bsdes with singular terminal condition and a control problem with constraints we provide a probabilistic solution of a not necessarily markovian control problem with a state constraint by means of a backward stochastic differential equation bsde the novelty of our solution approach is that the bsde possesses a singular terminal condition we prove that a solution of the bsde exists thus partly generalizing existence results obtained by popier in stochastic process appl 116 2006 pp 2014 2056 and ann probab 35 2007 pp 1071 1117 we perform a verification and discuss special cases for which the control problem has explicit solutions

on weak solutions of forward backward sdes in this paper we continue exploring the notion of weak solution of forward backward stochastic differential equations fbsdes and associated forward backward martingale problems fbmps the main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work ma et al in ann probab 36 6 2092 2125 2008 we consider a general class of non degenerate fbsdes in which all the coefficients are assumed to be essentially only bounded and uniformly continuous and the uniqueness is proved in the space of all the square integrable adapted solutions the standard solution space in the fbsde literature a new notion of semi strong solution is introduced to clarify the relations among different definitions of weak solution in the literature and it is in fact instrumental in our uniqueness proof as a by product we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear pde

meromorphic functions sharing values in an angular domain in this paper we study the problems of meromorphic functions sharing values in an angular domain and obtain some uniqueness results which are the improvement of the well known five value theorem and four value theorem given by nevanlinna

stochastic optimal control via forward and backward stochastic differential equations and importance sampling abstract the aim of this work is to present a novel sampling based numerical scheme designed to solve a certain class of stochastic optimal control problems utilizing forward and backward stochastic differential equations fbsdes by means of a nonlinear version of the feynman kac lemma we obtain a probabilistic representation of the solution to the nonlinear hamilton jacobi bellman equation expressed in the form of a system of decoupled fbsdes this system of fbsdes can be solved by employing linear regression techniques the proposed framework relaxes some of the restrictive conditions present in recent sampling based methods within the linearly solvable optimal control framework and furthermore addresses problems in which the time horizon is not prespecified to enhance the efficiency of the proposed scheme when treating more complex nonlinear systems we then derive an iterative algorithm based on girsanov s theorem on the change of measure which features importance sampling this scheme is shown to be capable of learning the optimal control without requiring an initial guess

uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions in this paper we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions

mean variance hedging under partial information we consider the mean variance hedging problem under partial information the underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available we show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term which is formulated in terms of observable processes we prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation

sharp derivative bounds for solutions of degenerate semi linear partial differential equations abstract the paper is a continuation of the kusuoka stroock programme of establishing smoothness properties of solutions of possibly degenerate partial differential equations by using probabilistic methods we analyze here a class of semi linear parabolic partial differential equations for which the linear part is a second order differential operator of the form v 0 i 1 n v i 2 where v 0 v n are first order differential operators that satisfy the so called ufg condition see kusuoka and stroock 1987 16 which is weaker than the hormander one specifically we prove that the bounds of the higher order derivatives of the solution along the vector fields coincide with those obtained in the linear case when the boundary condition is lipschitz continuous but that the asymptotic behavior of the derivatives may change because of the simultaneity of the nonlinearity and of the degeneracy when the boundary condition is of polynomial growth and measurable only

a multidimensional exponential utility indifference pricing model with applications to counterparty risk this paper considers exponential utility indifference pricing for a multidimensional nontraded assets model subject to intertemporal default risk and provides a semigroup approximation for the utility indifference price the key tool is the splitting method whose convergence is proved based on the barles souganidis monotone scheme and the convergence rate is derived based on krylov s shaking the coefficients technique we apply our methodology to study the counterparty risk of derivatives in incomplete markets

completeness of security markets and backward stochastic differential equations with unbounded coefficients abstract for a standard black scholes type security market completeness is equivalent to the solvability of a linear backward stochastic differential equation bsde when the interest rate is bounded there exists a bounded risk premium process and the volatility matrix has certain surjectivity then the bsde will be solvable and the market will be complete however if the risk premium process and or the interest rate is not bounded one gets a bsde with unbounded coefficients to solve in this paper we will discuss such a situation and will present some solvability results for the bsde which will lead to the completeness of the market

a stable multistep scheme for solving backward stochastic differential equations in this paper we propose a stable multistep scheme on time space grids for solving backward stochastic differential equations in our scheme the integrands which are conditional mathematical expectations derived from the original equations are approximated by using lagrange interpolating polynomials with values of the integrands at multiple time levels they are then numerically evaluated using the gauss hermite quadrature rules and polynomial interpolations on the spatial grids error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions finally various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme

existence results for some quasilinear parabolic equations on considere le probleme de cauchy pour une equation dont la partie principale est un operateur parabolique quasilineaire le terme non lineaire additionnel a au plus une croissance quadratique par rapport au gradient de la solution on demontre l existence d une solution bornee faible sous des hypotheses de regularite minimales en supposant l existence d une sous solution et d une supersolution pour le probleme

meromorphic functions in the unit disc that share values in an angular domain in this paper we investigate the uniqueness of meromorphic functions in the unit disc and consider the relation between the borel points and shared values of meromorphic functions in an angular domain

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 forward backward probabilistic algorithm for the incompressible navier stokes equations abstract a novel probabilistic scheme for solving the incompressible navier stokes equations is studied in which we approximate a generalized nonlinear feyman kac formula the velocity field is interpreted as the mean value of a stochastic process ruled by forward backward stochastic differential equations fbsdes driven by brownian motion following an approach by delbaen qiu and tang introduced in 2015 the pressure term is obtained from the velocity by solving a poisson problem as computing the expectation of an integral functional associated to an extra bsde the fbsdes components are numerically solved by using a forward backward algorithm based on euler type schemes for the local time integration and the quantization of the increments of brownian motion following the algorithm proposed by delarue and menozzi in 2006 numerical results are reported on spatially periodic analytic solutions of the navier stokes equations for incompressible fluids we illustrate the proposed algorithm on a two dimensional taylor green vortex and three dimensional beltrami flows

generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree abstract the set pol d r m n of m n complex matrix polynomials of grade d and normal rank at most r in a complex d 1 m n dimensional space is studied for r 1 min m n 1 we show that pol d r m n is the union of the closures of the r d 1 sets of matrix polynomials with rank r degree exactly d and explicitly described complete eigenstructures in addition for the full rank rectangular polynomials i e r min m n and m n we show that pol d r m n coincides with the closure of a single set of the polynomials with rank r degree exactly d and the described complete eigenstructure these complete eigenstructures correspond to generic m n matrix polynomials of grade d and rank at most r

convexity bounds for bsde solutions with applications to indifference valuation we consider backward stochastic dierential equations bsdes with a particular quadratic generator and study the behaviour of their solutions when the probability measure is changed the ltration is shrunk or the underlying probability space is transformed our main results are upper bounds for the solutions of the original bsdes in terms of solutions to other bsdes which are easier to solve we illustrate our results by applying them to exponential utility indierence valuation

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

numerical algorithms for forward backward stochastic differential equations efficient numerical algorithms are proposed for a class of forward backward stochastic differential equations fbsdes connected with semilinear parabolic partial differential equations as in j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 the algorithms are based on the known four step scheme for solving fbsdes the corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach the derivatives of the solution u of the semilinear equation are found by finite differences the forward equation is simulated by mean square methods of order 1 2 and 1 corresponding convergence theorems are proved along with the algorithms for fbsdes on a fixed finite time interval we also construct algorithms for fbsdes with random terminal time the results obtained are supported by numerical experiments

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

backward stochastic partial differential equations with quadratic growth abstract this paper is concerned with the weak solution in analytic sense to the cauchy dirichlet problem of a backward stochastic partial differential equation when the nonhomogeneous term has a quadratic growth in both the gradient of the first unknown and the second unknown existence and uniqueness results are obtained under separate conditions

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

backward stochastic differential equations associated with the vorticity equations abstract in this paper we derive a non linear version of the feynman kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions we prove the existence global in time and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2 a particular class of terminal values provide via these probabilistic methods solutions for the vorticity equation

meromorphic functions in the unit disc that share slowly growing functions in an angular domain in this paper we investigate the uniqueness of meromorphic functions in the unit disc that share five distinct meromorphic functions of slow growth in an angular domain

time inconsistent stochastic linear quadratic control in this paper we formulate a general time inconsistent stochastic linear quadratic lq control problem the time inconsistency arises from the presence of a quadratic term of the expected state as well as a state dependent term in the objective functional we define an equilibrium instead of optimal solution within the class of open loop controls and derive a sufficient condition for equilibrium controls via a flow of forward backward stochastic differential equations when the state is one dimensional and the coefficients in the problem are all deterministic we find an explicit equilibrium control as an application we then consider a mean variance portfolio selection model in a complete financial market where the risk free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes applying the general sufficient condition we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

bsdes with polynomial growth generators in a defaultable market abstract in this paper we study a backward stochastic differential equation bsde for short driven by a brownian motion and a jump martingale in a defaultable setting with a polynomial growth generator this kind of bsde has important applications to the defaultable market to demonstrate this in a strict way we first prove the existence uniqueness and uniform p p 2 estimate of its solution then two examples are given to illustrate its applications to recursive utility and contingent claim hedging

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

quadratic bsde with mathbb l 2 terminal data krylov s estimate ito krylov s formula and existence results we establish a krylov type estimate and an ito krylov change of variable formula for the solutions of one dimensional quadratic backward stochastic differential equations qbsdes with a measurable generator and an arbitrary terminal datum this allows us to prove various existence and uniqueness results for some classes of qbsdes with a square integrable terminal condition and sometimes a merely measurable generator it turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and or uniqueness of solutions we also establish a comparison theorem for solutions of a particular class of qbsdes with measurable generator as a byproduct we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations qpdes with a square integrable terminal datum

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 control for a class of nonlinear kernels and applications we consider a stochastic control problem for a class of nonlinear kernels more precisely our problem of interest consists in the optimization over a set of possibly non dominated probability measures of solutions of backward stochastic differential equations bsdes since bsdes are non linear generalizations of the traditional linear expectations this problem can be understood as stochastic control of a family of nonlinear expectations or equivalently of nonlinear kernels our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting which we then use to provide a semimartingale characterization of the value function we next explore several applications of our results we first obtain a wellposedness result for second order bsdes as introduced in 76 which does not require any regularity assumption on the terminal condition and the generator then we prove a non linear optional decomposition in a robust setting extending recent results of 63 which we then use to obtain a superhedging duality in uncertain incomplete and non linear financial markets finally we relate under additional regularity assumptions the value function to a viscosity solution of an appropriate path dependent partial differential equation ppde

on the backward stochastic riccati equation in infinite dimensions we study backward stochastic riccati equations bsres arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations we allow the coefficients both in the state equation and in the cost to be random in such a context bsres are backward stochastic differential equations existing in a non hilbert space and involving quadratic nonlinearities we propose two different notions of solutions to bsres and prove for both of them existence and uniqueness results we also show that such solutions allow us to perform the synthesis of the optimal control finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping

the dimension of matrices matrix pencils with given jordan kronecker canonical forms abstract the set of n by n matrices with a given jordan canonical form defines a subset of matrices in complex n 2 dimensional space we analyze one classical approach and one new approach to count the dimension of this set the new approach is based upon and meant to give insight into the staircase algorithm for the computation of the jordan canonical form as well as the occasional failures of this algorithm we extend both techniques to count the dimension of the more complicated set defined by the kronecker canonical form of an arbitrary rectangular matrix pencil a b

robustness and perturbations of minimal bases ii the case with given row degrees abstract this paper studies generic and perturbation properties inside the linear space of m m n polynomial matrices whose rows have degrees bounded by a given list d 1 d m of natural numbers which in the particular case d 1 d m d is just the set of m m n polynomial matrices with degree at most d thus the results in this paper extend to a much more general setting the results recently obtained in 29 only for polynomial matrices with degree at most d surprisingly most of the properties proved in 29 as well as their proofs remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well known sylvester matrices of polynomial matrices are replaced by the new trimmed sylvester matrices introduced in this paper the following results are presented among many others in this work 1 generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d 1 d m and with right minimal indices differing at most by one and having a sum equal to i 1 m d i and 2 under perturbations these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly

polynomial zigzag matrices dual minimal bases and the realization of completely singular polynomials minimal bases of rational vector spaces are a well known and important tool in systems theory if minimal bases for two subspaces of rational n space are displayed as the rows of polynomial matrices z 1 lambda k times n and z 2 lambda m times n respectively then z 1 and z 2 are said to be dual minimal bases if the subspaces have complementary dimension i e k m n and z 1 lambda z 2 t lambda 0 in other words each z j lambda provides a minimal basis for the nullspace of the other it has long been known that for any dual minimal bases z 1 lambda and z 2 lambda the row degree sums of z 1 and z 2 are the same in this paper we show that this is the only constraint on the row degrees thus characterizing the possible row degrees of dual minimal bases the proof is constructive making extensive use of a new class of sparse structured polynomial matrices that we have baptized zigzag matrices another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices given a list of left and right minimal indices and a desired degree d does there exist a completely singular matrix polynomial i e a matrix polynomial with no elementary divisors whatsoever of degree d having exactly the prescribed minimal indices we show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices the constructed realization is simple and explicitly displays the desired minimal indices in a fashion analogous to the classical kronecker canonical form of singular pencils

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

an interpolated stochastic algorithm for quasi linear pdes in this paper we improve the forward backward algorithm for quasi linear pdes introduced in a previous work the new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure for the convergence analysis we also exploit the optimality of the square gaussian quantization used to approximate the conditional expectations involved the resulting bound for the error is closely related to the holder exponent of the second order spatial derivatives of the true solution and turns out to be more satisfactory than the one previously established

well posedness and regularity of backward stochastic volterra integral equations backward stochastic volterra integral equations bsvies for short are studied notion of adapted m solution is introduced well posedness of bsvies is established and some regularity results are proved for the adapted m solutions via malliavin calculus a pontryagin type maximum principle is presented for optimal controls of stochastic volterra integral equations

robustness and perturbations of minimal bases abstract polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory linear systems theory and coding theory it is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis very recently minimal bases as well as the closely related pairs of dual minimal bases have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices the development of new classes of linearizations and l ifications of polynomial matrices and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations these new applications have revealed that although the algebraic properties of minimal bases are rather well understood their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature as far as we know therefore the main purpose of this paper is to study in detail when a minimal basis m is robust under perturbations i e when all the polynomial matrices in a neighborhood of m are minimal bases and in this case how perturbations of m change its dual minimal bases in order to study such problems a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and based on it we prove that polynomial matrices are generically minimal bases with very specific properties in addition some applications of the results of this paper are discussed

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

constructing strong l ifications from dual minimal bases abstract we provide an algorithm for constructing strong l ifications of a given matrix polynomial p of degree d and size m n using only the coefficients of the polynomial and the solution of linear systems of equations a strong l ification of p is a matrix polynomial of degree l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial p all explicit constructions of strong l ifications introduced so far in the literature have been limited to the case where l divides d though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible based on recent developments on dual polynomial minimal bases we present a general construction of strong l ifications for wider choices of the degree l namely when l divides one of nd or md and d l in the case where l divides nd respectively md the strong l ifications we construct allow us to easily recover the minimal indices of p in particular we show that they preserve the left resp right minimal indices of p and the right resp left minimal indices of the l ification are the ones of p increased by d l each moreover in the particular case l divides d the new method provides a companion l ification that resembles very much the companion l ifications already known in the literature

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

layer methods for stochastic navier stokes equations using simplest characteristics we propose and study a layer method for stochastic navier stokes equations snse with spatial periodic boundary conditions and additive noise the method is constructed using conditional probabilistic representations of solutions to snse and exploiting ideas of the weak sense numerical integration of stochastic differential equations we prove some convergence results for the proposed method including its first mean square order results of numerical experiments on two model problems are presented

numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations the probabilistic approach is used for constructing special layer methods to solve the cauchy problem for semilinear parabolic equations with small parameter despite their probabilistic nature these methods are nevertheless deterministic the algorithms are tested by simulating the burgers equation with small viscosity and the generalized kpp equation with a small parameter