connected graph representations of the quantum propagator and semiclassical expansions the time evolution operator u t s of a spinless nonrelativistic n body quantum system in euclidean space with real analytic time dependent scalar interaction v x t is studied a complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate and mixed coordinate momentum representation propagators the derivation is based on dyson s series for u t s and the combinatorics involved in the cluster expansion of the classical grand partition function these results provide an efficient means of generating nonperturbative propagator expansions in the physical variables mass m planck s constant h time displacement t s the structural bridge between the wkb and wigner kirkwood expansions is sketched for mixed representations of u t s in the heat equation context the graphical expansions are found for mixed representations of the density operator e beta h finally an explicit differential formula is obtained for wigner s distribution function f x p beta

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

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 the standard quantum brownian equation and an associated class of non autonomous master equations abstract it is shown that the standard quantum brownian equation qbe can violate positivity not only past the thermal correlation time but at arbitrarily long times at high system frequencies in an effort to improve the standard qbe exact operator solutions are provided for a class of non autonomous master equations these exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations

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

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

existence of solutions to a class of indefinite stochastic riccati equations an indefinite stochastic riccati equation is a matrix valued highly nonlinear backward stochastic differential equation together with an algebraic matrix positive definiteness constraint we introduce a new approach to solve a class of such equations including the existence of solutions driven by one dimensional brownian motion the idea is to replace the original equation by a system of backward stochastic differential equations without involving any algebraic constraint whose existence of solutions automatically enforces the original algebraic constraint to be satisfied

exact solutions for a wick type stochastic 2d kdv equation a wick type stochastic 2d kdv equation with variable coefficients is investigated the exact solutions are showed by using the homogeneous balance principle and the herimite transform in the white noise space

tree graphs and the solution to the hamilton jacobi equation a combinatorial method is used to construct solutions of the hamilton jacobi equation an exact expression for hamilton s principal function s is obtained for classical systems of finitely many particles interacting via a certain class of time dependent potentials if x p and t are the position momentum and time variables for n point particles of mass m it is shown that hamiltonians of the form h x p t 1 2m p2 v x t have complete integrals s that are analytic functions of the inverse mass parameter m 1 in a punctured disk about the origin if v x t is bounded c in the x variable and has controlled x derivative growth then the coefficients of the laurent expansion of s about m 1 0 may be expressed in terms of gradient structures associated with tree graphs this series expansion for s x t y t0 converges absolutely and uniformly for all x y for time displacements t t0 t 2k 1 m eu 1 2 where k and u are bounds associated with the space derivatives of the potential for t t0 t the classi

an asymptotic analysis of quantum evolution with electromagnetic fields a singular perturbation expansion of solutions to the schrodinger initial value problem is constructed using an approximate propagator for a nonrelativistic quantum system interacting with time dependent external electromagnetic fields this approximate propagator defines a gauge invariant semiclassical expansion that is realized by large mass scaling the asymptotic nature of this approximation is established by constructing error estimates that bound the hilbert space norm difference between the exact and approximate evolved states the maximum order of the approximation is determined explicitly as a function of the number of derivatives supported by the scalar and vector potentials the asymptotic expansion is obtained when the configuration space rd and also for problems where is a proper subset of rd and the self adjoint hamiltonian is defined using a supplementary boundary condition typically dirichlet or periodic

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

a generalized stochastic differential utility this paper generalizes in the setting of brownian information the duffie epstein 1992 stochastic differential formulation of intertemporal recursive utility sdu we provide a utility functional of state contingent consumption plans that exhibits a local dependency with respect to the utility intensity process the integrand of the quadratic variation and call it the generalized sdu this mathematical generalization of the sdu permits in fact more flexibility in the separation between risk aversion and intertemporal substitution and allows to model asymmetry in risk aversion we extensively use the backward stochastic differential equation theory to give sufficient conditions for comparative and absolute risk aversion behavior as well as aversion to specific directional risk additionally we discuss whether our functional exhibits monotonicity to its information filtration argument for purposes of illustration we provide some applications to the consumption portfolio strategy selection problem in a complete securities market

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

on the spanning property of risk bonds priced by equilibrium we propose a method of pricing financial securities written on nontradable underlyings such as temperature or precipitation levels to this end we analyze a financial market where agents are exposed to financial and nonfinancial risk factors the agents hedge their financial risk in the stock market and trade a risk bond issued by an insurance company from the issuer s point of view the bond s primary purpose is to shift insurance risks related to noncatastrophic weather events to financial markets as such its terminal payoff and yield curve depend on an underlying climate or temperature process whose dynamics are independent of the randomness driving stock prices we prove that if the bond s payoff function is monotone in the external risk process it can be priced by an equilibrium approach the equilibrium market price of climate risk and the equilibrium price process are characterized as solutions of nonlinear backward stochastic differential equations bsdes transferring the bsdes into partial differential equations pdes we represent the bond prices as smooth functions of the underlying risk factors our analytical results make the model amenable to a numerical analysis

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

backward stochastic dynamics on a filtered probability space we demonstrate that backward stochastic differential equations bsde may be reformulated as ordinary functional differential equations on certain path spaces in this framework neither ito s integrals nor martingale representation formulate are needed this approach provides new tools for the study of bsde and is particularly useful for the study of bsde with partial information the approach allows us to study the following type of backward stochastic differential equations with y t on a general filtered probability space f f t p where b is a d dimensional brownian motion l is a prescribed nonlinear mapping which sends a square integrable m to an adapted process l m and m a correction term is a square integrable martingale to be determined under certain technical conditions we prove that the system admits a unique solution y m in general the associated partial differential equations are not only nonlinear but also may be nonlocal and involve integral operators

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

a quantum wkb approximation without classical trajectories the time evolution of a coherent state is studied in the limit 0 for a spinless nonrelativistic quantum system in euclidean space with smooth bounded time dependent scalar interaction v x t the lowest order wkb approximation is derived rigorously complete with an explicit bound on its l2 norm error over a specified finite time interval the method is based on a constructive series representation of the classical action thereby avoiding the need to analyze the behavior of classical trajectories the amplitude of the initial states considered is neither assumed to be of rapid decrease nor to depend on

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

backward stochastic differential equations on manifolds the problem of finding a martingale on a manifold with a fixed random terminal value can be solved by considering bsdes with a generator with quadratic growth we study here a generalization of these equations and we give uniqueness and existence results in two different frameworks using differential geometry tools applications to pdes are given including a certain class of dirichlet problems on manifolds

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

path integrals and stationary phase approximation for quantum dynamical semigroups quadratic systems the solution of the markovian master equation for the quantum open system of n degrees of freedom is formally written in terms of a path integral and the stationary phase approximation is discussed the exactly soluble models with generators quadratic in position and momentum operators are investigated and the explicit expressions for the space time propagators for one dimensional systems are derived

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

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

exact constructions of square root helmholtz operator symbols the focusing quadratic profile operator symbols play a pivotal role in both the exact well posed one way reformulation of solving the elliptic helmholtz equation and the construction of the generalized bremmer coupling series the inverse square root and square root helmholtz operator symbols are the initial quantities of interest in both formulations in addition to providing the theoretical framework for the development and implementation of the parabolic equation pe method in wave propagation modeling exact standard left and weyl symbol constructions are presented for both the inverse square root and square root helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension extending and ultimately unifying the previously published corresponding results for the defocusing quadratic case j math phys 33 1887 1914 1992 both i spectral modal summation representations and ii contour integral representations exploiting the underlying periodicity of the associated quantum mechanical harmonic oscillator problem are derived and ultimately related through the propagating and nonpropagating contributions to the operator symbol high and low frequency asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the pe method moreover while the focusing quadratic profile is in some respects nonphysical the corresponding helmholtz operator symbols nevertheless establish canonical symbol features for more general profiles containing locally quadratic wells

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

phase space methods and path integration the analysis and computation of scalar wave equations abstract the scalar helmholtz equation plays a significant role in studies of electromagnetic seismic and acoustic direct wave propagation phase space or microscopic methods and path functional integral representations provide the appropriate framework to extend homogeneous fourier methods to inhomogeneous environments the two complementary approaches to this analysis and computation of the n dimensional helmholtz propagator are reviewed for the factorization one way path integration invariant imbedding approach the exact solution of the helmholtz composition equation for the weyl square root operator symbol is presented in the quadratic case the filtered one way phase space marching algorithm is examined in detail and compared numerically with wide angle one way partial differential wave equations formally derived from approximation theory for the second approach which directly constructs approximate two way path functionals the feasibility of a monte carlo statistical evaluation of the feynman garrod propagator is discussed

solvability conditions for indefinite linear quadratic optimal stochastic control problems and associated stochastic riccati equations a linear quadratic optimal stochastic control problem with random coefficients and indefinite state control weight costs is usually linked to an indefinite stochastic riccati equation sre which is a matrix valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown either the optimal control problem or the sre is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined in this paper by introducing a notion of subsolution for the sre we derive several novel sufficient conditions for the existence and uniqueness of the solution to the sre and for the solvability of the associated optimal stochastic control problem

quadratic bsdes with convex generators and unbounded terminal conditions in briand and hu probab theory relat fields 136 4 604 618 2006 the authors proved an existence result for bsdes with quadratic generators with respect to the variable z and with unbounded terminal conditions however no uniqueness result was stated in that work the main goal of this paper is to fill this gap in order to obtain a comparison theorem for this kind of bsdes we assume that the generator is convex with respect to the variable z under this assumption of convexity we are also able to prove a stability result in the spirit of the a priori estimates stated in karoui et al math finance 7 1 1 71 1997 with these tools in hands we can derive the nonlinear feynman kac formula in this context

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

on the wkb approximation to the propagator for arbitrary hamiltonians this paper presents a general expression for the wkb approximation to the propagator corresponding to an arbitrary hamiltonian operator h for example if the correspondence rule used to pass from the classical hamiltonian hc to h is such that it associates apiq j 1 a q jpi to piq j then the formula gives kwkb kvvexp 1 2 a ft 2hc qi pi qc t pc t t dt where kvv 2 ih n 2 det m 1 2exp isc h is van vleck s well known formula sc being the action functional evaluated at the classical path qc pc and mij 2sc qa i qbj more generally the formula presented here applies to any system with n degrees of freedom described by a function f x t whose time evolution is given by h x k x t k t f x t 0 regardless of the form of h the schrodinger equation of quantum mechanics and the fokker planck equation of diffusion are obvious examples many examples are discussed this generalizes results obtained in a previous publication j math phys 18 786 90 1977

on the validity of the wkb approximation we consider the wkb approximation to the non relativistic schrodinger equation for an n particle system with a possibly time dependent potential a recent rigorous and explicit error estimate given by molzahn 1988 is investigated by means of examples to this end we extend the class of potentials considered by molzahn and find that his estimates are almost optimal however these examples also reveal an inherent drawback of the wkb method

uniqueness results for second order bellman isaacs equations under quadratic growth assumptions and applications in this paper we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second order degenerate parabolic hamilton jacobi bellman and isaacs equations as an application we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation

a non linear parabolic pde with a distributional coefficient and its applications to stochastic analysis abstract we consider a non linear parabolic partial differential equation pde on r d with a distributional coefficient in the non linear term the distribution is an element of a besov space with negative regularity and the non linearity is of quadratic type in the gradient of the unknown under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the pde and investigate properties like continuity with respect to the initial condition and blow up times we prove a global existence and uniqueness result assuming further properties on the non linearity to conclude we consider an application of the pde to stochastic analysis in particular to a class of non linear backward stochastic differential equations with distributional drivers

feynman path integral for dissipative lagrangians for a general time dependent and the bateman morse feshbach bopp lagrangian the feynman path integral is evaluated by establishing a polynomial representation of the van vleck determinant

bsdes with terminal conditions that have bounded malliavin derivative abstract we show existence and uniqueness of solutions to bsdes of the form y t t t f s y s z s d s t t z s d w s in the case where the terminal condition has bounded malliavin derivative the driver f s y z is assumed to be lipschitz continuous in y but only locally lipschitz continuous in z in particular it can grow arbitrarily fast in z if in addition to having bounded malliavin derivative is bounded the driver needs only be locally lipschitz continuous in y in the special case where the bsde is markovian we obtain existence and uniqueness results for semilinear parabolic pdes with non lipschitz nonlinearities we discuss the case where there is no lateral boundary as well as lateral boundary conditions of dirichlet and neumann type

on the stochastic korteweg de vries equation abstract we consider a stochastic korteweg de vries equation forced by a random term of white noise type this can be a model of water waves on a fluid submitted to a random pressure we prove existence and uniqueness of solutions inh1 r in the case of additive noise and existence of martingales solutions inl2 r in the case of multiplicative noise copyright 1998 academic press nous etudions une equation de korteweg de vries stochastique comportant une force aleatoire de type bruit blanc cette equation peut decrire les ondes de surface sur un fluide soumis a un champ de pression aleatoire nous montrons l existence et l unicite de solutions dansh1 r lorsque le bruit est additif puis dans le cas du bruit multiplicatif nous etablissons l existence de solutions martingales dansl2 r

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

on a class of stochastic optimal control problems related to bsdes with quadratic growth in this paper we study a class of stochastic optimal control problems where the drift term of the equation has a linear growth on the control variable the cost functional has a quadratic growth and the control process takes values in a closed set not necessarily compact this problem is related to some backward stochastic differential equations bsdes with quadratic growth and unbounded terminal value we prove that the optimal feedback control exists and the optimal cost is given by the initial value of the solution of the related bsde

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

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

existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets a quasilinear parabolic equation with quadratic gradient terms is analyzed the equation models an optimal portfolio in so called incomplete financial markets consisting of risky assets and non tradable state variables its solution allows to compute an optimal portfolio strategy the quadratic gradient terms are essentially connected to the assumption that the so called relative risk aversion function is not logarithmic the existence of weak global in time solutions in any dimension is shown under natural hypotheses the proof is based on the monotonicity method of frehse furthermore the uniqueness of solutions is shown under a smallness condition on the derivatives of the covariance diffusion matrices using a nonlinear test function technique developed by barles and murat finally the influence of the non tradable state variables on the optimal value function is illustrated by a numerical example in three dimensions

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

scalar generalized screen algorithms in transversely isotropic media with a vertical symmetry axis the scalar generalized screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis vti although wave propagation in a transversely isotropic medium is essentially elastic we use an equivalent acoustic system of equations for the qp waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of mild anisotropy the enhanced accuracy of the generalized screen method as compared to the split step fourier methods allows the extension to vti media the generalized screen expansion of the one way propagator follows closely the method used in the isotropic case the medium is defined in terms of a background and a perturbation the generalized screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation we cast the theory into numerical algorithms and assess the accuracy of the generalized screen method in a partic

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

density analysis of bsdes in this paper we study the existence of densities with respect to the lebesgue measure for marginal laws of the solution y z to a quadratic growth bsde using the by now well established connection between these equations and their associated semi linear pdes together with the nourdin viens formula we provide estimates on these densities

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

linear forward backward stochastic differential equations with random coefficients solvability of linear forward backward stochastic differential equations fbsdes for short with random coefficients is studied a decoupling reduction method is introduced via which a large class of linear fbsdes with random or deterministic time varying coefficients is proved to be solvable on the other hand by means of four step scheme a riccati backward stochastic equation bsde for short for m n matrix valued processes is derived global solvability of such riccati bsdes is discussed for some special but nontrivial cases which leads to the solvability of the corresponding linear fbsdes

nonlinear fractional stochastic pdes and bdsdes with hurst parameter in 1 2 1 abstract we study a class of semilinear stochastic partial differential equations driven by a fractional brownian motion with hurst parameter h 1 2 1 for this end we use the doubly stochastic interpretation through a backward doubly stochastic differential equations driven by both a standard and an independent fractional brownian motion the doss sussmann transformation is employed to establish the link between the backward doubly stochastic differential equation and a backward stochastic differential equation driven only by the standard brownian motion through which the stochastic viscosity solution of the stochastic partial differential equation is studied

derivation and application of extended parabolic wave theories ii path integral representations the n dimensional reduced scalar helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through 1 the n dimensional extended parabolic weyl pseudodifferential equation and 2 an imbedding in an n 1 dimensional parabolic schrodinger equation the first relationship provides the basis for the parabolic based hamiltonian phase space path integral representation of the half space propagator the second relationship provides the basis for the elliptic based path integral representations associated with feynman and fradkin feynman and garrod and feynman and dewitt morette exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow and wide angle extended parabolic wave theories the path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process

viscosity solutions of general viscous hamilton jacobi equations we present comparison principles lipschitz estimates and study state constraints problems for degenerate second order hamilton jacobi equations

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

monotone stability of quadratic semimartingales with applications to unbounded general quadratic bsdes in this paper we study the stability and convergence of some general quadratic semimartingales motivated by financial applications we study simultaneously the semimartingale and its opposite their characterization and integrability properties are obtained through some useful exponential submartingale inequalities then a general stability result including the strong convergence of the martingale parts in various spaces ranging from h1 to bmo is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale this can be applied in particular to bsde like semimartingales this strong convergence result is then used to prove the existence of solutions of general quadratic bsdes under minimal exponential integrability assumptions relying on a regularization in both linear quadratic growth of the quadratic coefficient itself on the contrary to most of the existing literature it does not involve the seminal result of kobylanski ann probab 28 2010 558 602 on bounded solutions

constrained stochastic lq control with random coefficients and application to portfolio selection this paper is devoted to the study of a stochastic linear quadratic lq optimal control problem where the control variable is constrained in a cone and all the coefficients of the problem are random processes employing tanaka s formula optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic riccati equations esres the esres introduced for the first time in this paper are highly nonlinear backward stochastic differential equations bsdes whose solvability is proved based on a truncation function technique and kobylanski s results the general results obtained are then applied to a mean variance portfolio selection problem for a financial market with random appreciation and volatility rates and with short selling prohibited feasibility of the problem is characterized and efficient portfolios and efficient frontier are presented in closed forms

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

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 differential equations with rough drivers backward stochastic differential equations bsdes in the sense of pardoux peng lecture notes in control and inform sci 176 1992 200 217 provide a non markovian extension to certain classes of nonlinear partial differential equations the nonlinearity is expressed in the so called driver of the bsde our aim is to deal with drivers which have very little regularity in time to this end we establish continuity of bsde solutions with respect to rough path metrics in the sense of lyons rev mat iberoam 14 1998 215 310 and so obtain a notion of bsde with rough driver existence uniqueness and a version of lyons limit theorem in this context are established our main tool aside from rough path analysis is the stability theory for quadratic bsdes due to kobylanski ann probab 28 2000 558 602

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

numerical stability analysis of the euler scheme for bsdes in this paper we study the qualitative behavior of approximation schemes for backward stochastic differential equations bsdes by introducing a new notion of numerical stability for the euler scheme we provide sufficient conditions in the one dimensional and multidimensional cases to guarantee the numerical stability we then perform a classical von neumann stability analysis in the case of a linear driver f and exhibit necessary conditions to get stability in this case finally we illustrate our results with numerical applications

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

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

minimal supersolutions of convex bsdes we study the nonlinear operator of mapping the terminal value to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in y convex in z jointly lower semicontinuous and bounded below by an affine function of the control variable z we show existence uniqueness monotone convergence fatou s lemma and lower semicontinuity of this operator we provide a comparison principle for minimal supersolutions of bsdes

error estimates for approximate solutions of the riccati equation with real or complex potentials a method is presented for obtaining rigorous error estimates for approximate solutions of the riccati equation with real or complex potentials our main tool is to derive invariant region estimates for complex solutions of the riccati equation we explain the general strategy for applying these estimates and illustrate the method in typical examples where the approximate solutions are obtained by gluing together wkb and airy solutions of corresponding one dimensional schrodinger equations our method is motivated by and has applications to the analysis of linear wave equations in the geometry of a rotating black hole

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

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

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

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

bsde with quadratic growth and unbounded terminal value in this paper we study the existence of solution to bsde with quadratic growth and unbounded terminal value the main idea consists in using a localization procedure together with a priori bounds

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

exponential cluster solutions to quantum transport equations connected graph methods are used to solve the transport equation arising in a semiclassical representation of the schrodinger propagator k t s x y for a spinless quantum system whose hamiltonian is a smooth position and momentum dependent perturbation of the laplacian in rd exponential cluster expansions for k are obtained these formally exact solutions to the time dependent schrodinger equation employ explicit graphically determined derivatives of the hamiltonian s symbol integrated along geodesics for the special case of particles interacting with external electromagnetic fields the propagator s gauge invariant derivative expansion coefficients are determined in closed form in terms of the lorentz force a tree graph formula for hamilton s principal function is extracted from this result

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas

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

a representation theorem for generators of bsdes with continuous linear growth generators in the space of processes in this paper on the basis of some recent works of fan jiang and jia we establish a representation theorem in the space of processes for generators of bsdes with continuous linear growth generators which generalizes the corresponding results in fan 2006 2007 10 11 and fan and hu 2008 9

dynamic programming for general linear quadratic optimal stochastic control with random coefficients we are concerned with the linear quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable but essentially bounded processes and the terminal state weighting matrix in the cost functional is allowed to be random under suitable conditions we prove that the value field mathbb v t x omega t x omega in 0 t times r n times omega is quadratic in x and has the following form mathbb v t x langle mathbb k tx x rangle where mathbb k is an essentially bounded nonnegative symmetric matrix valued adapted processes using the dynamic programming principle dpp we prove that mathbb k is a continuous semimartingale of the form mathbb k t mathbb k 0 int 0 t dk s int 0 t sum i 1 d l s i dw s i t in 0 t with k being a continuous process of bounded variation and the stochastic integral int 0 cdot sum i 1 d l i s dw i s being a bound