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

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

asymptotic stability of linear neutral delay differential algebraic equations and runge kutta methods this paper is concerned with asymptotic stability of linear neutral delay differential algebraic equations and runge kutta methods first we give a new equivalent sufficient condition for the neutral delay differential algebraic equations to be delay independent asymptotically stable then we investigate the asymptotic stability of the numerical solutions generated by the runge kutta methods combined with lagrange interpolation some results on the asymptotic stability of runge kutta methods of high order are given finally numerical examples of index 1 and 2 are conducted to confirm our numerical stability result

optimal portfolio and consumption decisions for a small investor on a finite horizon a general consumption investment problem is considered for an agent whose actions cannot affect the market prices and who strives to maximize total expected discounted utility of both consumption

linear quadratic control of backward stochastic differential equations this paper is concerned with optimal control of linear backward stochastic differential equations bsdes with a quadratic cost criteria or backward linear quadratic blq control the solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion of squares technique two alternative though equivalent expressions for the optimal control are obtained the first of these involves a pair of riccati type equations an uncontrolled bsde and an uncontrolled forward stochastic differential equation sde while the second is in terms of a hamiltonian system contrary to the deterministic or stochastic forward case the optimal control is no longer a feedback of the current state rather it is a feedback of the entire history of the state a key step in our derivation is a proof of global solvability of the aforementioned riccati equations although of independent interest this issue has particular relevance to the blq problem since these riccati equations play a central role in our solution last but not least it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear quadratic lq problem this in turn reveals the origin of the riccati equations introduced

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

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

a bsde approach to a class of dependent risk model of mean variance insurers with stochastic volatility and no short selling abstract this paper studies the optimal reinsurance and investment strategy for an insurer with two dependent classes of insurance business where the claim number processes are correlated through a common shock it is assumed that the insurer also faces the decision making of investing in a financial market with one risk free asset and one risky asset following the heston stochastic volatility sv model the insurer is not allowed to short sell the risky asset under the mean variance criterion we consider the insurer s problem of maximizing the expected terminal wealth and at the same time minimizing the variance of the terminal wealth using the results of stochastic linear quadratic lq optimal control and backward stochastic differential equations bsdes we derive closed form expressions for the optimal strategies and the efficient frontiers in terms of solutions to the bsdes our approach shows how bsdes can be used to solve mean variance problems in insurance applications finally economic behavior of the efficient frontiers is analyzed by using some numerical examples

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

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

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

asymptotic stability of linear neutral delay differential algebraic equations and linear multistep methods this paper is concerned with delay independent asymptotic stability of linear neutral delay differential algebraic equations and linear multistep methods we first give some sufficient conditions for the delay independent asymptotic stability of these equations then we study and derive a sufficient and necessary condition for the delay independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with lagrange interpolation finally one numerical example is performed to confirm our theoretical result

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

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

a stochastic calculus model of continuous trading optimal portfolios the problem of choosing a portfolio of securities so as to maximize the expected utility of wealth at a terminal planning horizon is solved via stochastic calculus and convex analysis this problem is decomposed into two subproblems with security prices modeled as semimartingales and trading strategies modeled as predictable processes the set of terminal wealths is identified as a subspace in a space of integrable random variables the first subproblem is to find the terminal wealth that maximizes expected utility convex analysis is used to derive necessary and sufficient conditions for optimality and an existence result the second subproblem of finding the admissible trading strategy that generates the optimal terminal wealth is a martingale representation problem the primary advantage of this approach is that explicit formulas can readily be derived for the optimal terminal wealth and the corresponding expected utility as is shown for the case of an exponential utility function and a risky security modeled as geometric brownian motion

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

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

mean variance optimal portfolios in the presence of a benchmark with applications to fraud detection abstract we first study mean variance efficient portfolios when there are no trading constraints and show that optimal strategies perform poorly in bear markets we then assume that investors use a stochastic benchmark linked to the market as a reference portfolio we derive mean variance efficient portfolios when investors aim to achieve a given correlation or a given dependence structure with this benchmark we also provide upper bounds on sharpe ratios and show how these bounds can be useful for fraud detection for example it is shown that under some conditions it is not possible for investment funds to display a negative correlation with the financial market and to have a positive sharpe ratio all the results are illustrated in a black scholes market

mean variance portfolio selection under a non markovian regime switching model abstract in this paper we investigate the mean variance portfolio selection problem in a continuous time setting we assume that the coefficients in the model are random and adapted to the filtration generated by a markov chain instead of using the embedding approach which is widely adopted in the existing literature we study the problem from the viewpoint of mean field formulation and provide a distinctive and straightforward approach by introducing and discussing a new system of mean field backward stochastic differential equations driven by a markov chain we obtain both the optimal strategy and the efficient frontier in explicit forms in particular we revisit the markovian regime switching model in which the coefficients are deterministic functions of the markov chain

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

nonlinear regression in parametric activation studies parametric study designs can reveal information about the relationship between a study parameter e g word presentation rate and regional cerebral blood flow rcbf in functional imaging the brain s responses in relation to study parameters might be nonlinear therefore the linear correlation coefficient as often used in the analysis of parametric studies might not be a proper characterization we present a noninteractive method which fits nonlinear functions of stimulus or task parameters to rcbf responses using second order polynomial expansions this technique is implemented in the context of the general linear model and statistical parametric mapping we also consider the usefulness of statistical inferences based on f fields about similarities and differences of these nonlinear responses in different groups this approach is illustrated with a 12 run h 2 o pet activation study using an auditory paradigm of increasing word presentation rates a patient who had recovered from severe aphasia and a normal control were studied we demonstrate the ability of this new technique to identify brain regions where rcbf is closely related to increasing word presentation rate in both subjects without constraining the nature of this relationship and where these nonlinear responses differ r1996academic press inc

continuous time mean variance portfolio selection with random horizon in an incomplete market in this paper we consider a continuous time mean variance portfolio selection problem with random market parameters and random time horizon in an incomplete market this problem will be formulated as a linearly constrained stochastic linear quadratic lq optimal control problem the solvability of this lq problem will be reduced to the global solvability of two backward stochastic differential equations bsdes one is conventionally called a stochastic riccati equation sre and the other is referred to as an auxiliary bsde we shall apply the martingales of bounded mean oscillation briefly called bmo martingales to provide a direct and simplified proof of the solvability of the two bsdes we also derive closed form expressions for both the optimal portfolios and the efficient frontier in terms of the solutions of the two bsdes

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

activation detection in functional mri using subspace modeling and maximum likelihood estimation a statistical method for detecting activated pixels in functional mri fmri data is presented in this method the fmri time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activation baseline pattern a nuisance component representing effects of no interest and gaussian white noise for periodic activation baseline patterns the response signal is modeled by a truncated fourier series with a known fundamental frequency but unknown fourier coefficients the nuisance subspace is assumed to be unknown a maximum likelihood estimate is derived for the component of the nuisance subspace which is orthogonal to the response signal subspace an estimate for the order of the nuisance subspace is obtained from an information theoretic criterion a statistical test is derived and shown to be the uniformly most powerful ump test invariant to a group of transformations which are natural to the hypothesis testing problem the maximal invariant statistic used in this test has an f distribution the theoretical f distribution under the null hypothesis strongly concurred with the experimental frequency distribution obtained by performing null experiments in which the subjects did not perform any activation task applications of the theory to motor activation and visual stimulation fmri studies are presented

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

robust portfolio choice and indifference valuation we solve theoretically and numerically the problems of optimal portfolio choice and indifference valuation in a general continuous time setting the setting features i ambiguity and time consistent ambiguity averse preferences ii discontinuities in the asset price processes with a general and possibly infinite activity jump part next to a continuous diffusion part and iii general and possibly nonconvex trading constraints we characterize our solutions as solutions to backward stochastic differential equations bsdes generalizing kobylanski s result for quadratic bsdes to an infinite activity jump setting we prove existence and uniqueness of the solution to a general class of bsdes encompassing the solutions to our portfolio choice and valuation problems as special cases we provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the ju

mean variance portfolio selection in a complete market with unbounded random coefficients this paper concerns a mean variance portfolio selection problem in a complete market with unbounded random coefficients in particular we use adapted processes to model market coefficients and assume that only the interest rate is bounded while the appreciation rate volatility and market price of risk are unbounded under an exponential integrability assumption of the market price of risk process we first prove the uniqueness and existence of solutions to two backward stochastic differential equations with unbounded coefficients then we apply the stochastic linear quadratic control theory and the lagrangian method to solve the problem we represent the efficient portfolio and efficient frontier in terms of the unique solutions to the two backward stochastic differential equations to illustrate our results we derive explicit expressions for the efficient portfolio and efficient frontier in one example with markovian models of a bounded interest rate and an unbounded market price of risk

characterizing the response of pet and fmri data using multivariate linear models this paper presents a new method for characterizing brain responses in both pet and fmri data the aim is to capture the correlations between the scans of an experiment and a set of external predictor variables that are thought to affect the scans such as type intensity or shape of stimulus response its main feature is a canonical variates analysis cva of the estimated effects of the predictors from a multivariate linear model mlm the advantage of this over current methods is that temporal correlations can be incorporated into the model making the mlm method suitable for fmri as well as pet data moreover tests for the presence of any correlation and inference about the number of canonical variates needed to capture that correlation can be based on standard multivariate statistics rather than simulations when applied to an fmri data set previously analyzed by another cva method the mlm method reveals a pattern of responses that is closer to that detected in an earlier non cva analysis

dual method for continuous time markowitz s problems with nonlinear wealth equations continuous time mean variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method a necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique it is also shown that the optimal wealth and portfolio is the solution of a forward backward stochastic differential equation with constraints

characterizing stimulus response functions using nonlinear regressors in parametric fmri experiments parametric study designs proved very useful in characterizing the relationship between experimental parameters e g word presentation rate and regional cerebral blood flow in positron emission tomography studies in a previous paper we presented a method that fits nonlinear functions of stimulus or task parameters to hemodynamic responses using second order polynomial expansions here we expand this approach to model nonlinear relationships between bold responses and experimental parameters using fmri we present a framework that allows this technique to be implemented in the context of the general linear model employed by statistical parametric mapping spm statistical inferences in this instance are based on f statistics and in this respect we emphasize the use of corrected p values for f fields i e spm5f6 the approach is illustrated with a fmri study that looked at the effect of increasing auditory word presentation rate our parametric design allowed us to characterize different forms of rate dependent responses in three critical regions i bilateral frontal regions showed a categorical response to the presence of words irrespective of rate suggesting a role for this region in establishing cognitive e g attentional set ii in bilateral occipitotemporal regions activations increased linearly with increasing word rate and iii posterior auditory association cortex exhibited a nonlinear inverted u relationship to word rate r 1998 academic press

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

mean variance hedging when there are jumps in this paper we consider the problem of mean variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by brownian motion and doubly stochastic poisson processes this problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations the results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of poisson processes with stochastic intensities applications to the problem of hedging default risk are also discussed

on the structure of general mean variance hedging strategies we provide a new characterization of mean variance hedging strategies in a general semimartingale market the key point is the introduction of a new probability measure p which turns the dynamic asset allocation problem into a myopic one the minimal martingale measure relative to p coincides with the variance optimal martingale measure relative to the original probability measure p

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

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

approximation pricing and the variance optimal martingale measure let x be a semimartingale and let be the space of all predictable x integrable processes v such that so dx is in the space s 2 of semi martingales we consider the problem of approximating a given random variable h l 2 p by the sum of a constant c and a stochastic integral s 0 t v s dx s with respect to the l 2 p norm this problem comes from financial mathematics where the optimal constant v 0 can be interpreted as an approximation price for the contingent claim h an elementary computation yields v 0 as the expectation of h under the variance optimal signed martingale measure p and this leads us to study p in more detail in the case of finite discrete time we explicitly construct p by backward recursion and we show that p is typically not a probability but only a signed measure in a continuous time framework the situation becomes rather different we prove that p is nonnegative if x has continuous paths and satisfies a very mild no arbitrage condition as an application we show how to obtain the optimal integrand in feedback form with the help of a backward stochastic differential equation

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

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

delay dependent stability of runge kutta methods for linear delay differential algebraic equations abstract this paper is concerned with the stability analysis of numerical solutions for linear delay differential algebraic equations by means of the argument principle the delay dependent stability of runge kutta methods is investigated several numerical examples are given to illustrate the effectiveness of the proposed results

asymptotic and numerical stability of systems of neutral differential equations with many delays we are concerned with the asymptotic stability of a system of linear neutral differential equations with many delays in the form y t ly t email protected i 1dm iy email protected i email protected i 1dn iy email protected i where l m i n email protected c n x n i 1 2 d are constant complex matrices t i 0 i 1 2 d are constant delays and y t y 1 t y 2 t y n t t is an unknown vector valued function for t 0 we first establish a new result for the distribution of the roots of its characteristic function next we obtain a sufficient condition for its asymptotic stability and then we investigate the corresponding numerical stability of linear multistep methods applied to such systems one numerical example is given to testify our numerical analysis

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

stochastic hamilton jacobi bellman equations this paper studies the following form of nonlinear stochastic partial differential equation begin gathered d phi t mathop inf v in u left frac 1 2 sum i j left sigma sigma right ij x v t partial x i x j phi t x sum i b i x v t partial x i phi t x l x v t right hfill qquad qquad left sum i j sigma ij x v t partial x i psi j t x right dt psi t x dw t quad phi t x h x hfill end gathered where the coefficients sigma ij b i l and the final datum h may be random the problem is to find an adapted pair phi psi x t uniquely solving the equation the classical hamilton jacobi bellman hjb equation can be regarded as a special case of the above problem an existence and uniqueness theorem is obtained for the case where sigma does not contain the control variable v an optimal control interpretation is given the linear quadratic case is discussed as well

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

the numerical solution of delay differential algebraic equations of retarded and neutral type in this paper we consider the numerical solution of initial value delay differential algebraic equations ddaes of retarded and neutral types with a structure corresponding to that of hessenberg daes we give conditions under which the ddae is well conditioned and show how the ddae is related to an underlying retarded or neutral delay ordinary differential equation dode we present convergence results for linear multistep and runge kutta methods applied to ddaes of index 1 and 2 and show how higher index hessenberg ddaes can be formulated in as stable a way as index 2 hessenberg ddaes we also comment on some practical aspects of the numerical solution of these problems

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

mean variance portfolio selection with correlation risk the markowitz mean variance portfolio selection mvps problem is the building block of modern portfolio theory since markowitz 1952 published his seminal study there have been numerous extensions to the continuous time mvps problem under different market conditions this paper further enriches the literature by taking account of correlation risk among risky asset returns empirical studies reveal that correlations among economic variables change randomly over time and affect hedging and investment demand in different correlation regimes by incorporating correlation risk into the dynamic mvps through the wishart variance covariance matrix process this paper derives the explicit closed form solution to the optimal portfolio policy and determines the market regime in which the optimal policy is stable and well behaved this stable market regime is found to be fully characterized by the correlation between market returns and their variance covariance matrix or equivalently the effects of market leverage

mean variance portfolio selection under partial information this paper is concerned with a continuous time mean variance portfolio selection problem in a possibly incomplete market with multiple stocks and a bond only the past price movements of the stocks and the bond are the information available to the investors a separation principle is shown to hold in this setting efficient strategies based on the aforementioned partial information are derived which involve the optimal filter of the stock appreciation rate processes the main methodological contribution of the paper is to employ the particle system representation to develop analytical and numerical approaches in obtaining the filter as well as solving the related backward stochastic differential equation

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

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

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

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

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

mean variance portfolio selection with random parameters in a complete market this paper concerns the continuous time mean variance portfolio selection problem in a complete market with random interest rate appreciation rates and volatility coef cients the problem is tackled using the results of stochastic linear quadratic lq optimal control and backward stochastic differential equations bsdes two theories that have been extensively studied and developed in recent years specifically the mean variance problem is formulated as a linearly constrained stochastic lq control problem solvability of this lq problem is reduced in turn to proving global solvability of a stochastic riccati equation the proof of existence and uniqueness of this riccati equation which is a fully nonlinear and singular bsde with random coefficients is interesting in its own right and relies heavily on the structural properties of the equation efficient investment strategies as well as the mean variance efficient frontier are then analytically derived in terms of the solution of this equation in particular it is demonstrated that the efficient frontier in the mean standard deviation diagram is still a straight line or equivalently risk free investment is still possible even when the interest rate is random finally a version of the mutual fund theorem is presented

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

differential algebraic equations with after effect in this paper we are concerned with the solution of delay differential algebraic equations these are differential algebraic equations with after effect or constrained delay differential equations the general semi explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments even simply stated problems of this type can give rise to difficult analytical and numerical problems the more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems and to indicate problems that require further research

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

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

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

a kind of optimal investment problem under inflation and uncertain time horizon abstract in this paper we study a kind of optimal investment problem under inflation and uncertain time horizon it can be generally formulated into a stochastic optimal control problem in particular for the constant relative risk aversion utility we employ the method of completion of squares to give an explicit form of optimal portfolio and maximum utility by the solution of a stochastic riccati equation whose wellposedness is obtained and also of significance in its own right the most distinguishing result of our work is that the randomness of exit time actually affects not only the optimal portfolio but also the maximum utility in the case of stochastic market parameters moreover we present several numerical examples to show the application of theoretical results and further discuss the influence of inflation and random time horizon from the economic viewpoint

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

quadratic hedging and mean variance portfolio selection with random parameters in an incomplete market this paper concerns the problems ofquadratic hedging andpricing andmean variance portfolio selection in anincomplete market setting with continuous trading multiple assets and brownian information in particular we assume throughout that the parameters describing the market model may be random processes we approach these problems from the perspective oflinear quadratic lq optimal control and backward stochastic differential equations bsdes that is we focus on the so calledstochastic riccati equation sre associated with the problem excepting certain special cases solvability of the sre remains an open question our primary theoretical contribution is a proof of existence and uniqueness of solutions of the sre associated with the quadratic hedging and mean variance problems in addition we derive closed form expressions for the optimal portfolios and efficient frontier in terms of the solution of the sre a generalization of themutual fund theorem is also obtained

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

optimal control and hedging of operations in the presence of financial markets we consider the problem of dynamically hedging the profits of a corporation when these profits are correlated with returns in the financial markets in particular we consider the general problem of simultaneously optimizing over both the operating policy and the hedging strategy of the corporation we discuss how different informational assumptions give rise to different types of hedging and solution techniques finally we solve some problems commonly encountered in operations management to demonstrate the methodology

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

solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls the optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential riccati equation in this paper we provide general necessary and sufficient conditions for the solvability of this generalized differential riccati equation furthermore its asymptotic behavior is investigated along with its connection to the generalized algebraic riccati equation associated with the linear quadratic control problem in finite time horizon examples are presented to illustrate the results established

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

stability criteria for differential algebraic equations with multiple delays and their numerical solutions abstract this paper is concerned with the asymptotic stability of differential algebraic equations with multiple delays and their numerical solutions first we give a sufficient condition for delay independent stability after characterizing the coefficient matrices that satisfy this stability condition we propose some practical checkable criteria for asymptotic stability then we investigate the stability of numerical solutions obtained by methods and bdf methods finally solvability and stability of a class of weakly regular delay differential algebraic equations are analyzed

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