follmer schweizer decomposition and mean variance hedging for general claims let x be an r d valued special semimartingale on a probability space f f t 0 t t p with decomposition x x 0 m a and the space of all predictable x integrable processes such that dx is in the space s 2 of semimartingales if h is a random variable in l 2 we prove under additional assumptions on the process x that h can be written as the sum of an f 0 measurable random variable h 0 a stochastic integral of x and a martingale part orthogonal to m moreover this decomposition is unique and the function mapping h with its decomposition is continuous with respect to the l 2 norm finally we deduce from this continuity that the subspace of l 2 generated by dx where is closed in l 2 and we give some applications of this result to financial mathematics

harmonic analysis of stochastic equations and backward stochastic differential equations the bmo martingale theory is extensively used to study nonlinear multi dimensional stochastic equations in mathcal r p p in 1 infty and backward stochastic differential equations bsdes in mathcal r p times mathcal h p p in 1 infty and in mathcal r infty times overline l infty rm bmo with the coefficients being allowed to be unbounded in particular the probabilistic version of fefferman s inequality plays a crucial role in the development of our theory which seems to be new several new results are consequently obtained the particular multi dimensional linear cases for stochastic differential equations sdes and bsdes are separately investigated and the existence and uniqueness of a solution is connected to the property that the elementary solutions matrix for the associated homogeneous sde satisfies the reverse holder inequality for some suitable exponent p 1 finally some relations are established between kazamaki s quadratic critical exponent b m of a bmo martingale m and the spectral radius of the stochastic integral operator with respect to m which lead to a characterization of kazamaki s quadratic critical exponent of bmo martingales being infinite

variance optimal hedging in discrete time we solve the problem of approximating in 2 a given random variable h by stochastic integrals gt of a given discrete time process x we interpret h as a contingent claim to be paid out at time t x as the price evolution of some risky asset in a financial market and g as the cumulative gains from trade using the hedging strategy as an application we determine the variance optimal strategy which minimizes the variance of the net loss h gt over all strategies

a measurable group theoretic solution to von neumann s problem we give a positive answer in the measurable group theory context to von neumann s problem of knowing whether a non amenable countable discrete group contains a non cyclic free subgroup we also get an embedding result of the free group von neumann factor into restricted wreath product factors

infinite volume limit of the abelian sandpile model in dimensions d 3 we study the abelian sandpile model on mathbb z d in d 3 we prove existence of the infinite volume addition operator almost surely with respect to the infinite volume limit of the uniform measures on recurrent configurations we prove the existence of a markov process with stationary measure and study ergodic properties of this process the main techniques we use are a connection between the statistics of waves and uniform two component spanning trees and results on the uniform spanning forest measure on mathbb z d

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

a direct representation for the large sample maximum likelihood estimator of a gaussian autoregressive moving average process summary a simple procedure is described for obtaining the estimator of the parameters of an autoregressive moving average process the estimator is derived from the maximum likelihood procedure assuming gaussian residuals by making two simplifying assumptions which are negligible in large samples components of the estimator are expressed as linear combinations of a suitable set of sample serial correlations or as a rational function of such linear combinations

a semimartingale backward equation and the variance optimal martingale measure under general information flow we consider a financial market model where the dynamics of asset prices are given by an rd valued continuous semimartingale and the information flow is right continuous using the dynamic programming approach we express the variance optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation we consider two extreme cases when this equation admits an explicit solution in particular we give necessary and sufficient conditions in order that the variance optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case

scr e martingales and their applications in mathematical finance after introducing a new concept the notion of e martingale we extend the well known doob inequality for 1 p and the burkholder davis gundy inequalities for p 2 to e martingales by means of these inequalities we give sufficient conditions for the closedness of a space of stochastic integrals with respect to a fixed r d valued semimartingale a question which arises naturally in the applications to financial mathematics we also provide a necessary and sufficient condition for the existence and uniqueness of the follmer schweizer decomposition

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

robustness of quadratic hedging strategies in finance via fourier transforms in this paper we investigate the consequences of the choice of the model to partial hedging in incomplete markets in finance in fact we consider two models for the stock price process the first model is a geometric levy process in which the small jumps might have infinite activity the second model is a geometric levy process where the small jumps are truncated or replaced by a brownian motion which is appropriately scaled to prove the robustness of the quadratic hedging strategies we use pricing and hedging formulas based on fourier transform techniques we compute convergence rates and motivate the applicability of our results with examples

approximating random variables by stochastic integrals let x be a semimartingale and the space of all predictable x integrable processes such that dx is in the space 2 of semimartingales we consider the problem of approximating a given random variable h l 2 by a stochastic integral 0 t s dx s with respect to the l 2 norm if x is special and has the form x x 0 m d we construct a solution in feedback form under the assumptions that 2 d is deterministic and that h admits a strong f s decomposition into a constant a stochastic integral of x and a martingale part orthogonal to m we provide sufficient conditions for the existence of such a decomposition and we give several applications to quadratic optimization problems arising in financial mathematics

on the minimal entropy martingale measure let x be a locally bounded semimartingale using the theory of bmo martingales we give a sufficient criterion for a martingale measure for x to minimize relative entropy among all martingale measures this is applied to prove convergence of the q optimal martingale measure to the minimal entropy martingale measure in entropy for q 1 under the assumption that x is continuous and that the density process of some equivalent martingale measure satisfies a reverse llogl inequality

hedged monte carlo low variance derivative pricing with objective probabilities abstract we propose a new hedged monte carlo hmc method to price financial derivatives which allows to determine simultaneously the optimal hedge the inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading and for the very same reason reduces considerably the variance of our hmc scheme as compared to previous methods the explicit accounting of the hedging cost naturally converts the objective probability into the risk neutral one this allows a consistent use of purely historical time series to price derivatives and obtain their residual risk the method can be used to price a large class of exotic options including those with path dependent and early exercise features

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

kazhdan groups have cost 1 we prove that every countably infinite group with kazhdan s property t has cost 1 answering a well known question of gaboriau it remains open if they have fixed price 1

on the structure of general mean variance hedging strategies we provide a new characterization of mean variance hedging strategies in a general semimartingale market the key point is the introduction of a new probability measure p which turns the dynamic asset allocation problem into a myopic one the minimal martingale measure relative to p coincides with the variance optimal martingale measure relative to the original probability measure p

a transformation that will circumvent the problem of autocorrelation in an error component model abstract this paper derives a simple transformation which will transform serially correlated error components disturbances into spherical disturbances although and 1 are well known in the literature see lillard and willis 1978 the derivation of 1 2 has many advantages i it transforms gls into a wls procedure and therefore simplifies the computation ii it provides natural estimates of the variance components iii the transformation obtained can be easily extended to handle more general error processes on the remainder disturbances this is illustrated for the ar 1 model ar 2 model and the specialized ar 4 model for quarterly data also for the ar 1 model this transformation is extended to handle alternative assumptions on the initial observation this paper also shows that breusch s 1987 results on maximum likelihood estimation for the random error component model extend to the case of serial correlation in the remainder term this suggests an iterative gls procedure for obtaining the maximum likelihood estimates

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

model risk and discretisation of locally risk minimising strategies we consider two models for the price process a time continuous jump diffusion and a time discretisation of it then we study the robustness of the related locally risk minimising strategy to this model choice where we focus mainly on hedging asian and spread options using the discretisation scheme and the convergence results on backward stochastic differential equations as studied in khedher and vanmaele 2016 we show that the discrete time locally risk minimising strategies converge to the corresponding continuous time strategies in an l 2 sense we present different numerical examples to illustrate our results

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

estimation of a linear regression model with stationary arma p q errors abstract it is well known that consistent estimation of a linear regression model with a stationary gaussian arma process in the errors can be carried out by maximum likelihood or alternatively by two stage procedures involving estimation of the nuisance parameters followed by feasible generalized least squares for the model parameters we show that the estimators coincide up to o p t 3 2 and derive the variance to o t 2 which up to terms of this order is the same for both estimators considering the form of the error covariance matrix for an arma p q process allows us to examine a computationally convenient algorithm for estimation of the parameters of the regression model finally we provide a monte carlo comparison of the small sample properties of ols and two versions of the proposed estimator

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

testing for heteroskedasticity and serial correlation in a random effects panel data model this paper considers a panel data regression model with heteroskedastic as well as serially correlated disturbances and derives a joint lm test for homoskedasticity and no first order serial correlation the restricted model is the standard random individual error component model it also derives a conditional lm test for homoskedasticity given serial correlation as well as a conditional lm test for no first order serial correlation given heteroskedasticity all in the context of a random effects panel data model monte carlo results show that these tests along with their likelihood ratio alternatives have good size and power under various forms of heteroskedasticity including exponential and quadratic functional forms

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

the components of the wired spanning forest are recurrent we show that a s all of the connected components of the wired spanning forest are recurrent proving a conjecture of benjamini lyons peres and schramm our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree we believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests

on quadratic cost criteria for option hedging consider an option with maturity time t corresponding to a contingent claim h in an incomplete market a fair hedging price for h should take into account an optimal dynamical hedging plan against h let ct be the cumulative cost and t be the set of events of the history up to time t you can choose the plan at time t such that you minimize i e ct 1 ct 2 t ii e ct ct 2 t or iii e ct c0 2 sufficient conditions on the underlying stochastic process in discrete time are provided such that the fair hedging price does not depend on the choice of i ii or iii which fact should increase its acceptability

efficiency of iterative estimators in the regression model with ar 1 disturbances abstract the mean square error approximation method of nagar is applied to the iterated prais winsten and iterated maximum likelihood estimators of regression coefficients in the model with ar 1 disturbances their mean square errors are found to equal that of the two stage prais winsten estimator at the second order level of approximation

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

the stochastic difference between econometric statistics dans un contexte general et dans divers cas particuliers on calcule l ordre en probabilite de la difference entre des racines consistantes d equations d estimation rivales

on l 2 projections on a space of stochastic integrals let x be an ir d valued continuous semimartingale t a fixed time horizon and the space of all ir d valued predictable x integrable processes such that the stochastic integral g d xis a square integrable semimartingale a re cent paper of delbaen monat schachermayer schweizer stricker 1996 gives necessary and sufficient conditions on x for gt to be closed in l 2 p in this paper we describe the structure of the l 2 projection mapping an ft measurable random variable h l 2 p o ngt and provide the resulting integrand h in feedback form this is related to variance optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on x our proofs use the variance optimal martin gale measure p for x and weighted norm inequalities relating p to the original measure p

indistinguishability of the components of random spanning forests we prove that the infinite components of the free uniform spanning forest fusf of a cayley graph are indistinguishable by any invariant property given that the forest is different from its wired counterpart similar result is obtained for the free minimal spanning forest fmsf we also show that with the above assumptions there can only be 0 1 or infinitely many components which solves the problem for the fusf of caylay graphs completely these answer questions by benjamini lyons peres and schramm for cayley graphs which have been open up to now our methods apply to a more general class of percolations those satisfying weak insertion tolerance and work beyond cayley graphs in the more general setting of unimodular random graphs

on the maximum likelihood estimation of the parameters of a gaussian moving average process summary several authors have described methods for computing the exact maximum likelihood estimates for the parameters of a gaussian moving average process in this paper an iterative procedure for solving the likelihood equations is derived which includes the covariance determinant the procedure expresses the estimator of each parameter as a linear combination of a suitably large set of sample serial correlations which has many of the computational advantages of the direct representation of godolphin 1977 1978 it is also shown that the two procedures lead to virtually the same set of estimates for sample sizes likely to be encountered in practice these results imply that the additional effort required to compute exact maximum likelihood estimates is hard to justify illustrations of all three procedures are given separately for the moving average model of order one the iterative procedure for maximizing the exact likelihood function is less stable than the approximate procedures in certain cases

transforming the error components model for estimation with general arma disturbances abstract baltagi and li 1991 give a transformation which may be applied to certain autocorrelated disturbances in an error components model to yield spherical disturbances they derive the transformation for ar 1 and ar 2 cases we show that the results of galbraith and zinde walsh 1992 on the general arma form of the transformation matrix can be applied to allow estimation of the error components model with disturbances which follow any arma p q process we also show that for models with a cross sectional dimension which is large relative to the time dimension it is possible to use this method to perform a general autocorrelation transformation without specifying a particular arma process or estimating arma parameters

the component graph of the uniform spanning forest transitions in dimensions 9 10 11 ldots 9 10 11 we prove that the uniform spanning forests of mathbb z d and mathbb z ell have qualitatively different connectivity properties whenever ell d ge 4 in particular we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex which we call the component graph we introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8 to separate dimensions 5 6 7 and 8 we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph our result sharpens a theorem of benjamini kesten peres and schramm who proved that the diameter of the component graph increases by one every time the dimension increases by four