the ring of conditions of a semisimple group abstract in this paper we are going to give an explicit description of the so called ring of conditions of an adjoint semisimple algebraic group g originally introduced in c de concini c procesi complete symmetric varieties ii intersection theory in algebraic groups and related topics kyoto nagoya 1983 in adv stud pure math vol 6 north holland amsterdam 1985 pp 481 513 in order to study intersection theory in g

the adapted solution and comparison theorem for backward stochastic differential equations with poisson jumps and applications abstract this paper deals with a class of backward stochastic differential equations with poisson jumps and with random terminal times we prove the existence and uniqueness result of adapted solution for such a bsde under the assumption of non lipschitzian coefficient we also derive two comparison theorems by applying a general girsanov theorem and the linearized technique on the coefficient by these we first show the existence and uniqueness of minimal solution for one dimensional bsde with jumps when its coefficient is continuous and has a linear growth then we give a general feynman kac formula for a class of parabolic types of second order partial differential and integral equations pdies by using the solution of corresponding bsde with jumps finally we exploit above feynman kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi linear pdie of parabolic type

optimal control of mean field jump diffusion systems with delay this paper is concerned with an optimal control problem under mean field jump diffusion systems with delay firstly some existence and uniqueness results are proved for a jump diffusion mean field stochastic delay differential equation and a jump diffusion mean field advanced backward stochastic differential equation then necessary and sufficient maximum principles for control systems of mean field type and with delay are established under certain conditions a mean field delayed linear quadratic control problem is finally discussed using the obtained maximum principles

generalized reflected bsdes driven by a levy process and an obstacle problem for pdies with a nonlinear neumann boundary condition in this paper we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equations grbsdes in short driven by a levy process which involve the integral with respect to a continuous process by means of the snell envelope the penalization method and the fixed point theorem in addition we obtain the comparison theorem for the solutions of the grbsdes as an application we give a probabilistic formula for the viscosity solution of an obstacle problem for a class of partial differential integral equations pdies in short with a nonlinear neumann boundary condition

a markov partition that reflects the geometry of a hyperbolic toral automorphism we show how to construct a markov partition that reflects the geometrical action of a hyperbolic automorphism of the n torus the transition matrix is the transpose of the matrix induced by the automorphism in u dimensional homology provided this is non negative here u denotes the expanding dimension that condition is satisfied at least for some power of the original automorphism under a certain non degeneracy condition on the galois group of the characteristic polynomial the nu rectangles are constructed by an iterated function system and they resemble the product of the projection of a u dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal n u dimensional face onto the stable subspace

optimal stopping under uncertainty in drift and jump intensity this paper studies the optimal stopping problem in the presence of model uncertainty ambiguity we develop a numerically implementable method to solve this problem in a general setting allowing

an euler relation for valuations on polytopes abstract a locally finite point set such as the set z n of integral points gives rise to a lattice of polytopes in euclidean space taking vertices from the given point set we develop the combinatorial structure of this polytope lattice and derive euler type relations for valuations on polytopes using the language of mobius inversion in this context a new family of inversion relations is obtained thereby generalizing classical relations of euler dehn sommerville and macdonald

numerical approximation of doubly reflected bsdes with jumps and rcll obstacles we study a discrete time approximation scheme for the solution of a doubly reflected backward stochastic differential equation dbbsde in short with jumps driven by a brownian motion and an independent compensated poisson process moreover we suppose that the obstacles are right continuous and left limited rcll processes with predictable and totally inaccessible jumps and satisfy mokobodski s condition our main contribution consists in the construction of an implementable numerical sheme based on two random binomial trees and the penalization method which is shown to converge to the solution of the dbbsde finally we illustrate the theoretical results with some numerical examples in the case of general jumps

a note on the reflected backward stochastic differential equations driven by a levy process with stochastic lipschitz condition abstract in this note we prove the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations rbsdes in short related to the subdifferential operator of a lower semi continuous convex function driven by teugels martingales associated with a levy process some known results are generalized and improved

inflations of self affine tilings are integral algebraic perron we prove that any expanding linear map phi mathbb r mathtt d rightarrow mathbb r mathtt d that is the inflation map in an inflation substitution process generating a self affine tiling of mathbb r mathtt d is integral algebraic and perron this means that phi is linearly conjugate to a restriction of an integer matrix to a subspace mathcal e satisfying a maximal growth condition that generalizes the characterization of perron numbers as numbers that are larger than the moduli of their algebraic conjugates the case of diagonalizable phi has been previously resolved by richard kenyon and boris solomyak and it is rooted in thurston s idea of lifting the tiling from the physical space mathbb r mathtt d to a higher dimensional mathematical space where the tiles their control points sit on a lattice the main novelty of our approach is in lifting the inflation substitution process to the mathematical space and constructing a certain vector valued cocycle defined over the translation induced mathbb r mathtt d action on the tiling space the subspace mathcal e is obtained then by ergodic averaging of the cocycle more broadly we assemble a powerful framework for studying self affine tiling spaces

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

maximum principle for mean field jump diffusion stochastic delay differential equations and its application to finance abstract this paper investigates a stochastic optimal control problem with delay and of mean field type where the controlled state process is governed by a mean field jump diffusion stochastic delay differential equation two sufficient maximum principles and one necessary maximum principle are established for the underlying system as an application a bicriteria mean variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques under certain conditions explicit expressions are provided for the efficient portfolio and the efficient frontier which are as elegant as those in the classical mean variance problem without delays

the maximum principles for stochastic recursive optimal control problems under partial information a maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal agent problems by virtue of a classical spike variational approach and a filtering technique the maximum principle is obtained and the related adjoint processes are characterized by the solutions of forward backward stochastic differential equations in finite dimensional spaces then our theoretical results are applied to study a partially observed linear quadratic recursive optimal control problem in addition for the case with initial and terminal state constraints the corresponding maximum principle is also obtained by using ekeland s variational principle

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

a stochastic maximum principle for systems with jumps with applications to finance abstract we consider a stochastic control problem with linear dynamics with jumps convex cost criterion and convex state constraint in which the control enters the drift the diffusion and the jump coefficients we allow these coefficients to be random and do not impose any l p bounds on the control we obtain a stochastic maximum principle for this model that provides both necessary and sufficient conditions of optimality this is the first version of the stochastic maximum principle that covers the consumption investment problem in which there are jumps in the price system

a stochastic maximum principle for backward control systems with random default time this paper establishes a necessary and sufficient stochastic maximum principle for backward systems where the state processes are governed by jump diffusion backward stochastic differential equations with random default time an application of the sufficient stochastic maximum principle to an optimal investment and capital injection problem in the presence of default risk is discussed

a revisit to stochastic near optimal controls the critical case abstract this paper revisits the stochastic near optimal control problem considered in zhou 1998 where the stochastic system is given by a controlled stochastic differential equation with the control variable taking values in a general control space and entering both the drift and diffusion coefficients a necessary condition of near optimality is derived using ekeland s variational principle spike variation techniques and some delicate estimates for the state and the adjoint processes we improve the error bound of order from almost e 1 3 in zhou 1998 to exactly e 1 3

feynman kac representation for hamilton jacobi bellman ipde we aim to provide a feynman kac type representation for hamilton jacobi bellman equation in terms of forward backward stochastic differential equation fbsde with a simulatable forward process for this purpose we introduce a class of bsde where the jumps component of the solution is subject to a partial nonpositive constraint existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions we then show how minimal solution to this bsde class provides a new probabilistic representation for nonlinear integro partial differential equations ipdes of hamilton jacobi bellman hjb type when considering a regime switching forward sde in a markovian framework moreover we state a dual formula of this bsde minimal solution involving equivalent change of probability measures this gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

necessary and sufficient conditions for near optimality in stochastic control of fbsdes abstract we consider control problems for systems governed by a nonlinear forward backward stochastic differential equation fbsde we establish necessary as well as sufficient conditions for near optimality satisfied by all near optimal controls these conditions are described by two adjoint processes corresponding to the forward and backward components and a nearly maximum condition on the hamiltonian the proof of the main result is based on ekeland s variational principle and some estimates on the state and the adjoint processes with respect to the control variable as is well known optimal controls may fail to exist even in simple cases this justifies the use of near optimal controls which exist under minimal assumptions and are sufficient in most practical cases moreover since there are many nearly optimal controls it is possible to choose suitable ones that are convenient for implementation

algebraic numbers free group automorphisms and substitutions on the plane there has been much recent work on the geometric representation of pisot substitutions and the explicit construction of markov partitions for pisot toral automorphims we give a construction that extends this to the general hyperbolic case for the sake of simplicity we consider a simple example of an automorphism of the free group on 4 generators whose associated matrix has 4 distinct complex eigenvalues two of them of modulus larger than 1 and the other 2 of modulus smaller than 1 non pisot case using this generator we build substitution polygonal tilings of the contracting plane and the expanding plane of the matrix we prove that these substitution tilings can be lifted in a unique way to stepped surfaces approximating each of these planes the vertices of each of these stepped surfaces can be projected to an atomic surface a compact set with fractal boundary contained in the other plane we prove that both tilings can be refined to exact self similar tilings whose tiles have fractal boundaries and can be obtained by iteration or by a cut and project method by using the atomic surface as the window using the self similar tiling one can build a numeration system associated to a complex lambda expansion the natural extension of the lambda expansion associated with this number system is the linear map obtained by abelianization of the free group automorphism this gives an explicit markov partition of this hyperbolic toral automorphism the fractal domains can be used to define a pseudo group of translations which gives transversal dynamics in the sense of vershik 1994 or numeration systems in the sense of kamae 2005 the construction can be extended to a larger class of free group automorphisms each of which can be used to build substitution rules and dynamical systems

generalized translation invariant valuations and the polytope algebra abstract we study the space of generalized translation invariant valuations on a finite dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations our main theorem is that mcmullen s polytope algebra is a subalgebra of the partial convolution algebra of generalized translation invariant valuations more precisely we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position the convolution is defined and corresponds to the product in the polytope algebra

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

stochastic maximum principle for optimal control with multiple priors abstract the necessary condition is derived for optimal control with multiple priors which are mutually singular the tool we use is the theory of g expectation

stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of hamilton jacobi bellman equations abstract in this paper we study stochastic optimal control problems with jumps with the help of the theory of backward stochastic differential equations bsdes with jumps we generalize the results of peng s peng bsde and stochastic optimizations in j yan s peng s fang l wu topics in stochastic analysis science press beijing 1997 chapter 2 in chinese by considering cost functionals defined by controlled bsdes with jumps the application of bsde methods in particular the use of the notion of stochastic backward semigroups introduced by peng in the above mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps we prove that the value functions are the viscosity solutions of the associated generalized hamilton jacobi bellman equations with integral differential operators for this proof we adapt peng s bsde approach given in the above mentioned reference developed in the framework of stochastic control problems driven by brownian motion to that of stochastic control problems driven by brownian motion and poisson random measure

stochastic pdies and backward doubly stochastic differential equations driven by levy processes in this paper a new class of backward doubly stochastic differential equations driven by teugels martingales associated with a levy process satisfying some moment condition and an independent brownian motion is investigated we obtain the existence and uniqueness of solutions to these equations a probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given

discretisation of fbsdes driven by cadlag martingales abstract we study the discretisation of forward backward stochastic differential equations fbsdes driven by cadlag martingales we prove that under certain conditions imposed on the parameters of the fbsde the time discrete scheme we consider converges to the time continuous equation in the l 2 sense moreover we show that the l 2 norm of the error is of the order of the time step

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

on solutions to backward stochastic partial differential equations for levy processes in this paper we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations bspdes for short driven by the teugels martingales associated with a levy process satisfying some moment conditions and by an independent brownian motion an example is given to illustrate the theory

geometric theory of unimodular pisot substitutions we are concerned with the tiling flow t associated to a substitution over a finite alphabet our focus is on substitutions that are unimodular pisot i e their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the perron eigenvalue 1 the motivation is provided by the still open conjecture asserting that t has pure discrete spectrum for any such we develop a number of necessary and sufficient conditions for pure discrete spectrum including injectivity of the canonical torus map the geometric realization geometric coincidence condition partial commutation of t and the dual r d 1 action measure and tiling properties of rauzy fractals and concrete algorithms some of these are original and some have already appeared in the literature as sufficient conditions only but they all emerge from a unified approach based on the new device the strand space f of the proof of the necessity hinges on determination of the discrete spectrum of t as that of the associated kronecker toral flow

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

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

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

bsde driven by poisson point processes with discontinuous coefficient abstract in this paper we deal with the one dimensional backward stochastic differential equation bsde driven by poisson processes by means of the comparison theorem we first prove the existence of a minimal solution for bsde where the coefficient is continuous and satisfies an improved linear growth assumption then we extend the result to the case where the coefficient is left or right continuous

forward backward stochastic differential equations and quasilinear parabolic pdes this paper studies under some natural monotonicity conditions the theory existence and uniqueness a priori estimate continuous dependence on a parameter of forward backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations we use a purely probabilistic approach and allow the forward equation to be degenerate

infinite horizon optimal control of forward backward stochastic differential equations with delay we consider a problem of optimal control of an infinite horizon system governed by forward backward stochastic differential equations with delay sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived we illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay

general linear quadratic optimal stochastic control problems with random coefficients linear stochastic hamilton systems and backward stochastic riccati equations consider the minimization of the following quadratic cost functional j u e langle mx t x t rangle e int 0 t langle q sx s x s rangle langle n su s u s rangle ds where x is the solution of the following linear stochastic control system eq dx t a tx t b tu t dt sum i 1 d c t ix t d t iu t dw t i cr x 0 h cr u is a square integrable adapted process the problem is conventionally called the stochastic lq the abbreviation of linear quadratic problem we are concerned with the following general case the coefficients a b ci di q n and m are allowed to be adapted processes or random matrices we prove the existence and uniqueness result for the associated riccati equation which in our general case is a backward stochastic differential equation with the generator the drift term being highly nonlinear in the two unknown variables this solves bismut and peng s long standing open problem for the case of a brownian filtration which was initially proposed by the french mathematician j m bismut in seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 we also provide a rigorous derivation of the riccati equation from the stochastic hamilton system this completes the interrelationship between the riccati equation and the stochastic hamilton system as two different but equivalent tools for the stochastic lq problem there are two key points in our arguments the first one is to connect the existence of the solution of the riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system actually we establish their equivalence as a consequence we can construct solutions to a sequence of suitably modified riccati equations in terms of the associated stochastic hamilton systems and the optimal controls the second key point is to establish a new type of a priori estimate for solutions of riccati equations with which we show that the sequence of constructed solutions has a limit which is a solution to the original riccati equation

viscosity solutions of second order integral partial differential equations without monotonicity condition a new result abstract we show existence and uniqueness of a continuous with polynomial growth viscosity solution of a system of second order integral partial differential equations ipdes for short without assuming the usual monotonicity condition of the generator with respect to the jump component as in barles et al s article barles et al 1997 the levy measure is arbitrary and not necessarily finite in our study the main tool we used is the notion of backward stochastic differential equations with jumps

reflected scheme for doubly reflected bsdes with jumps and rcll obstacles we introduce a discrete time reflected scheme to solve doubly reflected backward stochastic differential equations with jumps in short drbsdes driven by a brownian motion and an independent compensated poisson process as in dumitrescu and labart 2014 we approximate the brownian motion and the poisson process by two random walks but contrary to this paper we discretize directly the drbsde without using a penalization step this gives us a fully implementable scheme which only depends on one parameter of approximation the number of time steps n contrary to the scheme proposed in dumitrescu and labart 2014 which also depends on the penalization parameter we prove the convergence of the scheme and give some numerical examples

brief infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems this paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation like deterministic control systems it is shown that the exact controllability of an open loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed loop stochastic system formed by means of suitable linear feedback of the states as an application a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward backward stochastic differential equations is provided

relationship between maximum principle and dynamic programming for stochastic differential games of jump diffusions this paper is concerned with the relationship between maximum principle and dynamic programming for zero sum stochastic differential games of jump diffusions under the assumption that the value function is smooth enough relations among the adjoint processes the generalised hamiltonian function and the value function are given a portfolio optimisation problem under model uncertainty in an incomplete financial market is discussed to show the applications of our result

valuations on lattice polytopes let be a lattice that is a module of finite rank and let denote the family of convex polytopes with vertices in here convexity refers to the underlying rational vector space in this paper it is shown that any valuation on satisfies the inclusion exclusion principle in the strong sense that appropriate extension properties of the valuation hold indeed the core result is that the class of a lattice polytope in the abstract group for valuations on can be identified with its characteristic function in in fact the same arguments are shown to apply to when is a module of finite rank over an ordered ring and more generally to appropriate families of not necessarily bounded polyhedra

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

backward doubly stochastic equations with jumps and comparison theorems abstract in this work we mainly prove the existence and pathwise uniqueness of solutions to general backward doubly stochastic differential equations with jumps appearing in both forward and backward integral parts several comparison theorems under some weak conditions are also given finally we apply comparison theorems in proving the existence of solution to some special backward doubly stochastic differential equations with drift coefficient increasing linearly

l p solutions of multidimensional bsdes with weak monotonicity and general growth generators abstract in this paper we first establish the existence and uniqueness of l p p 1 solutions for multidimensional backward stochastic differential equations bsdes under a weak monotonicity condition together with a general growth condition in y for the generator g then we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way finally we put forward and prove a stability theorem and a comparison theorem of l p p 1 solutions for this kind of bsdes

stochastic pdies with nonlinear neumann boundary conditions and generalized backward doubly stochastic differential equations driven by levy processes in this paper a new class of generalized backward doubly stochastic differential equations gbdsdes in short driven by teugels martingales associated with levy process and the integral with respect to an adapted continuous increasing process is investigated we obtain the existence and uniqueness of solutions to these equations a probabilistic interpretation for solutions to a class of stochastic partial differential integral equations pdies in short with a nonlinear neumann boundary condition is given

randomized dynamic programming principle and feynman kac representation for optimal control of mckean vlasov dynamics we analyze a stochastic optimal control problem where the state process follows a mckean vlasov dynamics and the diffusion coefficient can be degenerate we prove that its value function v admits a nonlinear feynman kac representation in terms of a class of forward backward stochastic differential equations with an autonomous forward process we exploit this probabilistic representation to rigorously prove the dynamic programming principle dpp for v the feynman kac representation we obtain has an important role beyond its intermediary role in obtaining our main result in fact it would be useful in developing probabilistic numerical schemes for v the dpp is important in obtaining a characterization of the value function as a solution of a non linear partial differential equation the so called hamilton jacobi belman equation in this case on the wasserstein space of measures we should note that the usual way of solving these equations is through the pontryagin maximum principle which requires some convexity assumptions there were attempts in using the dynamic programming approach before but these works assumed a priori that the controls were of markovian feedback type which helps write the problem only in terms of the distribution of the state process and the control problem becomes a deterministic problem in this paper we will consider open loop controls and derive the dynamic programming principle in this most general case in order to obtain the feynman kac representation and the randomized dynamic programming principle we implement the so called randomization method which consists in formulating a new mckean vlasov control problem expressed in weak form taking the supremum over a family of equivalent probability measures one of the main results of the paper is the proof that this latter control problem has the same value function v of the original control problem

bsdes with general filtration driven by levy processes and an application in stochastic controllability abstract in this paper we introduce a weak version of the strong solution the adapted solution used in pardoux and peng 1990 2 i e the transposition solution to the backward stochastic differential equation bsde with general filtration and random jumps and study the corresponding well posedness the main tools that we employ are the riesz representation theorem and the banach fixed point theorem without using the martingale representation theorem as an application we give a definition of controllability to the stochastic linear control system in the sense of the transposition solution and provide a kalman type rank condition to guarantee this property

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

near optimality conditions in stochastic control of jump diffusion processes abstract this paper is concerned with necessary as well as sufficient conditions for near optimality of controlled jump diffusion processes necessary conditions for a control to be near optimal are derived using ekeland s variational principle and some stability results on the state and adjoint processes with respect to the control variable in a second step we show that the necessary conditions for near optimality are in fact sufficient for near optimality provided some concavity conditions are fulfilled finally as an illustration some examples are solved explicitly

the k theory of assemblers abstract in this paper we introduce the notion of an assembler which formally encodes cutting and pasting data an assembler has an associated k theory spectrum in which 0 is the free abelian group of objects of the assembler modulo the cutting and pasting relations and in which the higher homotopy groups encode further geometric invariants the goal of this paper is to prove structural theorems about this k theory spectrum including analogs of quillen s localization and devissage theorems we demonstrate the uses of these theorems by analyzing the assembler associated to the grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes

a general maximum principle for optimal control of forward backward stochastic systems abstract a general maximum principle for optimal control problems derived by forward backward stochastic systems is established where control domains are non convex and forward diffusion coefficients explicitly depend on control variables these optimal control problems have broad applications in mathematical finance and economics such as the recursive mean variance portfolio choice problems the maximum principle is applied to study a forward backward linear quadratic optimal control problem with a non convex control domain an optimal solution is obtained

minimization of risk and linear quadratic optimal control theory this article is concerned with the optimal control problem for the linear stochastic system xt x t 0 asxs bsus fs ds t 0 d i 1 ci s xs di s us gi s dwi s with the convex risk functional j u em xt et 0 g t xt u t dt in order to guarantee the existence of an optimal control without any weak compactness assumption on the admissible control set we assume that the risk function m is coercive and thatd 1 d di is uniformlypositive rather than to assume like in the control literature that the running risk function g is coercive with respect to the control variable in this new setting the running risk function g maybe independent of the control variable and therefore the so called singular linear quadratic lq stochastic control problem is included a rigorous theoryis developed for the general stochastic lq problem with random coefficients and the bounded mean oscillation martingale theoryis used to account for the concerned integrability it plays a crucial role in the following exposition a to connect the stochastic lq problem to two associated backward stochastic differential equations bsdes one is an n n symmetric matrix valued nonlinear riccati bsde and the other is an n dimensional linear bsde with unbounded coefficients b to show that the latter bsde has an adapted solution pair of the suitablynecessaryregularity this seems to be the first application in a stochastic lq theory of the bmo martingale theory which roots in harmonic analysis furthermore with the help of an a priori estimate on the risk functional existence and uniqueness of the solutions of backward stochastic riccati differential equations bsrdes in the singular case is reduced to the regular case via a perturbation method and then a new existence and uniqueness result on bsrdes is obtained for the singular case

the maximum principle for partially observed optimal control of stochastic differential equations this paper concerns partially observed optimal control of possibly degenerate stochastic differential equations with correlated noises between the system and the observation the control is allowed to enter into all the coefficients a general maximum principle is proved for the partially observed optimal control and the relations among the adjoint processes are established adjoint vector fields which are adapted to the past and present observations are introduced as the solutions to some backward stochastic partial differential equations bspdes and their relations are established under suitable conditions the adjoint processes are characterized in terms of the adjoint vector fields their differentials and hessians along the optimal state process some other formulations of the partially observed stochastic maximum principle are then derived

the gram sommerville and gauss bonnet theorems and combinatorial geometric measures for noncompact polyhedra abstract some non numerical curvature functions are introduced for noncompact and unbounded polyhedcra in an affine space over an ordered field in this paper with these functions the generalized gram sommerville and gauss bonnet theorems are obtained and are found to be dual the euler characteristic can be interpreted as a zero dimensional curvature measure on the space of polyhedra and the higher dimensional geometric measures are similarly obtained the usual numerical curvatures are naturally derived from the non numerical curvature functions

combinatorial mixed valuations abstract combinatorial mixed valuations associated to translation invariant valuations on polytopes are introduced in contrast to the construction of mixed valuations via polarization combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations in particular it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative for combinatorially positive valuations this has strong computational implications applied to the discrete volume the results generalize and strengthen work of bihan 2015 on discrete mixed volumes for rational polytopes it is proved that combinatorial mixed monotonicity is equivalent to monotonicity stronger even a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of bernig fu 2011

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

optimality conditions for partial information stochastic control problems driven by levy processes abstract in this paper we consider a partial information stochastic control problem where the system is governed by a nonlinear stochastic differential equation driven by teugels martingales associated with some levy process and an independent brownian motion we prove optimality necessary conditions in the form of a maximum principle these conditions turn out to be sufficient under some convexity assumptions to illustrate the general results an example is solved

bse s bsde s and fixed point problems in this paper we introduce a class of backward stochastic equations bses that extend classical bsdes and include many interesting examples of generalized bsdes as well as semimartingale backward equations we show that a bse can be translated into a fixed point problem in a space of random vectors this makes it possible to employ general fixed point arguments to establish the existence of a solution for instance banach s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with lipschitz coefficients whereas schauder type fixed point arguments can be applied to non lipschitz equations the approach works equally well for multidimensional as for one dimensional equations and leads to results in several interesting cases such as equations with path dependent coefficients anticipating equations mckean vlasov type equations and equations with coefficients of superlinear growth

maximum principles for jump diffusion processes with infinite horizon abstract we prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information the results are applied to an optimal consumption and portfolio problem in infinite horizon

the maximum principle for a jump diffusion mean field model and its application to the mean variance problem abstract this paper establishes a necessary and sufficient stochastic maximum principle for a mean field model with randomness described by brownian motions and poisson jumps we also prove the existence and uniqueness of the solution to a jump diffusion mean field backward stochastic differential equation a new version of the sufficient stochastic maximum principle which only requires the terminal cost is convex in an expected sense is applied to solve a bicriteria mean variance portfolio selection problem

asymptotic stability of a stochastic may mutualism system abstract this paper deals with a mutualism system in random environments in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other mathematically we obtain the existence and uniqueness of a stable stationary distribution by means of markov semigroup theory and fokker planck equation in addition we prove that densities of the distributions for the solutions converge in l 1 to an invariant density under appropriate conditions numerical simulations are also presented to illustrate the theoretical results

triangulations of cayley and tutte polytopes cayley polytopes were defined recently as convex hulls of cayley compositions introduced by cayley in 1857 in this paper we resolve braun s conjecture which expresses the volume of cayley polytopes in terms of the number of connected graphs we extend this result to a two variable deformations which we call tutte polytopes the volume of the latter is given via an evaluation of the tutte polynomial of the complete graph our approach is based on an explicit triangulation of the cayley and tutte polytope we prove that simplices in the triangulations correspond to labeled trees and forests the heart of the proof is a direct bijection based on the neighbors first search graph traversal algorithm r esum

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

a valuation theoretic approach to translative equidecomposability abstract all simple translation invariant valuations on polytopes are classified as a direct consequence the well known conditions for translative equidecomposability are recovered furthermore a simplified proof of the classification of continuous simple translation invariant valuations is presented

a stochastic maximum principle for a markov regime switching jump diffusion model and its application to finance this paper develops a sufficient stochastic maximum principle for a stochastic optimal control problem where the state process is governed by a continuous time markov regime switching jump diffusion model we also establish the relationship between the stochastic maximum principle and the dynamic programming principle in a markovian case applications of the stochastic maximum principle to the mean variance portfolio selection problem are discussed

multiobjective control for nonlinear stochastic poisson jump diffusion systems via t s fuzzy interpolation and pareto optimal scheme abstract unlike the conventional mixed h 2 h control design method this study provides a multiobjective fuzzy control design method for nonlinear stochastic poisson jump diffusion systems to simultaneously achieve optimal cost and robustness performance in the pareto optimal sense via the proposed evolutionary algorithm for a nonlinear stochastic poisson jump diffusion system the poisson jumps cause its system behaviors to change intensely and discontinuously to design an efficient controller for a nonlinear stochastic jump diffusion system is much more difficult on the other hand the h 2 and h performance indices generally conflict with each other and can be regarded as a multiobjective optimization problem mop it is not easy to directly solve this mop owing to i the pareto front of the mop is difficult to obtain through direct calculation ii the mop is a hamilton jacobi inequalities hjis constrained mop to address these issues we use takagi sugeno t s interpolation scheme to transform the hjis constrained mop into a linear matrix inequality lmi constrained mop then we employ the proposed lmi constrained multiobjective optimization evolutionary algorithm lmi constrained moea to efficiently search for the pareto optimal solution from which the designer can select one kind of design according to their preference finally a design example is given to illustrate the design procedure and to verify our results