a stochastic target approach for p l matching problems within a brownian diffusion markovian framework we provide a direct pde characterization of the minimal initial endowment required so that the terminal wealth of a financial agent possibly diminished by the payoff of a random claim can match a set of constraints in probability such constraints should be interpreted as a rough description of a targeted profit and loss p l distribution this allows us to give a price to options under a p l constraint or to provide a description of the discrete p l profiles that can be achieved given an initial capital this approach provides an alternative to the standard utility indifference or marginal pricing rules that is better adapted to market practices from the mathematical point of view this is an extension of the stochastic target problem under controlled loss studied in bouchard touzi and elie bouchard b touzi n elie r 2009 stochastic target problems with controlled loss siam j control optim 48 5 3123 3150 to the case of multiple constraints although the associated hamilton jacobi bellman operator is fully discontinuous and the terminal condition is irregular we are able to construct a numerical scheme that converges at any continuity points of the pricing function

one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

a most compendious and facile quantum de finetti theorem in its most basic form the finite quantum de finetti theorem states that the reduced k partite density operator of an n partite symmetric state can be approximated by a convex combination of k fold product states variations of this result include renner s exponential approximation by almost product states a theorem which deals with certain triples of representations of the unitary group and the result of d cruz et al e print quant ph 0606139 phys rev lett 98 160406 2007 for infinite dimensional systems we show how these theorems follow from a single general de finetti theorem for representations of symmetry groups each instance corresponding to a particular choice of symmetry group and representation of that group this gives some insight into the nature of the set of approximating states and leads to some new results including an exponential theorem for infinite dimensional systems

solution of forward backward stochastic differential equations in this paper we study the existence and uniqueness of the solution to forward backward stochastic differential equations without the nondegeneracy condition for the forward equation under a certain monotonicity condition we prove the existence and uniqueness of the solution to forward backward stochastic differential equations

the stochastic reach avoid problem and set characterization for diffusions in this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost sure stochastic requirements in contrast we propose a methodology to tackle problems with less stringent requirements than almost sure to this end we first establish a connection between two distinct stochastic reach avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions subsequently we focus on solutions of one of the classes of stochastic optimal control problems the exit time problem which solves both the two reach avoid problems mentioned above we then derive a weak version of a dynamic programming principle dpp for the corresponding value function in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions moreover based on our dpp we provide an alternative characterization of the value function as a solution of a partial differential equation pde in the sense of discontinuous viscosity solutions along with boundary conditions both in dirichlet and viscosity senses theoretical justifications are also discussed to pave the way for deployment of off the shelf pde solvers for numerical computations finally we validate the performance of the proposed framework on the stochastic zermelo navigation problem

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

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

state constrained stochastic optimal control problems via reachability approach this paper deals with a class of stochastic optimal control problems socps in the presence of state constraints it is well known that for such problems the value function is in general discontinuous and its characterization by a hamilton jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system here we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions to this end the socp is first translated into a state constrained stochastic target problem then a level set approach is used to describe the backward reachable sets of the new target problem it turns out that these backward reachable sets describe the value function the main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization however the target problem involves a new state variable and a new control variable that is u

optimal control under stochastic target constraints we study a class of markovian optimal stochastic control problems in which the controlled process z nu is constrained to satisfy an almost sure constraint z nu t in g subset mathbb r d 1 mathbb p a s at some final time t 0 when the set is of the form g x y in mathbb r d times mathbb r g x y geq0 with g nondecreasing in y we provide a hamilton jacobi bellman characterization of the associated value function it gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function w which characterizes the set d t z nu t in 0 t times mathbb r d 1 z nu t in g a s for some nu contrary to standard state constraint problems the domain d is not given a priori and we do not need to impose conditions on its boundary it is naturally incorporated in the auxiliary value function w which itself is a viscosity solution of a nonlinear parabolic pde applying ideas recently developed in bouchard elie and touzi siam j control optim 48 2009 pp 3123 3150 our general result also allows us to consider optimal control problems with moment constraints of the form mathbb e g z nu t geq0 or mathbb p g z nu t geq0 geq p

critical path in an activity network with time constraints abstract an acyclic graph with nonnegative weights and with a unique source and destination is called an activity network a project comprised of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network can provide useful information for managing the project in a traditional activity network it is assumed that an activity can begin any time after all of its preceding activities have been completed this assumption does not adequately describe many practical situations in which some kinds of time constraint are usually associated with an activity in this paper we investigate two types of time constraint commonly encountered in project management the first is the time window constraint which assumes that an activity can begin its execution only in a specified time interval the second is the time schedule constraint which requires that an activity begin only at one of pre specified beginning times an efficient linear time algorithm for finding the longest path critical path and for analyzing the flow time of each arc is developed for activity networks with these time constraints

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

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

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

on exchangeable continuous variable systems we investigate permutation invariant continuous variable quantum states and their covariance matrices we provide a complete characterization of the latter with respect to permutation invariance and exchangeability and representing convex combinations of tensor power states on the level of the respective density operators this leads to necessary criteria for all these properties which become necessary and sufficient for gaussian states for these we use the derived results to provide de finetti type theorems for various distance measures

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

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

superreplication under gamma constraints in a financial market consisting of a nonrisky asset and a risky one we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint this is a constraint on the unbounded variation part of the hedging portfolio we first consider the case in which the prices are given as general markov diffusion processes and prove a verification theorem which characterizes the superreplication cost as the unique solution of a quasi variational inequality in the context of the black scholes model i e when volatility is constant this theorem allows us to derive an explicit solution of the problem these results are based on a new dynamic programming principle for general stochastic target problems

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

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

forward backward stochastic differential equations and quasilinear parabolic pdes this paper studies under some natural monotonicity conditions the theory existence and uniqueness a priori estimate continuous dependence on a parameter of forward backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations we use a purely probabilistic approach and allow the forward equation to be degenerate

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

finite de finetti theorem for conditional probability distributions describing physical theories we work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no signaling conditions we show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions generalizing the classical finite de finetti theorem of diaconis and freedman our results apply to correlations obtained from quantum states even when there is no bound on the local dimension so that known quantum de finetti theorems cannot be used

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

bound entangled gaussian states we discuss the entanglement properties of bipartite states with gaussian wigner functions for the separability and the positivity of the partial transpose we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state it is shown that for systems composed of a single oscillator for alice and an arbitrary number for bob positivity of the partial transpose implies separability however this implication fails with two oscillators on each side as we show by constructing a five parameter family of bound entangled gaussian states

finite size mean field models we characterize the two site marginals of exchangeable states of a system of quantum spins in terms of a simple positivity condition this result is used in two applications we first show that the distance between two site marginals of permutation invariant states on n spins and exchangeable states is of order 1 n the second application relates the mean ground state energy of a mean field model of composite spins interacting through a product pair interaction with the mean ground state energies of the components

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

a converse lyapunov theorem for almost sure stabilizability we prove a converse lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions given a stochastic system a s stochastic open loop stabilizable at the origin we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium this result provides with the direct lyapunov theorems proved in a companion paper a complete lyapunov like characterization of the a s stabilizability

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

minimizing task completion time with the execution set method we consider a partial ordered set poset of assembling operations with known execution durations that must be accomplished the assembling operations can be executed on an acyclic network with an identical set of robots on each conveyer arc the number of depots nodes is a known integer between each pair of depots we can locate only one conveyer we seek an arrangement of the network and a plan that divides the task operations among the conveyers minimizing the overall task completion time we use linear programming optimization subject to reasonably general rules for distributing the operation fragments among the conveyers

finding the critical path in an activity network with time switch constraints an activity network is an acyclic graph with non negative weights and with a unique source and destination a project consisting of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network provides useful information for managing the project in a traditional activity network it is assumed that an activity always begins after all of its preceding activities have been completed this assumption may not be adequate enough to describe some practical applications where some forms of time constraints are attached to an activity in this paper we investigate one type of time constraint called time switch constraint which assumes that an activity begins only in a specified time interval of a cycle with some pairs of exclusive components polynomial time algorithms are developed to find the critical path or longest path and analyze the float of each arc in this time constrained activity network the analysis shows that the critical path and float in this context diaer from those of a traditional activity network in some management perspectives and thus consideration of the time switch constraint leads to enhanced project management through more eaective use of budgets and resource allocation 2000 elsevier science b v all rights reserved

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

the dynamic programming equation for second order stochastic target problems motivated by applications in mathematical finance u cetin h m soner and n touzi options hedging for small investors under liquidity costs finance stoch to appear we continue our study of second order backward stochastic equations in this paper we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems in contrast with previous formulations of similar problems we restrict control processes to be continuous this new framework enables us to apply our results to a larger class of models also the resulting derivation is more transparent the main technical tool is the geometric dynamic programming principle in this context and it is proved by using the framework developed in h m soner and n touzi j eur math soc jems 8 2002 pp 201 236

a problem of singular stochastic control with optimal stopping in finite horizon in this paper we apply pde methods to solve a problem about minimizing the cost of a firm for operating a project which is a simple model of combined singular stochastic control and optimal stopping in finite horizon the value function v x t to the problem is governed by a time dependent hjb variational inequality with both gradient constraint and function constraint which leads to a time dependent stopping boundary and a moving boundary we will prove that two free boundaries are strictly monotonic and infinitely differentiable v partial xv partial tv are continuous and partial xx v is locally bounded based on these results we construct optimal control and optimal stopping

barrier option hedging under constraints a viscosity approach we study the problem of finding the minimal initial capital needed in order to hedge without risk a barrier option when the vector of proportions of wealth invested in each risky asset is constrained to lie in a closed convex domain in the context of a brownian diffusion model we provide a pde characterization of the superhedging price this extends the result of broadie cvitanicg and soner rev financial stud 11 1998 pp 59 79 and cvitanicg pham and touzi j appl probab 36 1999 pp 523 545 which was obtained for plain vanilla options and provides a natural numerical procedure for computing the corresponding superhedging price as a by product we obtain a comparison theorem for a class of parabolic pdes with relaxed dirichlet conditions involving a constraint on the gradient

optimal control versus stochastic target problems an equivalence result abstract within a general abstract framework we show that any optimal control problem in standard form can be translated into a stochastic target problem as defined in soner and touzi 2002 5 whenever the underlying filtered probability space admits a suitable martingale representation property this provides a unified way of treating these two classes of stochastic control problems as an illustration we show within a jump diffusion framework how the hamilton jacobi bellman equations associated to an optimal control problem in standard form can be easily retrieved from the partial differential equations associated to its stochastic target counterpart

stochastic target problems dynamic programming and viscosity solutions in this paper we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward sdes and forward backward sdes the controlled process x nu y nu takes values in mathbb r d times mathbb r and a given initial data for x nu 0 then the control problem is to find the minimal initial data for y nu so that it reaches a stochastic target at a specified terminal time t the main application is from financial mathematics in which the process x nu is related to stock price y nu is the wealth process and nu is the portfolio we introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation the boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense this provides a unique characterization of the value function which is the minimal initial data for y nu

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

the risk sensitive maximum principle for controlled forward backward stochastic differential equations abstract in this paper we consider the risk sensitive optimal control problem for forward backward stochastic differential equations fbsdes we consider two different cases case 1 the drift and diffusion terms of the forward part are independent of the backward part not fully coupled fbsde and the running cost does not include the backward part case 2 the fully coupled fbsde with terminal and initial costs in both cases the risk sensitive objective functional is considered the corresponding control domain is not necessarily convex and the diffusion term of the forward part depends on control and state variables in both cases by using the nonlinear transformations of the equivalent risk neutral problems we obtain two different risk sensitive maximum principles which are characterized in terms of the variational inequalities we show that the risk sensitive maximum principle of case 1 consists of the two coupled adjoint equations with an additional scalar adjoint equation which are backward stochastic differential equations bsdes the risk sensitive maximum principle of case 2 consists of the two coupled adjoint equations which are coupled fbsdes with an additional scalar adjoint equation and optimality conditions in both cases the additional scalar adjoint equations are bsdes which are induced due to the nonlinear transformations of the adjoint processes in the equivalent risk neutral problems through the application of results we consider the linear quadratic risk sensitive optimal control problem for the linear fbsde and by using the risk sensitive maximum principle of case 1 we characterize an explicit optimal solution

convex resource allocation problems on directed acyclic graphs duality complexity special cases and extensions consider the following resource allocation problem on a directed acyclic graph the precedence graph each vertex has a known work load and a fixed amount of total resource is available the time required to process a vertex is inversely proportional to the amount of the resource allocated to it the time to complete all of the work is the length of time to complete a longest chain in the graph the problem of finding an allocation which minimizes the time required to complete all of the work subject to the limited resource availability can be formulated as a separable convex programming problem we use results from lagrangian duality for convex programs the length width inequality and dilworth s theorem for directed acyclic graphs to obtain a strong relationship between optimal solutions of this problem and its dual this allows us to obtain closed form solutions for certain special classes of graphs and leads to a generalization of the lym property for partially ordered sets the computational complexity of the general problem is an open question however the ellipsoid method yields a fully polynomial approximation scheme and some light can be shed on the associated decision problem the results of this paper are shown to extend to resource allocation problems on perfect graphs

stochastic target problems with controlled loss we consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or more generally with a given level of expected loss by suitably increasing the state space and the controls we show that this problem can be converted into a stochastic target problem i e finding the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one unlike in the existing literature on stochastic target problems our increased controls are valued in an unbounded set in this paper we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls together with the appropriate boundary conditions these results are applied to the problem of quantile hedging in financial mathematics and are shown to recover the explicit solution of follmer and leukert finance stoch 3 1999 pp 251 273

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