation algorithm converges in norm it is proved that such a trigonometric basis can be quasi greedy only for p 2 and whenever the system forms a quasi greedy basis the basis must actually be a riesz basis

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

backward stochastic differential equations with constraints on the gains process we consider backward stochastic differential equations with convex constraints on the gains or intensity of noise process existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is lipschitz continuous in the state and gains processes and convex in the gains process it is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control type equation this representation is related to the penalization method for constructing solutions of stochastic differential equations involves change of measure techniques and employs notions and results from convex analysis such as the support function of the convex set of constraints and its various properties

bsdes with two reflecting barriers the general result in this paper we show the existence of a solution for the bsde with two reflecting barriers when those latter are completely separated neither mokobodzki s condition nor the regularity of the barriers are supposed the main tool is the notion of local solution of reflected bsdes applications related to dynkin games and double obstacle variational inequality are given

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

maximum principles for a class of partial information risk sensitive optimal controls in this technical note we study a class of partial information risk sensitive optimal controls the main results are some necessary and sufficient conditions for these partial information control problems for illustration one example is proposed and then solved using our results

a minimum principle for stochastic control problems with output feedback a minimum principle for stochastic control problems with output feedback is derived by applying bismut s minimum principle for stochastic control problems with full information about the past to the kushner stratonovitch equation describing the controlled evolution of the conditional density of the state the well known solution of the linear quadratic gaussian problem is obtained from the principle

solutions of compact and noncompact structures for nonlinear klein gordon type equation the focusing and the defocusing branches of nonlinear klein gordon type equation kg n n are considered a framework is implemented to show that the first model exhibits compactons solitons that do not have exponential tails whereas the second model demonstrates solitary patterns solutions the two variants of kg n n equation will be examined in one and higher dimensions general formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures

reflected scheme for doubly reflected bsdes with jumps and rcll obstacles we introduce a discrete time reflected scheme to solve doubly reflected backward stochastic differential equations with jumps in short drbsdes driven by a brownian motion and an independent compensated poisson process as in dumitrescu and labart 2014 we approximate the brownian motion and the poisson process by two random walks but contrary to this paper we discretize directly the drbsde without using a penalization step this gives us a fully implementable scheme which only depends on one parameter of approximation the number of time steps n contrary to the scheme proposed in dumitrescu and labart 2014 which also depends on the penalization parameter we prove the convergence of the scheme and give some numerical examples

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

dynkin game of convertible bonds and their optimal strategy abstract this paper studies the valuation and optimal strategy of convertible bonds as a dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method we first reduce such a dynkin game to an optimal stopping time problem with state constraint and then in a markovian setting we investigate the optimal strategy by analyzing the properties of the corresponding free boundary including its position asymptotics monotonicity and regularity we identify situations when call precedes conversion and vice versa moreover we show that the irregular payoff results in the possibly non monotonic conversion boundary surprisingly the price of the convertible bond is not necessarily monotonic in time it may even increase when time approaches maturity

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

the maximum principle for optimal control of diffusions with partial information we derive necessary conditions for the optimal control of a system that satisfies an ito equation with control entering the drift term the control is a function of a noise corrupted observation of the state and the cost is the expectation of an integral and a final term the robust form of the zakai equation from nonlinear filtering is used to compute the variation of the cost due to a strong or needle variation of a control this gives rise to an explicit formula for the adjoint process much as in the case with complete information

l p solutions of multidimensional bsdes with weak monotonicity and general growth generators abstract in this paper we first establish the existence and uniqueness of l p p 1 solutions for multidimensional backward stochastic differential equations bsdes under a weak monotonicity condition together with a general growth condition in y for the generator g then we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way finally we put forward and prove a stability theorem and a comparison theorem of l p p 1 solutions for this kind of bsdes

hilbert spaces of vector valued functions generated by quadratic forms abstract we study hilbert spaces l 2 e g where e r d is a measurable set e 0 and for almost every t e the matrix g t see 3 is a hermitian positive definite matrix we find necessary and sufficient conditions for which the projection operators t k f f k e k 1 k n are bounded the obtained results allow us to translate various questions in the spaces l 2 e g to weighted norm inequalities with weights which are the diagonal elements of the matrix g t in section 3 we study the properties of the system m t e j 1 j n m n in the space l 2 e g where m m 1 is a complete orthonormal system defined on a measurable set e r we concentrate our study on two classical systems the haar and the trigonometric systems simultaneous approximations of n elements f 1 f n of some banach spaces x 1 x n with respect to a system which is a basis in any of those spaces are studied

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

anticipated backward stochastic differential equations in this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations anticipated bsdes in these equations the generator includes not only the values of solutions of the present but also the future we show that these anticipated bsdes have unique solutions a comparison theorem for their solutions and a duality between them and stochastic differential delay equations

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

hamilton jacobi bellman equations for the optimal control of the duncan mortensen zakai equation abstract we study a class of hamilton jacobi bellman hjb equations associated to stochastic optimal control of the duncan mortensen zakai equation the equations are investigated in weighted l 2 spaces we introduce an appropriate notion of weak viscosity solution of such equations and prove that the value function is the unique solution of the hjb equation we apply the results to stochastic optimal control problems with partial observation and correlated noise

continuous time dynkin games with mixed strategies let x y z be a triple of payoff processes defining a dynkin game tilde r sigma tau e left x sigma 1 tau sigma y tau 1 tau sigma z tau 1 tau sigma right where sigma and tau are stopping times valued in 0 t in the case z y it is well known that the condition x leq y is needed in order to establish the existence of value for the game i e inf tau sup sigma tilde r sigma tau sup sigma inf tau tilde r sigma tau in order to remove the condition x leq y we introduce an extension of the dynkin game by allowing for an extended set of strategies namely the set of mixed strategies the main result of the paper is that the extended dynkin game has a value when z leq y and the processes x and y are restricted to be semimartingales continuous at the terminal time t

numerical algorithms and simulations for reflected backward stochastic differential equations with two continuous barriers in this paper we study different algorithms for reflected backward stochastic differential equations bsde in short with two continuous barriers based on the framework of using a binomial tree to simulate 1 d brownian motion we introduce numerical algorithms by the penalization method and the reflected method respectively in the end simulation results are also presented

strong solution of backward stochastic partial differential equations in c 2 domains this paper is concerned with the strong solution to the cauchy dirichlet problem for backward stochastic partial differential equations of parabolic type existence and uniqueness theorems are obtained due to an application of the continuation method under fairly weak conditions on variable coefficients and c2 domains the problem is also considered in weighted sobolev spaces which allow the derivatives of the solutions to blow up near the boundary as applications a comparison theorem is obtained and the semi linear equation is discussed in the c2 domain

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

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

optimal portfolio and consumption with habit formation in a jump diffusion market this paper studies optimal portfolio and consumption selection with habit formation in a jump diffusions incomplete market in continuous time the stochastic maximum principle for jump processes is applied to solve habit forming utility maximization problem we transform this problem into the case not involving habit formation in mechanically then the solution in the state feedback form is given the relationship between maximum principle and dynamic programming is employed to get the expression of the relative risk aversion rra coefficient and its distribution finally for a special case the stationary mean of the rra coefficient is obtained and the numerical experiment indicates our model with jump diffusions is better than the model in 1 to resolve the equity premium puzzle in a way

optimality conditions for partial information stochastic control problems driven by levy processes abstract in this paper we consider a partial information stochastic control problem where the system is governed by a nonlinear stochastic differential equation driven by teugels martingales associated with some levy process and an independent brownian motion we prove optimality necessary conditions in the form of a maximum principle these conditions turn out to be sufficient under some convexity assumptions to illustrate the general results an example is solved

stochastic maximum principle for mean field type optimal control under partial information this technical note is concerned with a partially observed optimal control problem whose novel feature is that the cost functional is of mean field type hence determining the optimal control is time inconsistent in the sense that bellman s dynamic programming principle does not hold a maximum principle is established using girsanov s theorem and convex variation some nonlinear filtering results for backward stochastic differential equations bsdes are developed by expressing the solutions of the bsdes as some ito s processes an illustrative example is demonstrated in terms of the maximum principle and the filtering

the partially observed stochastic minimum principle using stochastic flows and the generalized differentiation formula of bismut and kunita the change in cost due to a strong variation of an optimal control is explicitly calculated differentiating this expression gives a minimum principle in both the partially observed and stochastic open loop situations in the latter case the equation satisfied by the adjoint process is obtained by applying a martingale representation result

the maximum principle for global solutions of stochastic stackelberg differential games national science foundation dms 1303775 research grant council of the hksar cityu 500113

stochastic minimum energy control abstract we give the solution to the minimum energy control problem for linear stochastic systems the problem is as follows given an exactly controllable system find the control process with the minimum expected energy that transfers the system from a given initial state to a desired final state the solution is found in terms of a certain forward backward stochastic differential equation of hamiltonian type

second order backward stochastic differential equations and fully nonlinear parabolic pdes for a d dimensional diffusion of the form dxt xt dt xt dwt and continuous functions f and g we study the existence and uniqueness of adapted processes y z and a solving the second order backward stochastic differential equation 2bsde dyt f t xt yt zt t dt z dxt t 0 t dzt at dt t dxt