the neron component series of an abelian variety we introduce the neron component series of an abelian variety a over a complete discretely valued field this is a power series in mathbb z left left t right right which measures the behaviour of the number of components of the neron model of a under tame ramification of the base field if a is tamely ramified then we prove that the neron component series is rational it has a pole at t 1 whose order equals one plus the potential toric rank of a this result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties we expect that it extends to the wildly ramified case we prove this if a is an elliptic curve and if a has potential purely multiplicative reduction

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

stable reduction of curves and tame ramification we study stable reduction of curves in the case where a tamely ramified base extension is sufficient if x is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring there is a criterion due to saito that describes precisely in terms of the geometry of the minimal model with strict normal crossings of x when a tamely ramified extension suffices in order for x to obtain stable reduction for such curves we construct an explicit extension that realizes the stable reduction and we furthermore show that this extension is minimal we also obtain a new proof of saito s criterion avoiding the use of l adic cohomology and vanishing cycles

vanishing cycles and geometry of curves over a discrete valuation ring 0 introduction geometry of curves over a discrete valuation ring is reflected in the action of the monodromy on the f adic cohomology groups the stable reduction theorem d m gives an equivalent geometrical condition for the action of the inertia group to be unipotent in the global situation where global means that the curve in consideration is proper in this paper we will generalize it in two directions one is to the local situation with the isolated singularity the other is to consider the geometrical condition for the action of the wild ramification group to be trivial we will also give a purely cohomological proof without using picard schemes of the stable reduction theorem and a new proof of the formula of deligne la theorem 5 1 1 based only on the stable reduction theorem here we only treat a strictly local discrete valuation ring with algebraically closed residue field however the results apply to any discrete valuation ring with perfect residue field since they are etale local we introduce some notations and terminologies

a recursive algorithm for the inversion of matrices with circulant blocks we investigate the recursive inversion of matrices with circulant blocks matrices of this type appear in several applications of computational electromagnetics and in the numerical solution of integral equations with the boundary element method the inversion is based on the diagonalization of each circulant block by means of the discrete fourier transform and the application of a recursive algorithm for the inversion of the matrix with diagonal blocks determined by the eigenvalues of each block the efficiency of the recursive inversion is exhibited by determining its computational complexity an implementation of the algorithm in matlab is given and numerical results are presented to demonstrate the efficiency in terms of cpu time of our approach

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

the minimum maximum of a continuous martingale with given initial and terminal laws let m t 0 t 1 be a continuous martingale with initial law m 0 0 and terminal law m 1 1 and let s sup 0 t 1 m t in this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures on the law of s we give an explicit construction of this bound furthermore a martingale is constructed which attains this minimum by solving a skorokhod embedding problem the form of this martingale is motivated by a simple picture the result is applied to the robust hedging of a forward start digital option

motivic zeta functions of degenerating calabi yau varieties we study motivic zeta functions of degenerating families of calabi yau varieties our main result says that they satisfy an analog of igusa s monodromy conjecture if the family has a so called galois equivariant kulikov model we provide several classes of examples where this condition is verified we also establish a close relation between the zeta function and the skeleton that appeared in kontsevich and soibelman s non archimedean interpretation of the syz conjecture in mirror symmetry

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

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

geometry of distribution constrained optimal stopping problems we adapt ideas and concepts developed in optimal transport and its martingale variant to give a geometric description of optimal stopping times tau of brownian motion subject to the constraint that the distribution of tau is a given probability mu the methods work for a large class of cost processes at a minimum we need the cost process to be measurable and mathcal f 0 t t ge 0 adapted continuity assumptions can be used to guarantee existence of solutions we find that for many of the cost processes one can come up with the solution is given by the first hitting time of a barrier in a suitable phase space as a by product we recover classical solutions of the inverse first passage time problem shiryaev s problem

the geometry of multi marginal skorokhod embedding the skorokhod embedding problem is one of the classical problems in the theory of stochastic processes with applications in many different fields cf the surveys hobson in paris princeton lectures on mathematical finance 2010 volume 2003 of lecture notes in mathematics springer berlin 2011 obloj in probab surv 1 321 390 2004 many of these applications have natural multi marginal extensions leading to the optimal multi marginal skorokhod problem some of the first papers to consider this problem are brown et al probab theory relat fields 119 4 558 578 2001 hobson seminaire de probabilites xxxii volume 1686 of lecture notes in mathematics springer berlin 1998 madan and yor bernoulli 8 4 509 536 2002 however this turns out to be difficult using existing techniques only recently a complete solution was be obtained in cox et al probab theory relat fields 173 211 259 2018 establishing an extension of the root construction while other instances are only partially answered or remain wide open in this paper we extend the theory developed in beiglbock et al invent math 208 2 327 400 2017 to the multi marginal setup which is comparable to the extension of the optimal transport problem to the multi marginal optimal transport problem as for the one marginal case this viewpoint turns out to be very powerful in particular we are able to show that all classical optimal embeddings have natural multi marginal counterparts notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases moreover our results also have consequences for the study of the martingale transport problem as well as the peacock problem

model independent bounds for asian options a dynamic programming approach we consider the problem of finding model independent bounds on the price of an asian option when the call prices at the maturity date of the option are known our methods differ from most approach

martingale optimal transport duality we obtain a dual representation of the kantorovich functional defined for functions on the skorokhod space using quotient sets our representation takes the form of a choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets these sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given another important ingredient of our analysis is a regularized version of jakubowski s s topology on the skorokhod space

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

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

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