homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

generalized covariations local time and stratonovich ito s formula for fractional brownian motion with hurst index h 1 4 given a locally bounded real function g we examine the existence of a 4 covariation g b h b h b h b h where b h is a fractional brownian motion with a hurst index h ge tfrac 1 4 we provide two essential applications first we relate the 4 covariation to one expression involving the derivative of local time in the case h tfrac 1 4 generalizing an identity of bouleau yor type well known for the classical brownian motion a second application is an ito formula of stratonovich type for f b h the main difficulty comes from the fact b h has only a finite 4 variation

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

semimartingale integral representation we provide an integral representation for smooth functionals of continuous semimartingales the representation is related to an infinite dimensional nonautonomous parabolic equation semimartingale integral representations including a martingale representation are given in terms of a solution to this equation

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

linear neighborhood propagation and its applications in this paper a novel graph based transductive classification approach called linear neighborhood propagation is proposed the basic idea is to predict the label of a data point according to its neighbors in a linear way this method can be cast into a second order intrinsic gaussian markov random field framework its result corresponds to a solution to an approximate inhomogeneous biharmonic equation with dirichlet boundary conditions different from existing approaches our approach provides a novel graph structure construction method by introducing multiple wise edges instead of pairwise edges and presents an effective scheme to estimate the weights for such multiple wise edges to the best of our knowledge these two contributions are novel for semi supervised classification the experimental results on image segmentation and transductive classification demonstrate the effectiveness and efficiency of the proposed approach

periodic solutions and their connecting orbits of hamiltonian systems abstract this article deals with second order periodic hamiltonian systems we apply variational methods to obtain non constant periodic solutions for the time reversible hamiltonian systems there exist connecting orbits joining pairs of periodic solutions our methods can also be used to treat heteroclinic orbits connecting an equilibrium to a periodic solution

pathwise integration and change of variable formulas for continuous paths with arbitrary regularity we construct a pathwise integration theory associated with a change of variable formula for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p th variation along a sequence of time partitions for paths with finite p th variation along a sequence of time partitions we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated riemann sums results for functions are extended to regular path dependent functionals using the concept of vertical derivative of a functional we show that the pathwise integral satisfies an isometry formula in terms of p th order variation and obtain a signal plus noise decomposition for regular functionals of paths with strictly increasing p th variation for less regular functions we obtain a tanaka type change of variable formula using an appropriately defined notion of local time these results extend to multidimensional paths and yield a natural higher order extension of the concept of reduced rough path we show that while our integral coincides with a rough path integral for a certain rough path its construction is canonical and does not involve the specification of any rough path superstructure

two person zero sum game in weak formulation and path dependent bellman isaacs equation in this paper we study a two person zero sum stochastic differential game in weak formulation unlike the standard literature which uses strategy type controls the weak formulation allows us to consider the game with control against control we shall prove the existence of game value under natural conditions another main feature of the paper is that we allow for non markovian structure and thus the game value is a random process we characterize the value process as the unique viscosity solution of the corresponding path dependent bellman isaacs equation a notion recently introduced by ekren et al ann probab 42 2014 pp 204 236 and ekren touzi and zhang stochastic process to appear preprint arxiv 1210 0006v2 preprint arxiv 1210 0007v2

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

integration by parts and time reversal for diffusion processes in this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property assuming the existence of a density at every time t the proofs are based on techniques of the stochastic calculus of variations

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

infinite horizon backward stochastic differential equations and elliptic equations in hilbert spaces solutions of semilinear elliptic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations the backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition elliptic equations are intended in a mild sense suitable also for applications to optimal control we finally notice that due to the lack of smoothing properties the elliptic partial differential equation considered here could not be treated by analytic methods

integration with respect to the g brownian local time abstract let l be the local time of g brownian motion b in this paper we consider the existence of the quadratic covariation f b b t and the integral r f x l d x t under some suitable conditions we show that the sublinear version of the bouleau yor identity r f x l d x t f b b t holds these allow us to write the ito formula for c 1 functions

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

an infinite dimensional approach to path dependent kolmogorov equations in this paper a banach space framework is introduced in order to deal with finite dimensional path dependent stochastic differential equations a version of kolmogorov backward equation is formulated and solved both in the space of lplp paths and in the space of continuous paths using the associated stochastic differential equation thus establishing a relation between path dependent sdes and pdes in analogy with the classical case finally it is shown how to establish a connection between such kolmogorov equation and the analogue finite dimensional equation that can be formulated in terms of the path dependent derivatives recently introduced by dupire cont and fournie

supervised neighborhood graph construction for semi supervised classification graph based methods are among the most active and applicable approaches studied in semi supervised learning the problem of neighborhood graph construction for these methods is addressed in this paper neighborhood graph construction plays a key role in the quality of the classification in graph based methods several unsupervised graph construction methods have been proposed that have addressed issues such as data noise geometrical properties of the underlying manifold and graph hyper parameters selection in contrast in order to adapt the graph construction to the given classification task many of the recent graph construction methods take advantage of the data labels however these methods are not efficient since the hypothesis space of their possible neighborhood graphs is limited in this paper we first prove that the optimal neighborhood graph is a subgraph of a k nn graph for a large enough k which is much smaller than the total number of data points therefore we propose to use all the subgraphs of k nns graph as the hypothesis space in addition we show that most of the previous supervised graph construction methods are implicitly optimizing the smoothness functional with respect to the neighborhood graph parameters finally we provide an algorithm to optimize the smoothness functional with respect to the neighborhood graph in the proposed hypothesis space experimental results on various data sets show that the proposed graph construction algorithm mostly outperforms the popular k nn based construction and other state of the art methods

optimal control of systems with noisy memory and bsdes with malliavin derivatives abstract in this article we consider a stochastic optimal control problem where the dynamics of the state process x t is a controlled stochastic differential equation with jumps delay and noisy memory the term noisy memory is to the best of our knowledge new by this we mean that the dynamics of x t depend on t t x s d b s where b t is a brownian motion hence the dependence is noisy because of the brownian motion and it involves memory due to the influence from the previous values of the state process we derive necessary and sufficient maximum principles for this stochastic control problem in two different ways resulting in two sets of maximum principles the first set of maximum principles is derived using malliavin calculus techniques while the second set comes from reduction to a discrete delay optimal control problem and application of previously known results by oksendal sulem and zhang the maximum principles also apply to the case where the controller has only partial information in the sense that the admissible controls are adapted to a sub algebra of the natural filtration

stochastic optimal control problems and parabolic equations in banach spaces we consider stochastic optimal control problems in banach spaces these problems are related to nonlinear controlled equations with dissipative nonlinearity and are treated via the backward stochastic differential equation approach which also allows us to solve in a mild sense hamilton jacobi bellman equations in banach spaces we apply the results to controlled stochastic heat and wave equations with a cost functional that is well defined on continuous functions and to delay equations in spaces of p integrable functions

label information guided graph construction for semi supervised learning in the literature most existing graph based semi supervised learning methods only use the label information of observed samples in the label propagation stage while ignoring such valuable information when learning the graph in this paper we argue that it is beneficial to consider the label information in the graph learning stage specifically by enforcing the weight of edges between labeled samples of different classes to be zero we explicitly incorporate the label information into the state of the art graph learning methods such as the low rank representation lrr and propose a novel semi supervised graph learning method called semi supervised low rank representation this results in a convex optimization problem with linear constraints which can be solved by the linearized alternating direction method though we take lrr as an example our proposed method is in fact very general and can be applied to any self representation graph learning methods experiment results on both synthetic and real data sets demonstrate that the proposed graph learning method can better capture the global geometric structure of the data and therefore is more effective for semi supervised learning tasks

explicit form and robustness of martingale representations stochastic integral representation of martingales has been undergoing a renaissance due to questions motivated by stochastic finance theory in the brownian case one usually has formulas of differing degrees of exactness for the predictable integrands we extend some of these to markov cases where one does not necessarily have stochastic integral representation of all martingales moreover we study various convergence questions that arise naturally from for example approximations of price processes via euler schemes for solutions of stochastic differential equations we obtain general results of the following type let u u n be random variables with decompositions u alpha int 0 infty xi s dx s n infty u n alpha n int 0 infty xi n s dx n s n n infty where x n x n n n are martingales if x n to x and u n to u when and how does xi n rightarrow xi

existence of homoclinic solutions for a class of time dependent hamiltonian systems n 1 n n 2 where q s q q g r w g c r r r and l t g c r r 1 n is a positive definite symmetric matrix for all t g r the case where l t and w t q are either periodic in t or independent of t were studied by several authors see for instance ambrosetti and w x w x bertotti 1 coti zelati ekeland and sere 4 coti zelati and rabinowitz w x w x w x 5 rabinowitz 15 17 sere 19 and their references in this kind of problem the function l t plays an important role for example if l t s ai where a 0 and i denotes the identity matrix n n w x felmer and silva 8 have showed the existence of homoclinic orbits as the limit of subharmonic solutions provided that hs is autonomous see also w x w x rabinowitz and tanaka 18 serra tarallo and terracini 20 using an w x w x approach introduced by 4 see also 19 treated the hs problem

efficient graph based image segmentation this paper addresses the problem of segmenting an image into regions we define a predicate for measuring the evidence for a boundary between two regions using a graph based representation of the image we then develop an efficient segmentation algorithm based on this predicate and show that although this algorithm makes greedy decisions it produces segmentations that satisfy global properties we apply the algorithm to image segmentation using two different kinds of local neighborhoods in constructing the graph and illustrate the results with both real and synthetic images the algorithm runs in time nearly linear in the number of graph edges and is also fast in practice an important characteristic of the method is its ability to preserve detail in low variability image regions while ignoring detail in high variability regions

a variational approach to the construction and malliavin differentiability of strong solutions of sde s in this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients we aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by brownian motion an important and rather surprising consequence of our method which is based on malliavin calculus is that the solutions derived by veretennikov theory probab appl 24 354 366 1979 for brownian motion with bounded and measurable drift in mathbb r d are malliavin differentiable further a strength of our approach which does not rely on a pathwise uniqueness argument is that it can be transferred and applied to the analysis of various other types of stochastic equations we obtain a bismut elworthy li formula elworthy and li j funct anal 125 252 286 1994 for spatial derivatives of solutions to the kolmogorov equation with bounded and measurable drift coefficients to derive the formula we use that our approach can be applied to obtain sobolev differentiability in the initial condition in addition to malliavin differentiability of the associated stochastic differential equations another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields moreover our approach is also applicable to the construction of solutions of stochastic evolution equations on hilbert spaces

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

stability of stochastic functional differential systems using degenerate lyapunov functionals and applications motivated by the seminal work of dupire 2009 on functional ito formulas this work investigates asymptotic properties of systems represented by stochastic functional differential equations sfdes stability of general delay dependent sfdes is investigated using degenerate lyapunov functionals which are only positive semi definite rather than positive definite as used in the classical work this paper first establishes boundedness and regularity of sfdes by using degenerate lyapunov functionals then moment and almost sure exponential stabilities are obtained based on degenerate lyapunov functionals and the semi martingale convergence theorem as an application of the stability criteria consentability of stochastic multi agent systems with nonlinear dynamics is studied

on the drawdowns and drawups in diffusion type models with running maxima and minima we obtain closed form expressions for the values of joint laplace transforms of the running maximum and minimum of a diffusion type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels it is assumed that the coefficients of the diffusion type process are regular functions of the running values of the process itself its maximum and minimum as well as its maximum drawdown and maximum drawup processes the proof is based on the solution to the equivalent boundary value problems and application of the normal reflection conditions for the value functions at the edges of the state space of the resulting five dimensional markov process we show that the joint laplace transforms represent linear combinations of solutions to the systems of first order partial differential equations arising from the application of the normal reflection conditions

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

learning with ell 1 graph for image analysis the graph construction procedure essentially determines the potentials of those graph oriented learning algorithms for image analysis in this paper we propose a process to build the so called directed 1 graph in which the vertices involve all the samples and the ingoing edge weights to each vertex describe its 1 norm driven reconstruction from the remaining samples and the noise then a series of new algorithms for various machine learning tasks e g data clustering subspace learning and semi supervised learning are derived upon the 1 graphs compared with the conventional k nearest neighbor graph and ball graph the 1 graph possesses the advantages 1 greater robustness to data noise 2 automatic sparsity and 3 adaptive neighborhood for individual datum extensive experiments on three real world datasets show the consistent superiority of 1 graph over those classic graphs in data clustering subspace learning and semi supervised learning tasks

langevin formulation of a subdiffusive continuous time random walk in physical time systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power law scaling of the mean square displacement one of the most common models to describe such subdiffusive dynamics is the continuous time random walk ctrw stochastic trajectories of a ctrw can be described in terms of the subordination of a normal diffusive process by an inverse levy stable process here we propose an equivalent langevin formulation of a force free ctrw without subordination by introducing a different type of non gaussian noise we are able to express the ctrw dynamics in terms of a single langevin equation in physical time with additive noise we derive the full multipoint statistics of this noise and compare it with the scaled brownian motion sbm an alternative stochastic model describing subdiffusive dynamics interestingly these two noises are identical up to the second order correlation functions but different in the higher order statistics we extend our formalism to general waiting time distributions and force fields and compare our results with those of the sbm in the presence of external forces our proposed noise generates a different class of stochastic processes resembling a ctrw but with forces acting at all times

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

generalized integration and stochastic odes stochastic forward integrals for processes more general than semimartingales are shown to exist generalized forms of ito wentzell formula and covariation formula are proved and one dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved this generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

ito formula for c 1 functions of semimartingales we establish an ito formula forc1 functions of processes whose time reversal are semimartingales and forc1 functions whose first derivatives are holder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism

on a class of generalized takagi functions with linear pathwise quadratic variation abstract we consider a class x of continuous functions on 0 1 that is of interest from two different perspectives first it is closely related to sets of functions that have been studied as generalizations of the takagi function second each function in x admits a linear pathwise quadratic variation and can thus serve as an integrator in follmer s pathwise it calculus we derive several uniform properties of the class x for instance we compute the overall pointwise maximum the uniform maximal oscillation and the exact uniform modulus of continuity for all functions in x furthermore we give an example of a pair x y x for which the quadratic variation of the sum x y does not exist

viscosity solutions of fully nonlinear parabolic path dependent pdes part ii the main objective of this paper and the accompanying one viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint is to provide a notion of viscosity solutions for fully nonlinear parabolic path dependent pdes our definition extends our previous work ann probab 2014 42 204 236 focused on the semilinear case and is crucially based on the nonlinear optimal stopping problem analyzed in stochastic process appl 2014 124 3277 3311 we prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions and satisfies a stability property and a partial comparison result the latter is a key step for the well posedness results established in viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint we also show that the value processes of path dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equations

a stochastic portfolio optimization model with bounded memory this paper considers a portfolio management problem of merton s type in which the risky asset return is related to the return history the problem is modeled by a stochastic system with delay the investor s goal is to choose the investment control as well as the consumption control to maximize his total expected discounted utility under certain situations we derive the explicit solutions in a finite dimensional space

heteroclinic solutions for a class of the second order hamiltonian systems we shall be concerned with the existence of heteroclinic orbits for the second order hamiltonian system q vq t q 0 where q rn and v c1 r rn r v 0 we will assume that v and a certain subset m rn satisfy the following conditions m is a set of isolated points and m 2 for every sufficiently small e 0 there exists 0 such that for all t z r rn if d z m e then v t z the integrals v t z dt z m are equi bounded and v t z as t uniformly on compact subsets of rn m our result states that each point in m is joined to another point in m by a solution of our system

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

change of variable formulas for non anticipative functionals on path space we derive a change of variable formula for non anticipative functionals defined on the space of rd valued right continuous paths with left limits the functionals are only required to possess certain directional derivatives which may be computed pathwise our results lead to functional extensions of the ito formula for a large class of stochastic processes including semimartingales and dirichlet processes in particular we show the stability of the class of semimartingales under certain functional transformations

stochastic systems with memory and jumps abstract stochastic systems with memory naturally appear in life science economy and finance we take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps our results concern existence and uniqueness of strong solutions estimates for the moments and the fundamental tools of calculus such as the ito formula we study the robustness of the solution to the change of noises specifically we consider the noises with infinite activity jumps versus an adequately corrected gaussian noise the study is presented in two different frameworks we work with random variables in infinite dimensions where the values are considered either in an appropriate l p type space or in the space of cadlag paths the choice of the value space is crucial from the modelling point of view as the different settings allow for the treatment of different models of memory or delay our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation

functional ito calculus and stochastic integral representation of martingales we develop a non anticipative calculus for functionals of a continuous semimartingale using an extension of the ito formula to path dependent functionals which possess certain directional derivatives the construction is based on a pathwise derivative introduced by b dupire for functionals on the space of right continuous functions with left limits we show that this functional derivative admits a suitable extension to the space of square integrable martingales this extension denes a weak derivative which is shown to be the inverse of the ito integral and which may be viewed as a non anticipative lifting of the malliavin derivative these results lead to a constructive martingale representation formula for ito processes by contrast with the clark haussmann ocone formula this representation only involves nonanticipative quantities which may be computed pathwise

an averaging principle for two time scale stochastic functional differential equations abstract delays are ubiquitous pervasive and entrenched in everyday life thus taking it into consideration is necessary dupire recently developed a functional ito formula which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications based on the new development this work examines functional diffusions with two time scales in which the slow varying process includes path dependent functionals and the fast varying process is a rapidly changing diffusion the gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example this paper establishes mixed functional ito formulas and the corresponding martingale representation then it develops an averaging principle using weak convergence methods by treating the fast varying process as a random noise under appropriate conditions it is shown that the slow varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow varying process with respect to the invariant measure of the fast varying process

the full replica symmetry breaking in the ising spin glass on random regular graph in this paper we extend the full replica symmetry breaking scheme to the ising spin glass on a random regular graph we propose a new martingale approach that overcomes the limits of the parisi mezard cavity method providing a well defined formulation of the full replica symmetry breaking problem in random regular graphs finally we define the order parameters of the system and get a set of self consistency equations for the order parameters and the free energy we face up the problem only from a technical point of view the physical meaning of this approach and the quantitative evaluation of the solution of the self consistency equations will be discussed in next works

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

constructing functions with prescribed pathwise quadratic variation we construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations we also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function value these functions can then be used as integrators in follmer s pathwise it calculus our construction of the latter class of functions relies on an extension of the doss sussman method to a class of nonlinear it differential equations for the follmer integral as an application we provide a deterministic variant of the support theorem for diffusions we also establish that many of the constructed functions are nowhere differentiable

a functional ito s calculus approach to convex risk measures with jump diffusion abstract convex risk measures for european contingent claims are studied in a non markovian jump diffusion modeling framework using functional ito s calculus two representations for a convex risk measure are considered one based on a nonlinear g expectation and another one based on a representation theorem functional ito s calculus for cadlag processes backward stochastic differential equations bsdes with jumps and stochastic optimal control theory are used to discuss the evaluation of convex risk measures fpdies and pdies for convex risk measures are derived in the markovian and non markovian situations respectively an entropic risk measure which is a particular case of a convex risk measure is discussed

stochastic equations with delay optimal control via bsdes and regular solutions of hamilton jacobi bellman equations we consider an ito stochastic differential equation with delay driven by brownian motion whose solution by an appropriate reformulation defines a markov process x with values in a space of continuous functions mathbf c with generator mathcal l we then consider a backward stochastic differential equation depending on x with unknown processes y z and we study properties of the resulting system in particular we identify the process z as a deterministic functional of x we next prove that the forward backward system provides a suitable solution to a class of parabolic partial differential equations on the space mathbf c driven by mathcal l and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects we also include applications to optimal stochastic control of differential equation with delay in particular we characterize optimal controls as feedback laws in terms of the process x

a generalized formula of ito and some other properties of stochastic flows a stochastic differential equation with smooth coefficients is considered which defines a continuous flow t of c 8 mappings of rd in rd if zt is a continuous semi martingale t zt s is proved to be a semi martingale for which an ito type formula is established it is shown that a s for any t t is an onto diffeomorphism if zt is a continuous semi martingale t 1 zt is proved to be a semi martingale whose ito decomposition is explicitly found