martingale measures and hedging for discrete time financial markets the price of stocks is modelled by a discrete time square integrable vector valued process x no further boundedness condition on x is imposed contingent claims h are described by square integrable random variables one looks for values v of the initial wealth v that allow for super hedging h by some portfolio plan in several cases the smallest value v is known to coincide with the maximal expectation of h under equivalent martingale measures here within an l2 framework another sufficient condition is provided which can be looked upon as a stronger form of the no arbitrage condition the mathematical tool and one of the main contributions is an optional decomposition theorem for a process which is a supermartingale under any equivalent martingale measure the upper price process for a contingent claim is shown to be a typical example for such a process moreover it is shown that in a markovian model one can restrict attention to markovian portfolio plans and to markovian martingale measures

the fundamental theorem of asset pricing for unbounded stochastic processes the fundamental theorem of asset pricing states roughly speaking that the absence of arbitrage possibilities for a stochastic process s is equivalent to the existence of an equivalent martingale measure for s it turns out that it is quite hard to give precise and sharp versions of this theorem in proper generality if one insists on modifying the concept of no arbitrage as little as possible it was shown in ds94 that for a locally bounded r d valued semi martingale s the condition of no free lunch with vanishing risk is equivalent to the existence of an equivalent local martingale measure for the process s it was asked whether the local boundedness assumption on s may be dropped in the present paper we show that if we drop in this theorem the local boundedness assumption on s the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma martingale measure the concept of sigma martingales was introduced by chou and emery under the name of semimartingales de la classe sigma m we provide an example which shows that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigma martingales on the other hand we also observe that for the applications in mathematical finance the notion of sigma martingales provides a natural framework when working with non locally bounded processes s the duality results which we obtained earlier are also extended to the non locally bounded case as an application we characterize the hedgeable elements author s abstract

mean variance hedging under partial information we consider the mean variance hedging problem under partial information the underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available we show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term which is formulated in terms of observable processes we prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation

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

on the spanning property of risk bonds priced by equilibrium we propose a method of pricing financial securities written on nontradable underlyings such as temperature or precipitation levels to this end we analyze a financial market where agents are exposed to financial and nonfinancial risk factors the agents hedge their financial risk in the stock market and trade a risk bond issued by an insurance company from the issuer s point of view the bond s primary purpose is to shift insurance risks related to noncatastrophic weather events to financial markets as such its terminal payoff and yield curve depend on an underlying climate or temperature process whose dynamics are independent of the randomness driving stock prices we prove that if the bond s payoff function is monotone in the external risk process it can be priced by an equilibrium approach the equilibrium market price of climate risk and the equilibrium price process are characterized as solutions of nonlinear backward stochastic differential equations bsdes transferring the bsdes into partial differential equations pdes we represent the bond prices as smooth functions of the underlying risk factors our analytical results make the model amenable to a numerical analysis

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

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

some important classes of quantum measurements and their information gain the notions of ideal first kind and repeatable measurements are reformulated and their relations analyzed within a systematic theory of measurement in quantum mechanics this allows one to investigate new information theoretical characterizations of such measurements

pricing of american contingent claims with jump stock price and constrained portfolios in this paper we study the problem of pricing american contingent claims in an incomplete market where the stock price process is supposed to be driven by both a wiener process and a poisson random measure and the portfolios are constrained we formulate this problem as to find the minimal solution of a backward stochastic differential equation bsde with constraints we use the penalization method to construct a sequence of bsdes with respect to wiener process and poisson random measure and we show that the solutions of these equations converge to the minimal solution we are interested in finally in the markovian case we characterize the minimal hedging price as the minimal viscosity supersolution of an integral partial differential inequality with constraints

optimal consumption from investment and random endowment in incomplete semimartingale markets we consider the problem of maximizing expected utility from consumption in a con strained incomplete semimartingale market with a random endowment process and establish a general existence and uniqueness result using techniques from convex duality the notion of as ymptotic elasticity of kramkov and schachermayer is extended to the time dependent case by imposing no smoothness requirements on the utility function in the temporal argument we can treat both pure consumption and combined consumption terminal wealth problems in a com mon framework to make the duality approach possible we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of l 1 to its topological bidual l 1 a space of finitely additive measures as an application we treat a constrained ito process market model as well as a totally incomplete model

static arbitrage bounds on basket option prices we consider the problem of computing upper and lower bounds on the price of an european basket call option given prices on other similar options although this problem is hard to solve exactly in the general case we show that in some instances the upper and lower bounds can be computed via simple closed form expressions or linear programs we also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function we show that this relaxation is tight in some of the special cases examined before

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

numerical analysis of american option pricing in a jump diffusion model we discuss pricing formulae for american options in merton s jump diffusion model with the help of variational inequalities we derive some regularity properties of price functions using the finite difference method a discretization scheme is presented and a convergence theorem for the first order derivatives is proved numerical methods and results are also discussed

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

an infinite dimensional lp duality theorem this paper constructs an infinite dimensional version of the duality theorem for a linear program lp the algebraic dual lp is replaced with a new program called the topological dual lp that closes the range of the adjoint operator under some mild nondegeneracy conditions involving strict positivity the new duality theorem asserts that the optimal value of the primal lp equals the optimal value of the topological dual lp some applications to mathematical finance are also included

optimal stopping with multiple priors we develop a theory of optimal stopping under knightian uncertainty a suitable martingale theory for multiple priors is derived that extends the classical dynamic programming or snell envelope approach to multiple priors we relate the multiple prior theory to the classical setup via a minimax theorem in a multiple prior version of the classical model of independent and identically distributed random variables we discuss several examples from microeconomics operation research and finance for monotone payoffs the worst case prior can be identified quite easily with the help of stochastic dominance arguments for more complex payoff structures like barrier options model ambiguity leads to stochastic changes in the worst case beliefs

approximation pricing and the variance optimal martingale measure let x be a semimartingale and let be the space of all predictable x integrable processes v such that so dx is in the space s 2 of semi martingales we consider the problem of approximating a given random variable h l 2 p by the sum of a constant c and a stochastic integral s 0 t v s dx s with respect to the l 2 p norm this problem comes from financial mathematics where the optimal constant v 0 can be interpreted as an approximation price for the contingent claim h an elementary computation yields v 0 as the expectation of h under the variance optimal signed martingale measure p and this leads us to study p in more detail in the case of finite discrete time we explicitly construct p by backward recursion and we show that p is typically not a probability but only a signed measure in a continuous time framework the situation becomes rather different we prove that p is nonnegative if x has continuous paths and satisfies a very mild no arbitrage condition as an application we show how to obtain the optimal integrand in feedback form with the help of a backward stochastic differential equation

translationally invariant multipartite bell inequalities involving only two body correlators bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems the known constructions of multipartite bell inequalities contain however correlation functions involving all observers making their experimental implementation difficult the main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest to measure quantities that is the two body correlations in particular we determine all three and four partite bell inequalities constructed from one and two body expectation values that obey translational symmetry and show that they reveal nonlocality in multipartite states also by providing a particular example of a five partite bell inequality we show that nonlocality can be detected from two body correlators involving only nearest neighbours finally we demonstrate that any translationally invariant bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites we provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant bell inequality this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

bounding the set of classical correlations of a many body system we present a method to certify the presence of bell correlations in experimentally observed statistics and to obtain new bell inequalities our approach is based on relaxing the conditions defining the set of correlations obeying a local hidden variable model yielding a convergent hierarchy of semidefinite programs sdp s because the size of these sdp s is independent of the number of parties involved this technique allows us to characterize correlations in many body systems as an example we illustrate our method with the experimental data presented in science 352 441 2016 scieas0036 807510 1126 science aad8665

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

max plus decomposition of supermartingales and convex order application to american options and portfolio insurance we are concerned with a new type of supermartingale decomposition in the max plus algebra which essentially consists in expressing any supermartingale of class d as a conditional expectation of some running supremum process as an application we show how the max plus supermartingale decomposition allows in particular to solve the american optimal stopping problem without having to compute the option price some illustrative examples based on one dimensional diffusion processes are then provided another interesting application concerns the portfolio insurance hence based on the max plus martingale we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process on every intermediate date w r t the convex order on terminal values

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

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

unitary measurements of discrete quantities in quantum mechanics the pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given probabilistic aspects of measurements related to the so called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined the problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus as a by product a generalization of the wigner araki yanase theorem is given

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

fully coupled forward backward stochastic differential equations and applications to optimal control existence and uniqueness results of fully coupled forward backward stochastic differential equations with an arbitrarily large time duration are obtained some stochastic hamilton systems arising in stochastic optimal control systems and mathematical finance can be treated within our framework

minimal supersolutions of convex bsdes we study the nonlinear operator of mapping the terminal value to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in y convex in z jointly lower semicontinuous and bounded below by an affine function of the control variable z we show existence uniqueness monotone convergence fatou s lemma and lower semicontinuity of this operator we provide a comparison principle for minimal supersolutions of bsdes

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

tight multipartite bell s inequalities involving many measurement settings which quantum states do not allow a local realistic lr description this question still remains open mainly because our present tools to test local realism are not optimal most of bell s inequalities are for the case in which only two measurement settings can be chosen by each observer e g the clauser horneshimony holt chsh inequality 1 inequalities for bipartite higher dimensional systems 2 multipartite bell s inequalities 3 6 one can call such inequalities standard ones one can expect that allowing the observers to choose between more than two observables should give more stringent constraints on lr models thus new inequalities may extend the class of nonseparable states which cannot be described by lr variables violation of local realism is an important ingredient for building quantum information protocols that decrease the communication complexity 7 and is a criterion for the efficient quantum key distribution 8 multisetting bell s inequalities may lead to such novel schemes although various nonstandard bell s inequalities are known 9 13 we still lack an efficient method for their derivation a general way is to define the facets of the correlation polytope 9 13 yet this is a computationally hard np problem 14 recently wu and zong 15 derived an inequality for three parties which involves four local settings for two observers and two settings for the third one and showed that it is stronger than the standard three particle bell inequalities here by generalizing the method of ref 15 we derive a wide class of tight bell s inequalities for n 2 parties and many measurement settings we find a single general bell s inequality that generates all inequalities from this class we also give a necessary and sufficient condition for the violation of the new inequalities by quantum predictions most importantly the inequalities reveal violation of local realism for the classes of states for which all standard bell s inequalities 5 6 are satisfied we start with the case ofn 3 observers suppose that the first two observers can choose between four settings and the third one between two settings we denote such problem as 4 4 2 l etai withi 2f 1 2 3 4g stand for the predetermined local realistic values for the first observer under the local settingi bj withj 2f 1 2 3 4g for similar values for the second observer and ck withk 2 f1 2g for the values for the third observer for the given run of the experiment we assume thata i b j a ndc k can take values 1 or 1 then the lr values satisfy the following algebraic identity a12 s 0 x k l 1 2 s 0 k l a1 1 k a 2 b1 1 l b 2 4

backward sdes with constrained jumps and quasi variational inequalities we consider a class of backward stochastic differential equations bsdes driven by brownian motion and poisson random measure and subject to constraints on the jump component we prove the existence and uniqueness of the minimal solution for the bsdes by using a penalization approach moreover we show that under mild conditions the minimal solutions to these constrained bsdes can be characterized as the unique viscosity solution of quasi variational inequalities qvis which leads to a probabilistic representation for solutions to qvis such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems as a direct consequence this suggests a numerical scheme for the solution of such qvis via the simulation of the penalized bsdes

towards objectification of measurement in an orthodox but incomplete quantum mechanics the problem of objectification of measurement is investigated in orthodox quantum mechanics that is assuming the validity of the schrodinger equation and the other postulates for the composite object plus measuring instrument system it is argued that by giving physical meaning to basic entities the positivist and the realist positions must be explicitly distinguished it is shown that the objectification problem is satisfactorily solved within quantum mechanics for the positivist but not for the realist for the latter a partially satisfactory solution is obtained by assuming that the quantum mechanical description of individual systems is incomplete and by assuming further that the pointer observable has definite values prior to any measurement in any quantum state a so called beable