weak dynamic programming principle for viscosity solutions we prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control stopping problems which avoids the technical difficulties related to the measurable selection argument in the markov case our result is tailor made for the derivation of the dynamic programming equation in the sense of viscosity solutions

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

a pseudo markov property for controlled diffusion processes in this note we propose two different approaches to rigorously justify a pseudo markov property for controlled diffusion processes which is often explicitly or implicitly used to prove the dynamic programming principle in the stochastic control literature the first approach develops a sketch of proof proposed by fleming and souganidis indiana univ math j 38 1989 pp 293 314 the second approach is based on an enlargement of the original state space and a controlled martingale problem we clarify some measurability and topological issues raised by these two approaches

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

a weak dynamic programming principle for combined optimal stopping stochastic control with cal e f expectations we study a combined optimal control stopping problem under a nonlinear expectation e f induced by a bsde with jumps in a markovian framework the terminal reward function is only supposed to be borelian the value function u associated with this problem is generally irregular we first establish a sub resp super optimality principle of dynamic programming involving its upper resp lower semicontinuous envelope u resp u this result called the weak dynamic programming principle dpp extends that obtained in bouchard and touzi siam j control optim 49 2011 pp 948 962 in the case of a classical expectation to the case of an e f expectation and borelian terminal reward function using this weak dpp we then prove that u resp u is a viscosity sub resp super solution of a nonlinear hamilton jacobi bellman variational inequality

monotonic limit theorem of bsde and nonlinear decomposition theorem of doob meyers type we have obtained the following limit theorem if a sequence of rcll supersolutions of a backward stochastic differential equations bsde converges monotonically up to y t with e sup t y t 2 then y t itself is a rcll supersolution of the same bsde theorem 2 4 and 3 6 we apply this result to the following two problems 1 nonlinear doob meyer decomposition theorem 2 the smallest supersolution of a bsde with constraints on the solution y z the constraints may be non convex with respect to y z and may be only measurable with respect to the time variable t this result may be applied to the pricing of hedging contingent claims with constrained portfolios and or wealth processes

superhedging and dynamic risk measures under volatility uncertainty we consider dynamic sublinear expectations i e time consistent coherent risk mea sures whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty we derive a cadlag nonlinear martingale which is also the value process of a superhedg ing problem the superhedging strategy is obtained from a representation similar to the optional decomposition furthermore we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward sde the uniqueness of dynamic extensions of static sublinear expectations is also studied

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

linear quadratic optimal control problems for mean field stochastic differential equations linear quadratic optimal control problems are considered for mean field stochastic differential equations with deterministic coefficients by a variational method the optimality system is derived which is a linear mean field forward backward stochastic differential equation using a decoupling technique two riccati differential equations are obtained which are uniquely solvable under certain conditions then a feedback representation is obtained for the optimal control

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

an analysis of some fuzzy subset applications to information retrieval systems abstract several authors have attempted to improve the ability of an information retrieval system to process queries by allowing for a more general query and query processing structure than standard boolean expressions and ordinary set theory the first step in generalization from ordinary set theory to fuzzy subset theory is both philosophically and mathematically straightforward the next step that of generalizing the use of fuzzy subset theory has been more difficult of particular fundamental concern is the underlying lattice structure of the fuzzy subsets themselves and the possible loss of certain features of that structure in the second generalization step this paper will show that the mathematical requirements placed on the generalization in order that the lattice structures be preserved are really quite minimal

optional decompositions under constraints motivated by a hedging problem in mathematical finance el karoui and quenez 7 and kramkov 14 have developed optional versions of the doob meyer decomposition which hold simultaneously for all equivalent martingale measures we investigate the general structure of such optional decompositions both in additive and in multiplicative form and under constraints corresponding to different classes of equivalent measures as an application we extend results of karatzas and cvitanic 3 on hedging problems with constrained portfolios

dynamic programming for optimal control of stochastic mckean vlasov dynamics we study the optimal control of general stochastic mckean vlasov equation such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles players in mean field interaction under common noise our first main result is to state a dynamic programming principle for the value function in the wasserstein space of probability measures which is proved from a flow property of the conditional law of the controlled state process next by relying on the notion of differentiability with respect to probability measures due to p l lions 32 and ito s formula along a flow of conditional measures we derive the dynamic programming hamilton jacobi bellman equation and prove the viscosity property together with a uniqueness result for the value function finally we solve explicitly the linear quadratic stochastic mckean vlasov control problem and give an application to an interbank systemic risk model with common noise

on a certain parameter of the discretized extended linear quadratic problem of optimal control the number gamma vert hat q half hat r hat p half vert is an important parameter for the extended linear quadratic programming elqp problem associated with the lagrangian l hat u hat v hat p mdot hat u half hat u mdot hat p hat u hat q mdot hat v half hat v mdot hat q hat v hat v mdot hat r hat u over polyhedral sets hat u times hat v some fundamental properties of the problem as well as the convergence rates of certain newly developed algorithms for large scale elqp are all related to gamma in this paper we derive an estimate of gamma for the elqp problems resulting from discretization of an optimal control problem we prove that the parameter gamma of the discretized problem is bounded independently of the number of subintervals in the discretization

solving large scale minimax problems with the primal dual steepest descent algorithm this paper shows that the primal dual steepest descent algorithm developed by zhu and rockafellar for large scale extended linear quadratic programming can be used in solving constrained minimax problems related to a generalc2 saddle function it is proved that the algorithm converges linearly from the very beginning of the iteration if the related saddle function is strongly convex concave uniformly and the cross elements between the convex part and the concave part of the variables in its hessian are bounded on the feasible region better bounds for the asymptotic rates of convergence are also obtained the minimax problems where the saddle function has linear cross terms between the convex part and the concave part of the variables are discussed specifically as a generalization of the extended linear quadratic programming some fundamental features of these problems are laid out and analyzed

dynamic programming and pricing of contingent claims in an incomplete market the problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market when the market is incomplete prices cannot be derived from the absence of arbitrage since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities in this situation there is a price range for the actual market price of the contingent claim the maximum and minimum prices are studied using stochastic control methods the main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities a similar result is obtained for the minimum price which corresponds to the purchase price

martingale optimal transport and robust hedging in continuous time the duality between the robust or equivalently model independent hedging of path dependent european options and a martingale optimal transport problem is proved the financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time the hedging problem is to construct a minimal super hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given fixed maturity the dual is a monge kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity in addition to duality a family of simple piecewise constant super replication portfolios that asymptotically achieve the minimal super replication cost is constructed

on the primal dual steepest descent algorithm for extended linear quadratic programming the aim of this paper is two fold first new variants are proposed for the primal dual steepest descent algorithm as one in the family of primal dual projected gradient algorithms developed by zhu and rockafellar siam j optim 3 1993 pp 751 783 for large scale extended linear quadratic programming the variants include a second update scheme for the iterates where the primal dual feedback is arranged in a new pattern as well as alternatives for the perfect line search in the original version of the reference second new linear convergence results are proved for all these variants of the algorithm including the original version as a special case without the additional assumptions used by zhu and rockafellar for the variants with the second update scheme a much sharper estimation for the rate of convergence is obtained due to the new primal dual feedback pattern

a fuzzy document retrieval method based on two valued indexing abstract a fuzzy retrieval method that enables one to infer weights and ranking outputs is formulated in this paper still based on two valued indexing it provides a weighted system of document retrieval a keyword is represented by a set of keywords related to it as a document then a weight is calculated by the degree of matching between the representation of a keyword and that of a document this procedure can obviously be extended to boolean queries by fuzzy logic and hence to weighted boolean systems without assuming weighted indexing an interpretation of this method from the point of view of the possible world semantics in modal logic is further discussed

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

a new approach to information retrieval systems using fuzzy expressions a fuzzy document retrieval system has been realized that permits a reduction of the time needed for retrieving the set of documents fulfilling a user s query as well as the amount of core space needed for the document description relations the method described solves the problems that radecki who uses lambda level fuzzy sets met with trying to reduce time and gives rise to a good working information retrieval system as the examples show the retrieval process is described by an included algorithm

forward backward stochastic differential equations and controlled mckean vlasov dynamics the purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the mckean vlasov type motivated by the recent interest in mean field games we highlight the connection and the differences between the two sets of problems we prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control we also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions

stochastic perron s method for hamilton jacobi bellman equations we show that the value function of a stochastic control problem is the unique solution of the associated hamilton jacobi bellman equation completely avoiding the proof of the so called dynamic programming principle dpp using the stochastic perron s method we construct a supersolution lying below the value function and a subsolution dominating it a comparison argument easily closes the proof the program has the precise meaning of verification for viscosity solutions obtaining the dpp as a conclusion it also immediately follows that the weak and strong formulations of the stochastic control problem have the same value using this method we also capture the possible face lifting phenomenon in a straightforward manner

a modeling of search oriented thesaurus use based on multivalued logical inference abstract the recovery of incomplete information in an uncontrolled vocabulary system by the use of a search oriented thesaurus is formulated on the basis of multivalued logical inference the method consists of two parts to obtain several inference results for a given pair of a document and a keyword and to aggregate them by material implication as well as possibility implication newly introduced significantly the latter leads to less influence of meaningless evidence than the former

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

a general primal dual envelope method for convex programming problems a general decomposition framework for solving large scale convex programming problems is described new algorithms are obtained and several known techniques are recovered as special cases including dantzig wolfe column generation the finite envelope method of rockafellar and zhu s primal dual steepest descent method

viscosity solutions of fully nonlinear parabolic path dependent pdes part ii the main objective of this paper and the accompanying one viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint is to provide a notion of viscosity solutions for fully nonlinear parabolic path dependent pdes our definition extends our previous work ann probab 2014 42 204 236 focused on the semilinear case and is crucially based on the nonlinear optimal stopping problem analyzed in stochastic process appl 2014 124 3277 3311 we prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions and satisfies a stability property and a partial comparison result the latter is a key step for the well posedness results established in viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint we also show that the value processes of path dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equations

monotone stability of quadratic semimartingales with applications to unbounded general quadratic bsdes in this paper we study the stability and convergence of some general quadratic semimartingales motivated by financial applications we study simultaneously the semimartingale and its opposite their characterization and integrability properties are obtained through some useful exponential submartingale inequalities then a general stability result including the strong convergence of the martingale parts in various spaces ranging from h1 to bmo is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale this can be applied in particular to bsde like semimartingales this strong convergence result is then used to prove the existence of solutions of general quadratic bsdes under minimal exponential integrability assumptions relying on a regularization in both linear quadratic growth of the quadratic coefficient itself on the contrary to most of the existing literature it does not involve the seminal result of kobylanski ann probab 28 2010 558 602 on bounded solutions

stochastic target games and dynamic programming via regularized viscosity solutions we study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the opponent our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non linear partial differential equation because abstract measurable selection arguments cannot be used in this context the main obstacle is the construction of measurable almost optimal strategies we propose a novel approach where smooth supersolutions are used to define almost optimal strategies of markovian type similarly as in verification arguments for classical solutions of hamilton jacobi bellman equations the smooth supersolutions are constructed by an extension of krylov s method of shaken coefficients we apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending

wellposedness of second order backward sdes we provide an existence and uniqueness theory for an extension of backward sdes to the second order while standard backward sdes are naturally connected to semilinear pdes our second order extension is connected to fully nonlinear pdes as suggested in cheridito et al commun pure appl math 60 7 1081 1110 2007 in particular we provide a fully nonlinear extension of the feynman kac formula unlike cheridito et al in commun pure appl math 60 7 1081 1110 2007 the alternative formulation of this paper insists that the equation must hold under a non dominated family of mutually singular probability measures the key argument is a stochastic representation suggested by the optimal control interpretation and analyzed in the accompanying paper soner et al in dual formulation of second order target problems arxiv 1003 6050 2009

backwards sde with random terminal time and applications to semilinear elliptic pde suppose im t is the filtration induced by a wiener process w in r d tau is a finite im t stopping time terminal time xi is an im tau measurable random variable in r k terminal value and f cdot y z is a coefficient process depending on y in r k and z in l r d r k satisfying y tilde y f s y z f s tilde y z leq a y tilde y 2 f need not be lipschitz in y and f s y z f s y tilde z leq b z tilde z for some real a and b plus other mild conditions we identify a hilbert space depending on tau and on the number gamma equiv b 2 2a in which there exists a unique pair of adapted processes y z satisfying the stochastic differential equation dy s 1 s leq tau z s dw s f s y s z s ds with the given terminal condition y tau xi provided a certain integrability condition holds this result is applied to construct a continuous viscosity solution to the dirichlet problem for a class of semilinear elliptic pde s

weak dynamic programming for generalized state constraints we provide a dynamic programming principle for stochastic optimal control problems with expectation constraints a weak formulation using test functions and a probabilistic relaxation of the constraint avoids restrictions related to a measurable selection but still implies the hamilton jacobi bellman equation in the viscosity sense we treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints

stochastic control for a class of nonlinear kernels and applications we consider a stochastic control problem for a class of nonlinear kernels more precisely our problem of interest consists in the optimization over a set of possibly non dominated probability measures of solutions of backward stochastic differential equations bsdes since bsdes are non linear generalizations of the traditional linear expectations this problem can be understood as stochastic control of a family of nonlinear expectations or equivalently of nonlinear kernels our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting which we then use to provide a semimartingale characterization of the value function we next explore several applications of our results we first obtain a wellposedness result for second order bsdes as introduced in 76 which does not require any regularity assumption on the terminal condition and the generator then we prove a non linear optional decomposition in a robust setting extending recent results of 63 which we then use to obtain a superhedging duality in uncertain incomplete and non linear financial markets finally we relate under additional regularity assumptions the value function to a viscosity solution of an appropriate path dependent partial differential equation ppde

optional decomposition of supermartingales and hedging contingent claims in incomplete security markets letm x be the family of all equivalent local martingale measuresq for some locally boundedd dimensional processx andv be a positive process the main result of the paper theorem 2 1 states that the processv is a supermartingale whateverq m x if and only if this process admits the following decomposition v t whereh is an integrand forx andc is an adapted increasing process we call such a representationoptional because in contrast to the doob meyer decomposition it generally exists only with an adapted optional processc we apply this decomposition to the problem of hedging european and american style contingent claims in the setting ofincomplete security markets

filtration consistent nonlinear expectations and related g expectations from a general definition of nonlinear expectations viewed as operators preserving monotonicity and constants we derive under rather general assumptions the notions of conditional nonlinear expectation and nonlinear martingale we prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation and in particular admits continuous paths in other words it is a g martingale

probabilistic interpretation for solutions of fully nonlinear stochastic pdes in this article we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation 2bdsde we prove existence and uniqueness of the solution under a lipschitz type assumption on the generator and we investigate the links between the 2bdsdes and a class of parabolic fully nonlinear stochastic pdes precisely we show that the markovian solution of 2bdsdes provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear spdes

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

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