sequential quantum measurements a quantum effect is an operator a on a complex hilbert space h that satisfies 0 a i we denote the set of quantum effects by e h the set of self adjoint projection operators on h corresponds to sharp effects and is denoted by p h we define the sequential product of a b e h by a b a1 2ba1 2 the main purpose of this article is to study some of the algebraic properties of the sequential product many of our results show that algebraic conditions on a b imply that a and b commute for the usual operator product for example if a b satisfies certain distributive or associative laws then ab ba moreover if a b p h then ab ba and a b b or b a b if and only if ab ba b a natural definition of stochastic independence is introduced and briefly studied

another approach to the existence of value functions of stochastic differential games abstract the existence of value functions for general two player zero sum stochastic differential games has been obtained by fleming and souganidis in this paper we present a new approach to this problem we prove optimality inequalities of dynamic programming for viscosity sub and supersolutions of the associated bellman isaacs equations these inequalities are well known for deterministic differential games but are new for stochastic differential games it then easily follows that value functions are the unique viscosity solutions of the bellman isaacs equations and satisfy the principle of dynamic programming the results presented here are not the same as those of fleming and souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us

two player zero sum stochastic differential games with regime switching abstract this paper is concerned with the two player zero sum stochastic differential game in a regime switching model with an infinite horizon the state of the system is characterized by a number of diffusions coupled by a continuous time finite state markov chain based on the dynamic programming principle dpp the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper hamilton jacobi bellman isaacs hjbi equations respectively moreover the lower and upper value functions coincide under the isaacs condition which implies that the game admits a value all the proofs in this paper are markedly different from those for the case when there is no regime switching

two person zero sum game in weak formulation and path dependent bellman isaacs equation in this paper we study a two person zero sum stochastic differential game in weak formulation unlike the standard literature which uses strategy type controls the weak formulation allows us to consider the game with control against control we shall prove the existence of game value under natural conditions another main feature of the paper is that we allow for non markovian structure and thus the game value is a random process we characterize the value process as the unique viscosity solution of the corresponding path dependent bellman isaacs equation a notion recently introduced by ekren et al ann probab 42 2014 pp 204 236 and ekren touzi and zhang stochastic process to appear preprint arxiv 1210 0006v2 preprint arxiv 1210 0007v2

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

asymptotic perron s method and simple markov strategies in stochastic games and control we introduce a modification of perron s method where semisolutions are considered in a carefully defined asymptotic sense with this definition we can show in a rather elementary way that in a zero sum game or a control problem with or without model uncertainty the value function over all strategies coincides with the value function over markov strategies discretized in time therefore there are always discretized markov varepsilon optimal strategies uniform with respect to the bounded initial condition with a minor modification the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies

the second laws of quantum thermodynamics the second law of thermodynamics places constraints on state transformations it applies to systems composed of many particles however we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath is there a second law of thermodynamics in this regime here we find that for processes which are approximately cyclic the second law for microscopic systems takes on a different form compared to the macroscopic scale imposing not just one constraint on state transformations but an entire family of constraints we find a family of free energies which generalize the traditional one and show that they can never increase the ordinary second law relates to one of these with the remainder imposing additional constraints on thermodynamic transitions we find three regimes which determine which family of second laws govern state transitions depending on how cyclic the process is in one regime one can cause an apparent violation of the usual second law through a process of embezzling work from a large system which remains arbitrarily close to its original state these second laws are relevant for small systems and also apply to individual macroscopic systems interacting via long range interactions by making precise the definition of thermal operations the laws of thermodynamics are unified in this framework with the first law defining the class of operations the zeroth law emerging as an equivalence relation between thermal states and the remaining laws being monotonicity of our generalized free energies

the optimal control related to riemannian manifolds and the viscosity solutions to hamilton jacobi bellman equations abstract in this paper we study the optimal stochastic control problem for stochastic differential equations on riemannian manifolds the cost functional is specified by controlled backward stochastic differential equations in euclidean space under some suitable assumptions we conclude that the value function is the unique viscosity solution to the associated hamilton jacobi bellman equation which is a fully nonlinear parabolic partial differential equation on riemannian manifolds

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

stochastic perron s method for the probability of lifetime ruin problem under transaction costs we apply the stochastic perron s method to a singular control problem where an individual targets at a given consumption rate invests in a risky financial market in which trading is subject to proportional transaction costs and seeks to minimize her probability of lifetime ruin without relying on the dynamic programming principle we characterize the value function as the unique viscosity solution of an associated hamilton jacobi bellman variational inequality we also provide a complete proof of the comparison principle which is the main assumption of the stochastic perron s method

min max formulas for nonlocal elliptic operators on euclidean space abstract an operator satisfies the global comparison property if anytime a function touches another from above at some point then the operator preserves the ordering at the point of contact this is characteristic of degenerate elliptic operators including nonlocal and nonlinear ones in previous work the authors considered such operators in riemannian manifolds and proved they can be represented by a min max formula in terms of levy operators in this note we revisit this theory in the context of euclidean space with the intricacies of the general riemannian setting gone the ideas behind the original proof of the min max representation become clearer moreover we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity

stochastic verification theorem of forward backward controlled systems for viscosity solutions abstract in this paper we investigate the controlled systems described by forward backward stochastic differential equations with the control contained in drift diffusion and generator of bsdes a new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions it is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems as a relevant problem the optimal stochastic feedback controls for forward backward systems are discussed as well

maximum principle for differential games of forward backward stochastic systems with applications abstract this paper is concerned with a maximum principle for both zero sum and nonzero sum games the most distinguishing feature compared with the existing literature is that the game systems are described by forward backward stochastic differential equations this kind of games is motivated by linear quadratic differential game problems with generalized expectation we give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games finally an example of a nonzero sum game is worked out to illustrate that the theories may find interesting applications in practice in terms of the maximum principle the explicit form of an equilibrium point is obtained

one kind of linear quadratic zero sum stochastic differential game with jumps this paper is concerned with one kind of two person zero sum linear quadratic stochastic differential game with jumps a closed loop formulation for the saddle points of the game is established in

self catalytic conversion of pure quantum states conversion of entangled states under stochastic local operations and classical communication s locc admits the phenomenon of catalysis here we explore the possibility of a copy of the initial state itself performing as a catalyst which we call a self catalytic process we show explicit examples of self catalysis necessary and sufficient conditions for the phenomenon to take place are discussed we numerically estimate how frequent it is and we show that increasing the number of copies used as catalyst can increase the probability of conversion but does not make the process deterministic by the end we conjecture that under locc the probability of finding a self catalytic reaction does not increase monotonically with the dimensions whereas under slocc it does increase

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

risk sensitive zero sum differential games we consider two player risk sensitive zero sum differential games rszsdgs in our problem setup both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players and the objective functional is of the risk sensitive type first a stochastic maximum principle type necessary condition for an open loop saddle point of the rszsdg is established via nonlinear transformations of the adjoint processes of the equivalent risk neutral stochastic zero sum differential game in particular we obtain two variational inequalities namely the pair of saddle point inequalities of the rszsdg next we obtain the hamilton jacobi isaacs partial differential equation for the rszsdg which provides a sufficient condition for a feedback saddle point of the rszsdg using a logarithmic transformation of the associated value function finally we study the extended linear quadratic rszsdg lq rszsdg we show intractability of the extended lq rszsdg with the state and or controls of both players appearing in the diffusion term this unexpected intractability could lead to nonlinear open loop and feedback saddle points even if the problem itself is essentially lq and the isaacs condition holds

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

switching games of stochastic differential systems a two player zero sum switching game is formulated for general stochastic differential systems and is studied using a combined dynamic programming and viscosity solution approach the existence of the game value is proved for the proof of the related dynamic programming principle ddp for the lower and upper value functions the measurability problem of the same kind as mentioned in the paper of fleming and souganidis is also encountered and we are able to get around it via a delicate adaptation of their technique moreover the traditional direct method to prove the time continuity of lower and upper value functions also gives rise to a serious measurability problem to get around the new difficulty a subtle dynamic programming argument is developed to obtain the time continuity which in return is used to derive the ddp for random intermediate times from the ddp with deterministic intermediate times

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

differential games optimal control and directional derivatives of viscosity solutions of bellman s and isaacs equations recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two person zero sum differential games problems and the new notion of viscosity solutions of hamilton jacobi pde s introduced by m g crandall and p l lions in particular it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated hamilton jacobi bellman and isaacs equations in the present work it is shown that viscosity super and subsolutions of these equations must satisfy some inequalities called super and subdynamic programming principle respectively this is then used to prove the equivalence between the notion of viscosity solutions and the conditions introduced by a subbotin concerning the sign of certain generalized directional derivatives

approximate solutions of continuous time stochastic games the paper is concerned with a zero sum continuous time stochastic differential game with a dynamics controlled by a markov process and a terminal payoff the value function of the original game is estimated using the value function of a model game the dynamics of the model game differs from the original one the general result applied to differential games yields the approximation of the value function of the differential game by the solution of countable system of odes

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 generalization of gudder nagy s theorem with numerical ranges of operators in this paper we prove that for a b c which are bounded linear operators on a separable complex hilbert space h ax x bx x cx x for all x h with x 1 if and only if there exists a complex number c c such that a ci and c cb or b ci and c ca

linear quadratic mean field stochastic zero sum differential games abstract in this paper we consider the two player linear quadratic mean field stochastic differential game lq mf szsdg where the expected values of state and players control variables are included in the corresponding mean field stochastic system and the objective functional we characterize the explicit condition under which the feedback saddle point equilibrium for the lq mf szsdg exists the results are obtained by using the completion of squares method and the ordered interchangeability property of multiple saddle points we show that the corresponding saddle point equilibrium is linear in state which is characterized by the two coupled riccati differential equations crdes we also discuss the solvability conditions of the crdes numerical results are presented to verify the theoretical results of the paper

stochastic recursive optimal control problem of reflected stochastic differential systems in this paper we study one kind of stochastic recursive optimal control problem in which the control system is stochastic differential equations reflected in a domain and the cost functional is de

a pontryagin s maximum principle for non zero sum differential games of bsdes with applications this technical note is concerned with a maximum principle for a new class of non zero sum stochastic differential games the most distinguishing feature compared with the existing literature is that the game systems are described by backward stochastic differential equations bsdes this kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities we establish a necessary condition and a sufficient condition in the form of maximum principle for open loop equilibrium point of the foregoing games respectively to explain the theoretical results we use them to study a financial problem

stochastic differential switching game in infinite horizon abstract we study a zero sum stochastic differential switching game in infinite horizon we prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi variational inequalities with bilateral obstacles we also obtain a verification theorem which provides an optimal strategy of the game finally some numerical examples with two regimes are given

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

on zero sum stochastic differential games with jump diffusion driven state a viscosity solution framework a zero sum differential game with controlled jump diffusion driven state is considered and studied by a combination of dynamic programming and viscosity solution techniques we prove under certain conditions that the value of the game exists moreover the value function is shown to be the unique viscosity solution of a fully nonlinear integro partial differential equation in addition we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations

the existence of game value for path dependent stochastic differential game in this paper we study a two player zero sum stochastic differential game for a path dependent stochastic system under a recursive path dependent cost functional due to the typical non markovian structure the game value is a random field dividing the time horizontal into small intervals we approximate the path dependent game by a series of state dependent games we utilize the state dependent viscosity solution theory to prove that under isaacs condition the game value exists in our model coefficients of diffusion of the system contain control and strategy and could be degenerate the dimension of the state space is high the existence of approximating nash equilibrium varepsilon saddle point is given under the formula about nonanticipative strategy with delay

krylov and safonov estimates for degenerate quasilinear elliptic pdes we here establish an a priori holder estimate of krylov and safonov type for the viscosity solutions of a degenerate quasilinear elliptic pde of non divergence form the diffusion matrix may degenerate when the norm of the gradient of the solution is small the exhibited holder exponent and holder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient in particular the given estimate is independent of the regularity of the coefficients as in the original paper by krylov and safonov the proof relies on a probabilistic interpretation of the equation

regularity properties for general hjb equations a backward stochastic differential equation method in this work we investigate regularity properties of a large class of hamilton jacobi bellman hjb equations with or without obstacles which can be stochastically interpreted in the form of a stochastic control system in which nonlinear cost functional is defined with the help of a backward stochastic differential equation bsde or a reflected bsde more precisely we prove that first the unique viscosity solution v t x of an hjb equation over the time interval 0 t with or without an obstacle and with terminal condition at time t is jointly lipschitz in t x f ort running any compact subinterval of 0 t second for the case that v solves an hjb equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that v t x i s jointly semiconcave in t x these results extend earlier ones by buckdahn cannarsa and quincampoix nonlinear differential equations appl 17 2010 pp 715 728 our approach embeds their idea of time change into a bsde analysis we also provide an elementary counterexample which shows that in general for the case that v solves an hjb equation with a lower obstacle the semiconcavity doesn t hold true

a weak dynamic programming principle for zero sum stochastic differential games with unbounded controls we analyze a zero sum stochastic differential game between two competing players who can choose unbounded controls the payoffs of the game are defined through backward stochastic differential equations we prove that each player s priority value satisfies a weak dynamic programming principle and thus solves the associated fully nonlinear partial differential equation in the viscosity sense

on martingale problems with continuous time mixing and values of zero sum games without the isaacs condition we consider a zero sum stochastic differential game over elementary mixed feedback strategies these are strategies based only on the knowledge of the past state randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules once both players choose such strategies the state equation admits a unique solution in the sense of the martingale problem of stroock and varadhan we show that the game defined over martingale solutions has a value which is the unique continuous viscosity solution of the randomized isaacs equation

numerical ranges of the powers of an operator abstract the numerical range w a of a bounded linear operator a on a hilbert space is the collection of complex numbers of the form a v v with v ranging over the unit vectors in the hilbert space in terms of the location of w a inclusion regions are obtained for w a k for positive integers k and also for negative integers k if a 1 exists related inequalities on the numerical radius w a sup w a and the crawford number c a inf w a are deduced

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

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

value in mixed strategies for zero sum stochastic differential games without isaacs condition in the present work we consider 2 person zero sum stochastic differential games with a nonlinear pay off functional which is defined through a backward stochastic differential equation our main objective is to study for such a game the problem of the existence of a value without isaacs condition not surprising this requires a suitable concept of mixed strategies which to the authors best knowledge was not known in the context of stochastic differential games for this we consider nonanticipative strategies with a delay defined through a partition pi of the time interval 0 t the underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing brownian motion and knowing the information available at the left time point t j 1 of the subintervals generated by pi the controls of players 1 and 2 are conditionally independent over t j 1 t j it is shown that the associated lower and upper value functions w pi and u pi converge uniformly on compacts to a function v the so called value in mixed strategies as the mesh of pi tends to zero this function v is characterized as the unique viscosity solution of the associated hamilton jacobi bellman isaacs equation

a note on operator probability theory involving numerical ranges abstract in this note the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered

multiscale problems and homogenization for second order hamilton jacobi equations we prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic pdes as a special case we cover the iterated homogenization for such equations with oscillating initial data explicit examples among others are the two scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n scale homogenization of uniformly parabolic fully nonlinear pdes

the controller and stopper game for a linear diffusion consider a process x x t 0 t 1 with values in the interval i 0 1 absorption at the boundary points of i and dynamics dx t t dt t dw t x 0 x the values t t are selected by a controller from a subset of 0 1 that depends on the current position x t for every t 0 at any stopping rule of his choice a second player called stopper can halt the evolution of the process x upon which he receives from the controller the amount e u x here 2 0 1 is a discount factor and u 0 1 is a continuous reward function under appropriate conditions on this function and on the contoller s set of choices it is shown that the two players have a saddle point of optimal strategies these can be described fairly explicitly by reduction to a suitable problem of optimal stopping whose maximal expected reward v coincides with the value of the game v sup inf x e e u x inf x sup e e u x

on the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the isaacs equations we prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time homogeneous diffusion processes considered up to the first exit time from a domain in contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well there is no assumption about solvability of the the isaacs equation in any sense classical or viscosity the zeroth order coefficient and the free term are only assumed to be measurable in the space variable we also prove that value functions are uniquely determined by the functions defining the corresponding isaacs equations and thus stochastic games with the same isaacs equation have the same value functions

entanglement of pure states for a single copy an optimal local conversion strategy between any two pure states of a bipartite system is presented it is optimal in that the probability of success is the largest achievable if the parties which share the system and which can communicate classically are only allowed to act locally on it the study of optimal local conversions sheds some light on the entanglement of a single copy of a pure state we propose a quantification of such an entanglement by means of a finite minimal set of new measures from which the optimal probability of conversion follows

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

catalyst assisted probabilistic entanglement transformation we are concerned with catalyst assisted probabilistic entanglement transformations a necessary and sufficient condition is presented under which there exist partial catalysts that can increase the maximal transforming probability of a given entanglement transformation we also design an algorithm which leads to an efficient method for finding the most economical partial catalysts with minimal dimension the mathematical structure of catalyst assisted probabilistic transformation is carefully investigated

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

local unitary classification of arbitrary dimensional multipartite pure states we propose a practical entanglement classification scheme for general multipartite pure states in arbitrary dimensions under local unitary equivalence by exploiting the high order singular value decomposition technique and local symmetries of the states by virtue of this scheme the method of determining the local unitary equivalence of n qubit states proposed by kraus is extended to the case for arbitrary dimensional multipartite states

stochastic perron s method and elementary strategies for zero sum differential games we develop here the stochastic perron method in the framework of two player zero sum differential games we consider the formulation of the game where both players play symmetrically feedback strategies as opposed to the elliott kalton formulation prevalent in the literature the class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic perron method carry through in a rather simple way more precisely we define the game over elementary strategies which are well motivated by intuition within this framework the stochastic perron method produces a viscosity subsolution of the upper isaacs equation dominating the upper value of the game and a viscosity supersolution of the upper isaacs equation lying below the upper value of the game using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper isaacs equation an identical statement hol

robust feedback switching control dynamic programming and viscosity solutions we consider a robust switching control problem the controller only observes the evolution of the state process and thus uses feedback closed loop switching strategies a nonstandard class of switching controls introduced in this paper the adverse player nature chooses open loop controls that represent the so called knightian uncertainty i e misspecifications of the model the half game switcher versus nature is then formulated as a two step robust optimization problem we develop the stochastic perron s method in this framework and prove that it produces a viscosity subsolution and supersolution to a system of hjb variational inequalities which envelop the value function together with a comparison principle this characterizes the value function of the game as the unique viscosity solution to the hjb equation and shows as a by product the dynamic programming principle for the robust feedback switching control problem

functional ito calculus and stochastic integral representation of martingales we develop a non anticipative calculus for functionals of a continuous semimartingale using an extension of the ito formula to path dependent functionals which possess certain directional derivatives the construction is based on a pathwise derivative introduced by b dupire for functionals on the space of right continuous functions with left limits we show that this functional derivative admits a suitable extension to the space of square integrable martingales this extension denes a weak derivative which is shown to be the inverse of the ito integral and which may be viewed as a non anticipative lifting of the malliavin derivative these results lead to a constructive martingale representation formula for ito processes by contrast with the clark haussmann ocone formula this representation only involves nonanticipative quantities which may be computed pathwise

a probabilistic representation for the value of zero sum differential games with incomplete information on both sides we prove that for a class of zero sum differential games with incomplete information on both sides the value admits a probabilistic representation as the value of a zero sum stochastic differential game with complete information where both players control a continuous martingale a similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side see cardaliaguet rainer 8 and our result is a continuous time analog of the so called splitting game introduced in laraki 20 and sorin 27 in order to analyze discrete time models it was proved by cardaliaguet 4 5 that the value of the games we consider is the unique solution of some hamilton jacobi equation with convexity constraints our result provides therefore a new probabilistic representation for solutions of hamilton jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

everything you always wanted to know about locc but were afraid to ask in this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication locc while locc emerges as the natural class of operations in many important quantum information tasks its mathematical structure is complex and difficult to characterize here we provide a precise description of locc and related operational classes in terms of quantum instruments our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds while the set of locc is not topologically closed we show that finite round locc constitutes a compact subset of quantum operations additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of locc implementable maps finally we demonstrate a two qubit map whose action can be approached arbitrarily close using locc but nevertheless cannot be implemented perfectly

exponentially growing solutions of parabolic isaacs equations abstract this paper contributes to the literature on unbounded viscosity solutions of fully nonlinear and possibly degenerate parabolic equations under natural assumptions it is established that the initial value problem for a degenerate parabolic isaacs equation set in 0 t r n has a unique continuous viscosity solution with at most exponential growth at infinity

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

comparison of viscosity solutions of semi linear path dependent pdes this paper provides a probabilistic proof of the comparison result for viscosity solutions of path dependent semilinear pdes we consider the notion of viscosity solutions introduced in 8 which considers as test functions all those smooth processes which are tangent in mean when restricted to the markovian case this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in 1 2 our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison this is most remarkable in the context of the linear path dependent heat equation as a key ingredient for our methodology we introduce a notion of punctual differentiation similar to the corresponding concept in the standard viscosity solutions 3 and we prove that semimartingales are almost everywhere punctually differentiable this smoothness result can be viewed as the counterpart of the aleksandroff smoothness result for convex functions a similar comparison result was established earlier in 8 the result of this paper is more general and more importantly the arguments that we develop do not rely on any representation of the solution

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

continuous dependence estimates and homogenization of quasi monotone systems of fully nonlinear second order parabolic equations abstract the aim of this paper is to extend the continuous dependence estimates proved by jakobsen and karlsen 2002 3 to quasi monotone systems of fully nonlinear second order parabolic equations as a by product of these estimates we get an holder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation in the second part of the paper we employ similar techniques to study the periodic homogenization of quasi monotone systems of fully nonlinear second order uniformly parabolic equations finally some examples are discussed

value functions and the dirichlet problem for isaacs equation in a smooth domain in this paper we investigate probabilistic solutions of the dirichlet problem for the elliptic isaacs equation in a smooth bounded domain in euclidean space

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

bsdes with weak terminal condition we introduce a new class of backward stochastic differential equations in which the t terminal value y t of the solution y z is not fixed as a random variable but only satisfies a weak constraint of the form e psi y t ge m for some possibly random non decreasing map psi and some threshold m we name them textit bsdes with weak terminal condition and obtain a representation of the minimal time t values y t such that y z is a supersolution of the bsde with weak terminal condition it provides a non markovian bsde formulation of the pde characterization obtained for markovian stochastic target problems under controlled loss in bouchard elie and touzi we then study the main properties of this minimal value in particular we analyze its continuity and convexity with respect to the m parameter appearing in the weak terminal condition and show how it can be related to a dual optimal control problem in meyer form these last properties generalize to a non markovian framework previous results on quantile hedging and hedging under loss constraints obtained in follmer and leukert and in bouchard elie and touzi finally we observe a surprisingly strong connection between bsdes with weak terminal condition and 2nd order bsdes in the quasi linear case