approximation of cvar minimization for hedging under exponential levy models in this paper we study the hedging problem based on the cvar in incomplete markets as the superhedging is quite expensive in terms of initial capital we construct a self financing strategy that minimizes the cvar of hedging risk under a budget constraint on the initial capital in incomplete markets no explicit solution can be provided to approximate the problem we apply the neyman pearson lemma approach with a specific equivalent martingale measure afterwards we explicit the solution for call options hedging under the exponential levy class of price models this approach leads to an efficient and easy to implement method using the fast fourier transform we illustrate numerical results for the merton model

some ideas about quantitative convergence of collision models to their mean field limit we consider a stochastic n particle model for the spatially homogeneous boltzmann evolution and prove its convergence to the associated boltzmann equation when n with non asymptotic estimates for any time t 0 we bound the distance between the empirical measure of the particle system and the measure given by the boltzmann evolution in a relevant hilbert space the control got is gaussian i e we prove that the distance is bigger than xn 1 2 with a probability of type o e x 2 the two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of lipschitz continuity for the boltzmann collision kernel we study more extensively the case where our hilbert space is the homogeneous negative sobolev space smash dot h s then we are only able to give bounds for maxwellian models however numerical computations tend to show that our results are useful in practice

a phase transition behavior for brownian motions interacting through their ranks consider a time varying collection of n points on the positive real axis modeled as exponentials of n brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time if at each time point we divide the points by their sum under suitable assumptions the rescaled point process converges to a stationary distribution depending on n and the vector of drifts as time goes to infinity this stationary distribution can be exactly computed using a recent result of pal and pitman the model and the rescaled point process are both central objects of study in models of equity markets introduced by banner fernholz and karatzas in this paper we look at the behavior of this point process under the stationary measure as n tends to infinity under a certain continuity at the edge condition on the drifts we show that one of the following must happen either i all points converge to 0 or ii the maximum goes to 1 and the rest go to 0 or iii the processes converge in law to a non trivial poisson dirichlet distribution the underlying idea of the proof is inspired by talagrand s analysis of the low temperature phase of derrida s random energy model of spin glasses the main result establishes a universality property for the bfk models and aids in explicit asymptotic computations using known results about the poisson dirichlet law

quantum fluctuation relations for the lindblad master equation an open quantum system interacting with its environment can be modeled under suitable assumptions as a markov process described by a lindblad master equation in this work we derive a general set of fluctuation relations for systems governed by a lindblad equation these identities provide quantum versions of jarzynski hatano sasa and crooks relations in the linear response regime these fluctuation relations yield a fluctuation dissipation theorem fdt valid for a stationary state arbitrarily far from equilibrium for a closed system this fdt reduces to the celebrated callen welton kubo formula

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

on the martingale property of certain local martingales the stochastic exponential z t rm exp m t m 0 1 2 langle m m rangle t of a continuous local martingale m is itself a continuous local martingale we give a necessary and sufficient condition for the process z to be a true martingale in the case where m t int 0 t b y u dw u and y is a one dimensional diffusion driven by a brownian motion w furthermore we provide a necessary and sufficient condition for z to be a uniformly integrable martingale in the same setting these conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of y as an application we provide a deterministic criterion for the absence of bubbles in a one dimensional setting

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

convergence to equilibrium in wasserstein distance for fokker planck equations we describe conditions on non gradient drift diffusion fokker planck equations for its solutions to converge to equilibrium with a uniform exponential rate in wasserstein distance this asymptotic behaviour is related to a functional inequality which links the distance with its dissipation and ensures a spectral gap in wasserstein distance we give practical criteria for this inequality and compare it to classical ones the key point is to quantify the contribution of the diffusion term to the rate of convergence which to our knowledge is a novelty

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

nonconvex optimization for pricing and hedging in imperfect markets the paper deals with imperfect financial markets and provides new methods to overcome many inefficiencies caused by frictions transaction costs are quite general and far from linear or convex the concepts of pseudoarbitrage and efficiency are introduced and deeply analyzed by means of both scalar and vector optimization problems their optimality conditions and solutions yield strategies to invest and hedging portfolios as well as bid ask spread improvements they also point out the role of coalitions when dealing with these markets several sensitivity results will permit us to show that a significant transaction costs reduction is very often feasible in practice as well as to measure its effect on the general efficiency of the market all these findings may be especially important for many emerging and still illiquid spot or derivative markets electricity markets commodity markets markets related to weather inflation linked or insurance linked derivatives etc

fluctuation theorems work is not an observable the characteristic function of the work performed by an external time dependent force on a hamiltonian quantum system is identified with the time ordered correlation function of the exponentiated system s hamiltonian a similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the jarzynski work theorem

the tasaki crooks quantum fluctuation theorem starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system in talkner et al 2007 phys rev e 75 050102 preprint cond mat 0703213 the quantum version of the crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse fourier transformation

modified log sobolev inequality for a compact pure jump markov process with degenerate jumps we study the modified log sobolev inequality for a class of pure jump markov processes that describe the interactions between brain neurons as a result we obtain concentration properties for empirical approximations of the process in particular we focus on a finite and compact process with degenerate jumps inspired by the model introduced by galves and locherbach j stat phys 151 896 921 2013

an algorithm for multistage dynamic networks with random arc capacities with an application to dynamic fleet management we consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems we propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty this method decomposes the network in each stage into tree subproblems whose expected recourse functions are easy to obtain we also compare this method with two alternative methods on a set of dynamic fleet management problems the numerical results show that this method is superior to the two alternative methods

on the best constant of hardy sobolev inequalities abstract we obtain the sharp constant for the hardy sobolev inequality involving the distance to the origin this inequality is equivalent to a limiting caffarelli kohn nirenberg inequality in three dimensions in certain cases the sharp constant coincides with the best sobolev constant

sharp hardy inequalities in the half space with trace remainder term abstract in this paper we deal with a class of inequalities which interpolate the kato inequality and the hardy inequality in the half space starting from the classical hardy s inequality in the half space r n r n 1 0 we show that if we replace the optimal constant n 2 2 4 with a smaller one 2 2 4 2 n then we can add an extra trace term similar to that one that appears in the kato inequality the constant in the trace remainder term is optimal and it tends to zero when goes to n while it is equal to the optimal constant in kato s inequality when 2 the approach is based on a very classical method of calculus of variation due to weierstrass and developed by hilbert that usually is considered to prove that the solutions of the euler lagrange equation associated to a functional are in fact extremals in this paper we will show how this method is well suited also to functionals that have no extremals

some trace hardy type inequalities and trace hardy sobolev maz ya type inequalities abstract we prove a trace hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of alvino et al and of tzirakis on the upper half space we also prove some trace hardy sobolev maz ya type inequalities which generalize the recent results of filippas et al in applications we derive some hardy type inequalities and hardy sobolev maz ya type inequalities for fractional laplacian finally we prove the logarithmic sobolev trace inequalities and logarithmic hardy trace inequalities on the upper half spaces the best constants in these inequalities are explicitly computed in the radial case

efficient hedging with coherent risk measure abstract the idea of efficient hedging has been introduced by follmer and leukert they defined the shortfall risk as the expectation of the shortfall weighted by a loss function and looked for strategies that minimize the shortfall risk under a capital constraint in this paper to measure the shortfall risk we use the coherent risk measures introduced by artzner delbaen eber and heath we show that for a given contingent claim h the optimal strategy consists in hedging a modified claim h for some randomized test this is an analogue of the results by follmer and leukert

convergence to the equilibria for self stabilizing processes in double well landscape we investigate the convergence of mckean vlasov diffusions in a nonconvex landscape these processes are linked to nonlinear partial differential equations according to our previous results there are at least three stationary measures under simple assumptions hence the convergence problem is not classical like in the convex case by using the method in benedetto et al j statist phys 91 1998 1261 1271 about the monotonicity of the free energy and combining this with a complete description of the set of the stationary measures we prove the global convergence of the self stabilizing processes

all adapted topologies are equal a number of researchers have introduced topological structures on the set of laws of stochastic processes a unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes aldous defines an extended weak topology based on the weak convergence of prediction processes in the economic literature hellwig introduced the information topology to study the stability of equilibrium problems bion nadal and talay introduce a version of the wasserstein distance between the laws of diffusion processes pflug and pichler consider the nested distance and the weak nested topology to obtain continuity of stochastic multistage programming problems these distances can be seen as a symmetrization of lassalle s causal transport problem but there are also further natural ways to derive a topology from causal transport our main result is that all of these seemingly independent approaches define the same topology in finite discrete time moreover we show that this weak adapted topology is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions

unified treatment of the quantum fluctuation theorem and the jarzynski equality in terms of microscopic reversibility there are two related theorems which hold even in far from equilibrium namely fluctuation theorem and jarzynski equality fluctuation theorem states the existence of symmetry of fluctuation of entropy production while the jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes on the other hand the relationship between these theorems was investigated by crooks phys rev e 60 2721 1999 for the classical stochastic systems in this paper we derive quantum analogues of fluctuation theorem and jarzynski equality in terms of microscopic reversibility in other words the quantum analog of the work by crooks is presented also for the quasiclassical langevin system microscopically reversible condition is confirmed

optimizing improved hardy inequalities let be a bounded domain in rn n 3 containing the origin motivated by a question of brezis and vazquez we consider an improved hardy inequality with best constant b that we formally write as n 22 21 x 2 bv x we first give necessary conditions on the potential v under which the previous inequality can or cannot be further improved we show that the best constant b is never achieved in h01 and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in h01 our analysis reveals that the original inequality can be repeatedly improved by adding on the right hand side specific potentials this leads to an infinite series expansion of hardy s inequality the series obtained is in some sense optimal in establishing these results we derive various sharp improved hardy sobolev inequalities

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

weighted finsler trace hardy inequality on half spaces abstract we prove the sharp weighted trace hardy inequality on the half space in the finsler context the obtained inequality extends the results obtained in 23 29 in the euclidean context to the finsler context it also generalizes the recent result in 5 to the weight context

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

duality and martingales a stochastic programming perspective on contingent claims the hedging of contingent claims in the discrete time discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale the model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims however we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options an extension of the model to incorporate pre existing liabilities and endowments reveals the reasons why buyers and sellers trade in options the model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets

empirical processes indexed by lipschitz functions on obtient des conditions necessaires et suffisantes sur p pour que les boules unites de bl r et lip r soient des classes de p donsker fonctionnelles

minimal energy cost for thermodynamic information processing measurement and information erasure the fundamental lower bounds on the thermodynamic energy cost of measurement and information erasure are determined the lower bound on the erasure validates landauer s principle for a symmetric memory for other cases the bound indicates the breakdown of the principle our results constitute the second law of information thermodynamics in which information content and thermodynamic variables are treated on an equal footing

distances between transition probabilities of diffusions and applications to nonlinear fokker planck kolmogorov equations abstract we estimate the total variation and kantorovich distances between transition probabilities of two diffusions with different diffusion matrices and drifts via a natural quadratic distance between the drifts and diffusion matrices applications to nonlinear fokker planck kolmogorov equations optimal control and mean field games are given

stability of multistage stochastic programs quantitative stability of linear multistage stochastic programs is studied it is shown that the infima of such programs behave locally lipschitz continuous with respect to the sum of an l r distance and of a distance measure for the filtrations of the original and approximate stochastic input processes various issues of the result are discussed and an illustrative example is given consequences for the reduction of scenario trees are also discussed

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

on information gain by quantum measurements of continuous observables a generalization of shannon s amount of information into quantum measurements of continuous observables is introduced a necessary and sufficient condition for measuring processes to have a non negative amount of information is obtained this resolves groenewold s conjecture completely including the case of measurements of continuous observables as an application the approximate position measuring process considered by von neumann and later by davies is shown to have a non negative amount of information

improving interpolated hardy and trace hardy inequalities on bounded domains abstract in this work we study inequalities which interpolate weighted hardy and trace hardy inequalities on bounded domains we establish improvements of these inequalities by adding correction terms of hardy type with a singular logarithmic weight we show that this weight is optimal in the sense that the inequality fails for more singular weights moreover we determine the best values of the constants for the remaining terms we also show that the aforementioned inequalities can be repeatedly improved obtaining an infinite series of correction terms involving specific hardy type potentials in the two borderline cases of these interpolation inequalities we obtain refinements of the weighted hardy and the trace weighted hardy inequality respectively it follows that the trace hardy and the hardy weighted inequalities share the same optimal constant for the specific remaining terms these results constitute an extension of a well known result when the interpolation inequality reduces to the classical nonweighted hardy inequality

free transport entropy inequalities for non convex potentials and application to concentration for random matrices talagrand s inequalities provide a link between two fundamentals concepts of probability transportation and entropy the study of the counterpart of these inequalities in the context of free probability has been initiated by biane and voiculescu and later extended by hiai petz and ueda for convex potentials in this work we prove a free analogue of a result of bobkov and gotze in the classical setting thus providing free transport entropy inequalities for a very natural class of measures appearing in random matrix theory these inequalities are weaker than the ones of hiai petz and ueda but still hold beyond the convex case we then use this result to get a concentration estimate for beta ensembles under mild assumptions on the potential

sharp trace hardy sobolev inequalities and fractional hardy sobolev inequalities in this work we establish sharp weighted trace hardy inequalities with trace remainder terms involving the critical sobolev exponent corrected by a singular logarithmic weight we show that this weight is optimal in the sense that the inequality fails for more singular weights then we apply these results to derive sharp hardy inequalities and relative improvements associated with fractional s th powers of the laplacian s 0 1 in particular we deal with two different operators of this type defined on bounded domains it follows that hardy inequalities associated with two different fractional laplacians share the same best constant as well as they can be both sharpened by adding sobolev type remainder term involving the same optimal logarithmic correction hardy type remainder terms are also considered our results are in direct accordance with earlier results for their non fractional counterpart where s 1 that is the standard laplacian

scenario tree modeling for multistage stochastic programs an important issue for solving multistage stochastic programs consists in the approximate representation of the multivariate stochastic input process in the form of a scenario tree in this paper we develop stability theory based heuristics for generating scenario trees out of an initial set of scenarios they are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps conditions are established implying closeness of optimal values of the original process and its tree approximation respectively by relying on a recent stability result in heitsch romisch and strugarek siam j optim 17 511 525 2006 for multistage stochastic programs numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management

incorporating statistical model error into the calculation of acceptability prices of contingent claims the determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics given this model optimal bid and ask prices can be found by stochastic optimization however the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure we define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model this neighborhood serves as ambiguity set for a multistage stochastic optimization problem under model uncertainty we obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation moreover we prove a general large deviations result for the nested distance which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data

portfolio choice and optimal hedging with general risk functions a simplex like algorithm abstract the minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging there are two major reasons firstly heavy tails and the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies from the point of view of the second order stochastic dominance secondly but not less important many institutional investors must respect legal capital requirements which may be more easily studied if one deals with a risk measure related to capital losses this paper proposes a new method to simultaneously minimize several general risk or dispersion measures the representation theorems of risk functions are applied to transform the general risk minimization problem in a minimax problem and later in a linear programming problem between infinite dimensional banach spaces then new necessary and sufficient optimality conditions are stated and a simplex like algorithm is developed the algorithm solves the dual problem and provides both optimal portfolios and their sensitivities the approach is general enough and does not depend on any particular risk measure but some of the most important cases are specially analyzed a final real data numerical example illustrates the practical performance of the proposed methodology

dynamic asset allocation with uncertain jump risks a pathwise optimization approach this paper studies the dynamic portfolio choice problem with ambiguous jump risks in a multidimensional jump diffusion framework we formulate a continuous time model of incomplete market with uncertain jumps we develop an efficient pathwise optimization procedure based on the martingale methods and minimax results to obtain closed form solutions for the indirect utility function and the probability of the worst scenario we then introduce an orthogonal decomposition method for the multidimensional problem to derive the optimal portfolio strategy explicitly under ambiguity aversion to jump risks finally we calibrate our model to real market data drawn from 10 international indices and illustrate our results by numerical examples the certainty equivalent losses affirm the importance of jump uncertainty in optimal portfolio choice

convergence rates for empirical barycenters in metric spaces curvature convexity and extendable geodesics this paper provides rates of convergence for empirical generalised barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry we study the validity of variance inequalities in spaces of non positive and non negative aleksandrov curvature in this last scenario we show that variance inequalities hold provided geodesics emanating from a barycenter can be extended by a constant factor we also relate variance inequalities to strong geodesic convexity while not restricted to this setting our results are largely discussed in the context of the 2 wasserstein space

thermodynamic uncertainty relations including measurement and feedback thermodynamic uncertainty relations quantify how the signal to noise ratio of a given observable is constrained by dissipation fluctuation relations generalize the second law of thermodynamics to stochastic processes we show that any fluctuation relation directly implies a thermodynamic uncertainty relation considerably increasing their range of applicability in particular we extend thermodynamic uncertainty relations to scenarios which include measurement and feedback since feedback generally breaks time reversal invariance the uncertainty relations involve quantities averaged over the forward and the backward experiment defined by the associated fluctuation relation this implies that the signal to noise ratio of a given experiment can in principle become arbitrarily large as long as the corresponding backward experiment compensates e g by being sufficiently noisy we illustrate our results with the szilard engine as well as work extraction by free energy reduction in a quantum dot

a bound on the mutual information and properties of entropy reduction for quantum channels with inefficient measurements the holevo bound is a bound on the mutual information for a given quantum encoding in 1996 schumacher westmoreland and wootters phys rev lett 76 3452 1996 derived a bound that reduces to the holevo bound for complete measurements but that is tighter for incomplete measurements the most general quantum operations may be both incomplete and inefficient here we show that the bound derived by sww can be further extended to obtain one that is yet again tighter for inefficient measurements this allows us in addition to obtain a generalization of a bound derived by hall and to show that the average reduction in the von neumann entropy during a quantum operation is concave in the initial state for all quantum operations this is a quantum version of the concavity of the mutual information we also show that both this average entropy reduction and the mutual information for pure state ensembles are schur concave for unitarily covariant measurements that is for these measurements information

integral quantum fluctuation theorems under measurement and feedback control we derive integral quantum fluctuation theorems and quantum jarzynski equalities for a feedback controlled system and a memory which registers outcomes of the measurement the obtained equalities involve the information content which reflects the information exchange between the system and the memory and take into account the back action of a general measurement contrary to the classical case the generalized second law of thermodynamics under measurement and feedback control is reproduced from these equalities

from empirical observations to tree models for stochastic optimization convergence properties in multistage stochastic optimization we use stylized processes to model the relevant stochastic data processes the basis for building these models is empirical observations it is well known that the determining distance concept for multistage stochastic optimization problems is the nested distance and not the distance in distribution in this paper we investigate the question of how to generate models out of empirical data which approximate well the underlying stochastic processes in nested distance we demonstrate first that the empirical measure which is built from observed sample paths does not converge in nested distance to the pertaining distribution if the latter has a density on the other hand we show that smoothing convolutions which are appropriately adapted from classical kernel density estimation can be employed to modify the empirical measure in order to obtain stochastic processes which converge in nested distance to the underlying process we employ the results to estimate the cond

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

martingale and duality methods for utility maximization in a incomplete market the problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional brownian motion process w the coefficients of the bond and stock processes are adapted to the filtration history of w and incompleteness arises when the number of stocks is strictly smaller than the dimension of w it is shown that there is a way to complete the market by introducing additional fictitious stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market the notion of a least favorable completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market this notion is expounded upon using martingale techniques several equivalent characterizations are provided

random lsc functions an ergodic theorem an ergodic theorem for random lsc lower semicontinuous functions is obtained by relying on a scalarization of such functions in the process kolmogorov s extension theorem for random lsc functions is established applications to statistical estimation problems composite materials and stochastic optimization problems are briefly noted

sharp trace hardy sobolev maz ya inequalities and the fractional laplacian in this work we establish trace hardy and trace hardy sobolev maz ya inequalities with best hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity we then use them to produce fractional hardy sobolev maz ya inequalities with best hardy constants for various fractional laplacians in the case where the domain is the half space our results cover the full range of the exponent s in 0 1 of the fractional laplacians in particular we give a complete answer in the l2 setting to an open problem raised by frank and seiringer sharp fractional hardy inequalities in half spaces in around the research of vladimir maz ya international mathematical series 2010

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

log optimal and rapid paths in von neumann gale dynamical systems abstract von neumann gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors possessing certain properties of convexity and homogeneity a central role in the theory of such systems is played by a special class of paths trajectories called rapid they grow over each time period t 1 t in a sense faster than others the paper establishes existence and characterization theorems for such paths showing in particular that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid the proof of this result is based on the methods of convex analysis in spaces of measurable functions the study is motivated by the applications of the theory of von neumann gale dynamical systems to the modeling of capital growth in financial markets with friction

evaluations of risk measures for different probability measures stochastic optimization problems which arise in different areas such as simple asset allocation problems or problems in insurance often involve coherent risk measures in these real world problems the considered risk measure is frequently built on empirical distributions it is therefore of interest to understand the potential deviation which occurs when evaluating a risk measure for a perturbed or slightly changed probability measure this paper addresses the potential deviation it turns out that the wasserstein distance a well known distance for probability measures provides a valuable notion of distance in the present context many risk measures allow a precise quantification in terms of the wasserstein distance and important risk measures are continuous with respect to this distance for specific random variables which often occur in concrete real world problems it is moreover demonstrated that the derived constants describing the continuity properties cannot be improved the associated

shadow price of information in discrete time stochastic optimization the shadow price of information has played a central role in stochastic optimization ever since its introduction by rockafellar and wets in the mid seventies this article studies the concept in an extended formulation of the problem and gives relaxed sufficient conditions for its existence we allow for general adapted decision strategies which enables one to establish the existence of solutions and the absence of a duality gap e g in various problems of financial mathematics where the usual boundedness assumptions fail as applications we calculate conjugates and subdifferentials of integral functionals and conditional expectations of normal integrands we also give a dual form of the general dynamic programming recursion that characterizes shadow prices of information

infinite dimensional extension of a theorem of komlos a well known theorem of komlos is extended to integrable functions taking values in a reflexive banach space

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

fluctuation theorem with information exchange role of correlations in stochastic thermodynamics we establish the fluctuation theorem in the presence of information exchange between a nonequilibrium system and other degrees of freedom such as an observer and a feedback controller where the amount of information exchange is added to the entropy production the resulting generalized second law sets the fundamental limit of energy dissipation and energy cost during the information exchange our results apply not only to feedback controlled processes but also to a much broader class of information exchanges and provide a unified framework of nonequilibrium thermodynamics of measurement and feedback control

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

probabilistic approach for granular media equations in the non uniformly convex case we use here a particle system to prove both a convergence result with convergence rate and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex the proof of convergence is simpler than the one in carrillo mccann villani rev mat iberoamericana 19 971 1018 2003 arch rat mech anal 179 217 263 2006 all the results complete former results of malrieu ann appl probab 13 540 560 2003 in the uniformly convex case the main tool is an uniform propagation of chaos property and a direct control in wasserstein distance of solutions starting with different initial measures the deviation inequality is obtained via a t 1 transportation cost inequality replacing the logarithmic sobolev inequality which is no more clearly dimension free

on the scenario tree optimal value error for stochastic programming problems stochastic programming problems generally lead to large scale programs if the number of random outcomes is large or if the problem has many stages a way to tackle them is provided by scenario tree

convex duality in stochastic optimization and mathematical finance this paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration the framework unifies many well known duality frameworks from operations research and mathematical finance the unification allows the extension of some useful techniques from these two fields to a much wider class of problems in particular combining certain finite dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance we are able to close the duality gap in some situations where traditional topological arguments fail

risk measures on ordered non reflexive banach spaces this work is devoted to the study of coherent and convex risk measure on non reflexive banach spaces an extension of dual representation and continuity results which hold in the case of reflexive spaces is established in this paper for the class of non reflexive banach spaces this study also relies on the fact that the riskless bond is replaced by some numeraire asset which defines a base on the cone of the spot price functionals

version independence and nested distributions in multistage stochastic optimization the amount of stagewise available information is crucial in multistage stochastic optimization but unlike data which directly enter the profit loss functions of a decision problem information is invariant w r t bijective transformations the usual concept to deal with information in multistage stochastic programming is by introducing filtrations i e increasing sequences of sigma algebras to which the decisions must be adapted for the definition of filtrations one has to fix a certain probability space while random variables are typically given by their distributions only and all realizations of this distribution on some probability space are equivalent we introduce here the new concept of nested distributions to describe the information structure as well as the scenario process of a stochastic optimization program in a way which is independent of specific versions of probability spaces and random variables the setting is totally in distribution two stochastic programs with identical objective function and constraints are equivalent if and only if the scenario processes have the same nested distribution as a byproduct we analyze the question of whether introducing extra randomness by defining randomized decisions would lead to improvement in the objective value in the language of information this would mean that enlarging the filtration based on available information by conditionally independent additional random variables would have a positive effect we show that in general the answer is yes while for compound convex objectives the answer is no finally we define a distance between nested distributions which generalizes the well known kantorovich distance of probability distributions and demonstrates that this distance may be used in quantifying the quality of approximation between a continuous stochastic program and a tree discretization or between two tree discretizations

minimizing measures of risk by saddle point conditions the minimization of risk functions is becoming a very important topic due to its interesting applications in mathematical finance and actuarial mathematics this paper addresses this issue in a general framework many types of risk function may be involved a general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions this new problem involves banach spaces but a mean value theorem for risk measures is stated and this simplifies the dual problem then optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable this characterization is significantly different if one compares it with previous literature furthermore the saddle point condition very easily applies in practice four applications in finance and insurance are presented

generalized jarzynski equality under nonequilibrium feedback control the jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control the new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control our results lead to a generalized fluctuation dissipation theorem that reflects the readout information and they can be experimentally tested using small thermodynamic systems we illustrate our general results by introducing an information ratchet which can transport a brownian particle in one direction and extract positive work from the particle

nonlinear stochastic programming with a case study in continuous switching the optimal solution as well as the objective of stochastic programming problems vary with the underlying probability measure this paper addresses stability with respect to the underlying probability measure and stability of the objective the techniques presented are employed to make problems numerically tractable which are formulated by involving numerous scenarios or even by involving a continuous probability measure the results justify clustering techniques which significantly reduce computation times while guaranteeing a desired approximation quality the second part of the paper highlights newton s method to solve the reduced stochastic recourse problems the techniques presented exploit the particular structure of the recourse function of the stochastic optimization problem the tools are finally demonstrated on a benchmark problem which is taken from electrical power flows

convex duality in optimal investment under illiquidity we study the problem of optimal investment by embedding it in the general conjugate duality framework of convex analysis this allows for various extensions to classical models of liquid markets in particular we obtain a dual representation for the optimum value function in the presence of portfolio constraints and nonlinear trading costs that are encountered e g in modern limit order markets the optimization problem is parameterized by a sequence of financial claims such a parameterization is essential in markets without a numeraire asset when pricing swap contracts and other financial products with multiple payout dates in the special case of perfectly liquid markets or markets with proportional transaction costs we recover well known dual expressions in terms of martingale measures

stochastic programs without duality gaps this paper studies dynamic stochastic optimization problems parameterized by a random variable such problems arise in many applications in operations research and mathematical finance we give sufficient conditions for the existence of solutions and the absence of a duality gap our proof uses extended dynamic programming equations whose validity is established under new relaxed conditions that generalize certain no arbitrage conditions from mathematical finance

probability distance inequalities on riemannian manifolds and path spaces abstract we construct otto villani s coupling for general reversible diffusion processes on a riemannian manifold as an application some new estimates are obtained for wasserstein distances by using a sobolev poincare type inequality introduced by latala and oleszkiewicz the corresponding concentration estimates of the measure are presented finally our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the l2 distance and the intrinsic distance in particular talagrand s inequality holds on the path space over a compact manifold

a functional version of the birkhoff ergodic theorem for a normal integrand a variational approach in this paper we prove a new version of the birkhoff ergodic theorem bet for random variables depending on a parameter alias integrands this involves variational convergences namely epigraphical hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands we also have to establish the measurability of the epigraphical lower and upper limits with respect to the field of invariant subsets from the main result applications to uniform versions of the bet to sequences of random sets and to the strong consistency of estimators are briefly derived

a large deviation approach to some transportation cost inequalities new transportation cost inequalities are derived by means of elementary large deviation reasonings their dual characterization is proved this provides an extension of a well known result of s bobkov and f gotze their tensorization properties are investigated sufficient conditions and necessary conditions too for these inequalities are stated in terms of the integrability of the reference measure applying these results leads to new deviation results concentration of measure and deviations of empirical processes

stochastic programming duality multipliers for unbounded constraints with an application to mathematical finance a new duality theory is developed for a class of stochastic programs in which the probability distribution is not necessarily discrete this provides a new framework for problems which are not necessarily bounded are not required to have relatively complete recourse and do not satisfy the typical slater condition of strict feasibility these problems instead satisfy a different constraint qualification called direction free feasibility to deal with possibly unbounded constraint sets and calmness of a certain finite dimensional value function to serve as a weaker condition than strict feasibility to obtain the existence of dual multipliers in this way strong duality results are established in which the dual variables are finite dimensional despite the possible infinite dimensional character of the second stage constraints from this infinite dimensional dual problems are obtained in the space of essentially bounded functions it is then shown how this framework could be used to obtain duality results in the setting of mathematical finance

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

a stability result for linear markovian stochastic optimization problems in this paper we propose a semi metric for markov processes that allows to bound optimal values of linear markovian stochastic optimization problems similar to existing notions of distance for general stochastic processes our distance is based on transportation metrics as opposed to the extant literature the proposed distance is problem specific i e dependent on the data of the problem whose objective value we want to bound as a result we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature we derive several properties of the proposed semi metric and demonstrate its use in a stylized numerical example

a distance for multistage stochastic optimization models we describe multistage stochastic programs in a purely in distribution setting i e without any reference to a concrete probability space the concept is based on the notion of nested distributions which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made the nested distance between these distributions is introduced and turns out to be a generalization of the wasserstein distance for stochastic two stage problems we give characterizations of this distance and show its usefulness in examples the main result states that the difference of the optimal values of two multistage stochastic programs which are lipschitz and differ only in the nested distribution of the stochastic parameters can be bounded by the nested distance of these distributions this theorem generalizes the well known kantorovich rubinstein theorem which is applicable only in two stage situations to multistage moreover a dual characterization for the nested distance is established the setup is applicable both for general stochastic processes and for finite scenario trees in particular the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation minimizing at least heuristically this distance is what good scenario tree generation is all about

on the worst conditional expectation abstract we study continuous coherent risk measures on l p in particular the worst conditional expectations we show some representation theorems for them extending the results of artzner delbaen eber heath and kusuoka