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

linear quadratic partially observed forward backward stochastic differential games and its application in finance abstract this paper is concerned with a partially observed linear quadratic game problem driven by forward backward stochastic differential equations where the forward diffusion coefficients do not contain control variables and the control domains are not necessarily convex the drift term of the observation equation is linear with respect to the state and there is correlated noise between the state and the observation equation by virtue of the classical spike variational method and the backward separation technique we derive a necessary and a sufficient condition of the stochastic differential game problem then we obtain filtering equations and present a feedback representation form of the equilibrium point through riccati equations as a practical application we solve a partial information investment problem involving g expectation as a convex risk measurement and give the numerical simulation to show the explicit investment strategy and illustrate some reasonable phenomena influenced by common financial parameters

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

reflected solutions of backward sde s and related obstacle problems for pde s we study reflected solutions of one dimensional backward stochastic differential equations the reflection keeps the solution above a given stochastic process we prove uniqueness and existence both by a fixed point argument and by approximation via penalization we show that when the coefficient has a special form then the solution of our problem is the value function of a mixed optimal stopping optimal stochastic control problem we finally show that when put in a markovian framework the solution of our reflected bsde provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation

sub concave and sub convex capacities abstract in this study we propose the concepts of sub concave and sub convex capacities first we investigate some properties of sub concave and sub convex capacities second we discuss the choquet integrals with respect to sub concave and sub convex capacities we then show that the upper and lower probabilities comprising the supremum and infimum over the set of risk neutral martingale measures in the asymmetrical case are sub concave and sub convex capacities respectively

on the rate of convergence of the binomial tree scheme for american options an american put option can be modelled as a variational inequality with a penalization approximation to this variational inequality the convergence rate o big delta x 2 3 big of the binomial tree scheme is obtained in this paper

existence conditions and variational approach for adapted solutions of the two point boundary value problem of stochastic differential equations abstract this paper considers the two point boundary value problem of stochastic differential equation with the following form dx t f t x t dt t x t dw t ax 0 bx t the sufficient and necessary conditions are given for the existence of the adapted solutions in the simple case that f t xt ft the solution can be obtained by introducing a control term ft extending the solution from xt to xt ft and constructing a process sequence all the results obtained are compared with those related to the eigenvalue problems and the martingale approach solution proposed by pardoux and peng for backward stochastic differential equations

counterexamples to q matrix conjectures abstract a matrix m r n n is in the class q if for all q r n there exist w z r n such that w mz q wtz 0 it has been conjectured that it is possible to tell if a matrix m is in q solely by considering the signs of subdeterminants of m we present two matrices that have the same signs of corresponding subdeterminants but are such that one is in q and the other is not the second of these two matrices has the property that every column of the matrix i m is in the interior of the union of the complementary cones associated with i m

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

numerical methods for a class of nonlocal diffusion problems with the use of backward sdes we propose a novel numerical approach for nonlocal diffusion equations du et al 2012 with integrable kernels based on the relationship between the backward kolmogorov equation and backward stochastic differential equations bsdes driven by levy jumps processes the nonlocal diffusion problem under consideration is converted to a bsde for which numerical schemes are developed as a stochastic approach the proposed method completely avoids the challenge of iteratively solving non sparse linear systems arising from the nature of nonlocality this allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion moreover our method recovers the convergence rates of classic deterministic approaches e g finite element or collocation methods due to the use of high order temporal and spatial discretization schemes in addition our approach can handle a broad class of problems with general inhomogeneous forcing terms as long as they are globally lipschitz continuous rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach

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

monotonic limit theorem of bsde and nonlinear decomposition theorem of doob meyers type we have obtained the following limit theorem if a sequence of rcll supersolutions of a backward stochastic differential equations bsde converges monotonically up to y t with e sup t y t 2 then y t itself is a rcll supersolution of the same bsde theorem 2 4 and 3 6 we apply this result to the following two problems 1 nonlinear doob meyer decomposition theorem 2 the smallest supersolution of a bsde with constraints on the solution y z the constraints may be non convex with respect to y z and may be only measurable with respect to the time variable t this result may be applied to the pricing of hedging contingent claims with constrained portfolios and or wealth processes

finding adapted solutions of forward backward stochastic differential equations method of continuation the notion of bridge is introduced for systems of coupled forward backward stochastic differential equations fbsdes for short this notion helps us to unify the method of continuation in finding adapted solutions to such fbsdes over any finite time durations it is proved that if two fbsdes are linked by a bridge then they have the same unique solvability consequently by constructing appropriate bridges we obtain several classes of uniquely solvable fbsdes

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

optimal convergence rate of the explicit finite difference scheme for american option valuation an optimal convergence rate o dx for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition s email protected dx 2 1 using completely pde methods as a corollary a binomial tree scheme of an american put option where s email protected dx 2 1 is convergent unconditionally with the rate o dt 1 2

convergence of binomial tree methods for european american path dependent options the binomial tree method first proposed by cox ross and rubinstein journal of financial economics 7 1979 pp 229 263 is one of the most popular approaches to pricing options by introducing an additional path dependent variable such methods can be readily extended to the valuation of path dependent options in this paper using numerical analysis and the notion of viscosity solutions we present a unifying theoretical framework to show the uniform convergence of binomial tree methods for european american path dependent options including arithmetic average options geometric average options and lookback options

a partial characterization of a class of matrices defined by solutions to the linear complementarity problem in this paper we analyze the class of matrices q0 characterized as all n n matrices m for which the linear complementarity problem w mz q w z 0 w z 0 has a solution whenever q is feasible the method is to analyze the faces of the linear cone cone i m where i is the n n identity essentially a labelling requirement for the faces is shown necessary for m to be in q0 with proper restrictions and nondegenerate assumptions sufficiency is also shown these results arc related to the signs of the minors of m and some conjectures in this area are also discussed

on the equality of solutions of max min and min max systems of variational inequalities with interconnected bilateral obstacles abstract in this paper we deal with the solutions of systems of pdes with bilateral interconnected obstacles of min max and max min types these systems arise naturally in stochastic switching zero sum game problems we show that when the switching costs of one side are regular the solutions of the min max and max min systems coincide then this common viscosity solution is related to a multi dimensional doubly reflected bsde with bilateral interconnected obstacles finally its relationship with the values of a zero sum switching game is studied

generalized bsde with 2 reflecting barriers and stochastic quadratic growth abstract we study the existence of a solution for a one dimensional generalized backward stochastic differential equation with two reflecting barriers grbsde for short under assumptions on the input data which are weaker than that on the current literature in particular we construct a maximal solution for such a grbsde when the terminal condition is only f t measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any p integrability conditions the work is suggested by the interest the results might have in dynkin game problem and american game option

laguerre type special functions and population dynamics abstract we introduce new laguerre type population dynamics models these models arise quite naturally by substituting in classical models the ordinary derivatives with the laguerre derivatives and therefore by using the so called laguerre type exponentials instead of the ordinary exponential the l exponentials e n t are increasing convex functions for t 0 but increasing slower with respect to exp t for this reason these functions are useful in order to approximate different behaviors of population growth we consider mainly the laguerre type derivative d t t d t connected with the l exponential e 1 t and investigate the corresponding modified logistic bernoulli and gompertz models invariance of the volterra lotka model is mentioned

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

on semi linear degenerate backward stochastic partial differential equations in this paper we study a class of one dimensional degenerate semilinear backward stochastic partial differential equations bspdes for short of parabolic type by establishing some new a priori estimates for both linear and semilinear bspdes we show that the regularity and uniform boundedness of the adapted solution to the semilinear bspde can be determined by those of the coefficients a special feature that one usually does not expect from a stochastic differential equation the proof follows the idea of the so called bootstrap method which enables us to analyze each of the derivatives of the solution under consideration some related results including some comparison theorems of the adapted solutions for semilinear bspdes as well as a nonlinear stochastic feynman kac formula are also given

pricing of american contingent claims with jump stock price and constrained portfolios in this paper we study the problem of pricing american contingent claims in an incomplete market where the stock price process is supposed to be driven by both a wiener process and a poisson random measure and the portfolios are constrained we formulate this problem as to find the minimal solution of a backward stochastic differential equation bsde with constraints we use the penalization method to construct a sequence of bsdes with respect to wiener process and poisson random measure and we show that the solutions of these equations converge to the minimal solution we are interested in finally in the markovian case we characterize the minimal hedging price as the minimal viscosity supersolution of an integral partial differential inequality with constraints

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

anticipated backward stochastic differential equations driven by the teugels martingales abstract in this paper a class of anticipated backward stochastic differential equations driven by teugels martingales associated with levy process is investigated we obtain the existence and uniqueness of solutions to these equations by means of the fixed point theorem we show that a comparison theorem for this type of absdes also holds under some slight stronger conditions

on the integral representation of g expectations with terminal constraints abstract in this paper we study the integral representation of g expectations with two kinds of terminal constraints and obtain the corresponding necessary and sufficient conditions

a finite characterization of k matrices in dimensions less than four the class of real n n matrices m known as k matrices for which the linear complementarity problem w mz q w 1 o z 1 o wtz 0 has a solution whenever w mz q w i o z 1 0 has a solution is characterized for dimensions n 4 the characterization is finite and practical several necessary conditions sufficient conditions and counterexamples pertaining to k matrices are also given a finite characterization of completely k matrices k matrices all of whose principal submatrices are also k matrices is proved for dimensions 4

well posedness of backward stochastic differential equations with general filtration abstract this paper is addressed to the well posedness of some linear and semilinear backward stochastic differential equations with general filtration without using the martingale representation theorem the point of our approach is to introduce a new notion of solution i e the transposition solution which coincides with the usual strong solution when the filtration is natural but it is more flexible for the case of general filtration than the existing notion of solutions a comparison theorem for transposition solutions and a pontryagin type stochastic maximum principle are also presented

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

q matrix recognition via secondary and universal polytopes mx q 0 mx q 0 x 0 has a solution we explain how one can use the polyhedral structure of the set of all triangulations of a finite point set to determine if an n n matrix m is a q matrix our implementation of the algorithm is practical for deciding the q nature for all m with n 8

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

anticipated backward doubly stochastic differential equations in this paper we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations anticipated bdsdes the coefficients of these bdsdes depend on the future value of the solution y z we obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations besides as an application we also establish a duality between the anticipated bdsdes and the delayed doubly stochastic differential equations delayed dsdes

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

a copositive q matrix which is not r 0 jeter and pye gave an example to show that pang s conjecture thatl1 q r0 is false while seetharama gowda showed that the conjecture is true for symmetric matrices it is known thatl1 symmetric matrices are copositive matrices jeter and pye as well as seetharama gowda raised the following question is it truec0 q r0 in this note we present an example of a copositive q matrix which is notr0 the example is based on the following elementary proposition leta be a square matrix of ordern supposer1 r2 whereri stands for theith row ofa further supposea11 anda22 are q matrices whereaii stands for the principal submatrix omitting theith row andith column froma thena is a q matrix

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

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

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

a homological characterization of q matrices a real square matrix m is said to be a q matrix if the linear complementarity problem q m has a solution for every vector q there is as yet no characterization of q matrices which makes it easy to determine whether or not a given matrix is q ideas from topology in particular degreetheory have previously been used to obtain sufficient conditions for when a matrix is q in this paper we will apply some other ideas from topology to give a homological characterization of q matrices continuing to borrow from topology we define the nerve of a matrix which along with our characterization leads to an algorithm for checking whether or not a matrix is q this algorithm has smaller bounds on its worst case time complexity than previous methods

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

backward stochastic differential equations with reflection and dynkin games we establish existence and uniqueness results for adapted solutions of backward stochastic differential equations bsde s with two reflecting barriers generalizing the work of el karoui kapoudjian pardoux peng and quenez existence is proved first by solving a related pair of coupled optimal stopping problems and then under different conditions via a penalization method it is also shown that the solution coincides with the value of a certain dynkin game a stochastic game of optimal stopping moreover the connection with the backward sde enables us to provide a pathwise deterministic approach to the game

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

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

a pontryagin s maximum principle for non zero sum differential games of bsdes with applications this technical note is concerned with a maximum principle for a new class of non zero sum stochastic differential games the most distinguishing feature compared with the existing literature is that the game systems are described by backward stochastic differential equations bsdes this kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities we establish a necessary condition and a sufficient condition in the form of maximum principle for open loop equilibrium point of the foregoing games respectively to explain the theoretical results we use them to study a financial problem

forward backward stochastic differential equations and controlled mckean vlasov dynamics the purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the mckean vlasov type motivated by the recent interest in mean field games we highlight the connection and the differences between the two sets of problems we prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control we also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions

backward stochastic differential equations with constraints on the gains process we consider backward stochastic differential equations with convex constraints on the gains or intensity of noise process existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is lipschitz continuous in the state and gains processes and convex in the gains process it is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control type equation this representation is related to the penalization method for constructing solutions of stochastic differential equations involves change of measure techniques and employs notions and results from convex analysis such as the support function of the convex set of constraints and its various properties

bsdes with two reflecting barriers the general result in this paper we show the existence of a solution for the bsde with two reflecting barriers when those latter are completely separated neither mokobodzki s condition nor the regularity of the barriers are supposed the main tool is the notion of local solution of reflected bsdes applications related to dynkin games and double obstacle variational inequality are given

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

dynkin game of stochastic differential equations with random coefficients and associated backward stochastic partial differential variational inequality a dynkin game is considered for stochastic differential equations with random coefficients we first apply qiu and tang s maximum principle for backward stochastic partial differential equations to generalize a krylov estimate for the distribution of a markov process to that of a non markov process and establish a generalized ito kunita wentzell formula which allows the test function as an ito process to take values in a sobolev space we then prove the verification theorem that the nash equilibrium point and the value of the dynkin game are characterized by the strong solution of the associated hamilton jacobi bellman isaacs equation which is currently a backward stochastic partial differential variational inequality bspdvi with two obstacles finally we prove the existence and uniqueness result and a comparison theorem for strong solution of the bspdvi

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

the laguerre transform in two variables and its application in partial differential equations an attempt is made to investigate some properties of the laguerre transform in two variables this technique is very effective for solving certain types of partial differential equations

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

some results on quasi monomiality a polynomial set p n n 0 is called quasi monomial if and only if it is possible to define two operators p and m independent of n such thatp p n x np n 1 x andm p n x p n 1 x in this paper we show that every polynomial set is quasi monomial and we present some useful tools to explicitly express the p and m operators for some polynomial families given by their generating functions the obtained results are applied to boas buck polynomial sets

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

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

dynkin game of convertible bonds and their optimal strategy abstract this paper studies the valuation and optimal strategy of convertible bonds as a dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method we first reduce such a dynkin game to an optimal stopping time problem with state constraint and then in a markovian setting we investigate the optimal strategy by analyzing the properties of the corresponding free boundary including its position asymptotics monotonicity and regularity we identify situations when call precedes conversion and vice versa moreover we show that the irregular payoff results in the possibly non monotonic conversion boundary surprisingly the price of the convertible bond is not necessarily monotonic in time it may even increase when time approaches maturity

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

backward stochastic optimal control with mixed deterministic controller and random controller and its applications in linear quadratic control abstract this paper investigates an optimal control problem driven by backward stochastic differential equation bsde with two controllers one is called deterministic controller and the other one is called random controller necessary and sufficient conditions for the mixed optimal control problem are derived a linear quadratic lq case of the mixed optimal control problem is also studied the mixed optimal controllers are explicitly expressed by the solution of a fully coupled mean field forward backward stochastic differential equation sde one of novel features is that a kind of mean field bsde naturally arises from the research on the mixed optimal control problem finally a product management problem is used to illustrate the theoretical results

on the quasi linear reflected backward stochastic partial differential equations abstract this paper is concerned with the quasi linear reflected backward stochastic partial differential equation rbspde for short basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential we first associate rbspde to a variational problem and via the penalization method we prove the existence and uniqueness of the solution for linear rbspde with laplacian leading coefficients with the continuity approach we further obtain the well posedness of general quasi linear rbspdes related results including ito formulas for backward stochastic partial differential equations with stochastic regular measures the comparison principle for solutions of rbspdes and the connections with reflected backward stochastic differential equations and optimal stopping problems are addressed as well

the dynamic programming equation for second order stochastic target problems motivated by applications in mathematical finance u cetin h m soner and n touzi options hedging for small investors under liquidity costs finance stoch to appear we continue our study of second order backward stochastic equations in this paper we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems in contrast with previous formulations of similar problems we restrict control processes to be continuous this new framework enables us to apply our results to a larger class of models also the resulting derivation is more transparent the main technical tool is the geometric dynamic programming principle in this context and it is proved by using the framework developed in h m soner and n touzi j eur math soc jems 8 2002 pp 201 236

anticipated backward stochastic differential equations in this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations anticipated bsdes in these equations the generator includes not only the values of solutions of the present but also the future we show that these anticipated bsdes have unique solutions a comparison theorem for their solutions and a duality between them and stochastic differential delay equations

numerical algorithms and simulations for reflected backward stochastic differential equations with two continuous barriers in this paper we study different algorithms for reflected backward stochastic differential equations bsde in short with two continuous barriers based on the framework of using a binomial tree to simulate 1 d brownian motion we introduce numerical algorithms by the penalization method and the reflected method respectively in the end simulation results are also presented

continuous time dynkin games with mixed strategies let x y z be a triple of payoff processes defining a dynkin game tilde r sigma tau e left x sigma 1 tau sigma y tau 1 tau sigma z tau 1 tau sigma right where sigma and tau are stopping times valued in 0 t in the case z y it is well known that the condition x leq y is needed in order to establish the existence of value for the game i e inf tau sup sigma tilde r sigma tau sup sigma inf tau tilde r sigma tau in order to remove the condition x leq y we introduce an extension of the dynkin game by allowing for an extended set of strategies namely the set of mixed strategies the main result of the paper is that the extended dynkin game has a value when z leq y and the processes x and y are restricted to be semimartingales continuous at the terminal time t

numerical analysis on binomial tree methods for a jump diffusion model this paper studies the numerical approximation for an european option pricing model with jump diffusion equivalence of the binomial tree method and an explicit difference scheme is discussed the optimal error estimation of the binomial tree approximation is also given another explicit difference scheme is constructed which has higher accuracy than the binomial tree method numerical results coincide with the theoretical results

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

the deformed exponential functions of two variables in the context of various statistical mechanics in the recent development in various disciplines of physics it is noted the need for including the deformed versions of the exponential functions in last two decades the tsallis and kaniadakis versions have found a lot of applications in this paper we consider the deformations which have two purposes first we introduce them like beginning of a more general mathematical approach where the tsallis and kaniadakis exponential functions are the special cases then we wish to pay attention to the mathematical community that they have a lot of interesting properties from mathematical point of view and possibilities in applications really we will show the differential and difference properties of our deformations which are important for the formation and explanation of continuous and discrete models of numerous phenomena

generalized tricomi and hermite tricomi functions abstract general classes of tricomi and hermite tricomi functions are introduced by exploiting properties of an iterated isomorphism related to the so called laguerre type exponentials and we mainly consider the properties of the general classes of 3 variable 2 index tricomi functions and 2 index 4 variable 1 parameter hermite tricomi functions

the maximum principle for global solutions of stochastic stackelberg differential games national science foundation dms 1303775 research grant council of the hksar cityu 500113

bsdes with regime switching weak convergence and applications abstract this paper is concerned with a system of backward stochastic differential equations bsdes with regime switching the bsdes are coupled by a finite state markov chain the underlying markov chain is assumed to have a two time scale or weak and strong interactions structure namely the states of the markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups it is shown in this paper that the original bsde system can be approximated by a limit system in which the states in each group are aggregated out and replaced by a single state in particular it is proved that the solution of the original bsde system converges weakly under the meyer zheng topology as the fast jump rate goes to infinity the limit process is a solution of aggregated bsdes which can be determined by the corresponding martingale problem the results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system finally a numerical example is given to demonstrate the approximation results

second order backward stochastic differential equations and fully nonlinear parabolic pdes for a d dimensional diffusion of the form dxt xt dt xt dwt and continuous functions f and g we study the existence and uniqueness of adapted processes y z and a solving the second order backward stochastic differential equation 2bsde dyt f t xt yt zt t dt z dxt t 0 t dzt at dt t dxt