blind image deconvolution through bezoutians in this paper we introduce a fast algorithm for computing the univariate gcd of several polynomials not pairwise based on the generalized bezout matrix by using barnett s method this novel approach is devoted to presenting an algorithm that permits to solve the problem of blind image deconvolution by computing greatest common divisors gcd of several polynomials specifically we design a specialized algorithm for computing the gcd of bivariate polynomials of blurred images which correspond to z transforms to recover the original image all algorithms have been implemented in matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach

follmer schweizer decomposition and mean variance hedging for general claims let x be an r d valued special semimartingale on a probability space f f t 0 t t p with decomposition x x 0 m a and the space of all predictable x integrable processes such that dx is in the space s 2 of semimartingales if h is a random variable in l 2 we prove under additional assumptions on the process x that h can be written as the sum of an f 0 measurable random variable h 0 a stochastic integral of x and a martingale part orthogonal to m moreover this decomposition is unique and the function mapping h with its decomposition is continuous with respect to the l 2 norm finally we deduce from this continuity that the subspace of l 2 generated by dx where is closed in l 2 and we give some applications of this result to financial mathematics

martingale measures and hedging for discrete time financial markets the price of stocks is modelled by a discrete time square integrable vector valued process x no further boundedness condition on x is imposed contingent claims h are described by square integrable random variables one looks for values v of the initial wealth v that allow for super hedging h by some portfolio plan in several cases the smallest value v is known to coincide with the maximal expectation of h under equivalent martingale measures here within an l2 framework another sufficient condition is provided which can be looked upon as a stronger form of the no arbitrage condition the mathematical tool and one of the main contributions is an optional decomposition theorem for a process which is a supermartingale under any equivalent martingale measure the upper price process for a contingent claim is shown to be a typical example for such a process moreover it is shown that in a markovian model one can restrict attention to markovian portfolio plans and to markovian martingale measures

markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

blind image deconvolution using a robust gcd approach in this correspondence a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions in z domain the desired image can be regarded as the greatest common polynomial divisor among the distorted versions with the assumption that the distortion filters are finite impulse response fir and relatively coprime in the absence of noise this becomes a problem of taking the greatest common divisor gcd of two or more two dimensional 2 d polynomials exact gcd is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2 d gcd method based on a one dimensional 1 d sylvester type gcd algorithm experimental results with both synthetically blurred images and real motion blurred pictures show that it is computationally efficient and moderately noise robust

variance optimal hedging in discrete time we solve the problem of approximating in 2 a given random variable h by stochastic integrals gt of a given discrete time process x we interpret h as a contingent claim to be paid out at time t x as the price evolution of some risky asset in a financial market and g as the cumulative gains from trade using the hedging strategy as an application we determine the variance optimal strategy which minimizes the variance of the net loss h gt over all strategies

utility based indifference pricing in regime switching models article deposited according to the policy found on the elsevier website http www elsevier com wps find authorsview authors postingpolicy june 28 2012

a semimartingale backward equation and the variance optimal martingale measure under general information flow we consider a financial market model where the dynamics of asset prices are given by an rd valued continuous semimartingale and the information flow is right continuous using the dynamic programming approach we express the variance optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation we consider two extreme cases when this equation admits an explicit solution in particular we give necessary and sufficient conditions in order that the variance optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

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

approximating random variables by stochastic integrals let x be a semimartingale and the space of all predictable x integrable processes such that dx is in the space 2 of semimartingales we consider the problem of approximating a given random variable h l 2 by a stochastic integral 0 t s dx s with respect to the l 2 norm if x is special and has the form x x 0 m d we construct a solution in feedback form under the assumptions that 2 d is deterministic and that h admits a strong f s decomposition into a constant a stochastic integral of x and a martingale part orthogonal to m we provide sufficient conditions for the existence of such a decomposition and we give several applications to quadratic optimization problems arising in financial mathematics

on the minimal entropy martingale measure let x be a locally bounded semimartingale using the theory of bmo martingales we give a sufficient criterion for a martingale measure for x to minimize relative entropy among all martingale measures this is applied to prove convergence of the q optimal martingale measure to the minimal entropy martingale measure in entropy for q 1 under the assumption that x is continuous and that the density process of some equivalent martingale measure satisfies a reverse llogl inequality

dynamic programming and pricing of contingent claims in an incomplete market the problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market when the market is incomplete prices cannot be derived from the absence of arbitrage since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities in this situation there is a price range for the actual market price of the contingent claim the maximum and minimum prices are studied using stochastic control methods the main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities a similar result is obtained for the minimum price which corresponds to the purchase price

minimal martingale measure pricing and hedging in a pure jump model under restricted information abstract financial asset price movements are considered in a model with partial information the risky assets price is modeled as a marked point process the underlying event arrival process which is again a marked point process is assumed to be unobserved by the market agents taking into account the presence of catastrophic events the possibility of common jump times between the risky assets price process and the arrival process is allowed the problem of characterizing the equivalent martingale measures is investigated and in particular the minimal martingale measure is studied the hedging and the pricing problem of a european contingent claim are discussed the arbitrage free price is identified with the conditional expectation with respect to the observations under the minimal martingale measure

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

equilibrium pricing in incomplete markets under translation invariant preferences we propose a general discrete time framework for deriving equilibrium prices of financial securities it allows for heterogeneous agents unspanned random endowments and convex trading constraints we give a dual characterization of equilibria and provide general results on their existence and uniqueness in the special case where all agents have preferences of the same type and in equilibrium all random endowments are replicable by trading in the financial market we show that a one fund theorem holds and give an explicit expression for the equilibrium pricing kernel

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

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

polynomial computations for blind image deconvolution this paper considers the problem of blind image deconvolution bid when the blur arises from a spatially invariant point spread function psf h which implies that a blurred image g is formed by the convolution of h and the exact form f of g since the multiplication of two bivariate polynomials is performed by convolving their coefficient matrices the equivalence of the formation of a blurred image and the product of two bivariate polynomials implies that bid can be performed by considering f g and h to be bivariate polynomials on which polynomial operations are performed these operations allow the psf to be computed which is then deconvolved from the blurred image g thereby obtaining a deblurred image that is a good approximation of the exact image f computational results show that the deblurred image obtained using polynomial computations is better than the deblurred image obtained using other methods for blind image deconvolution

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

on quadratic cost criteria for option hedging consider an option with maturity time t corresponding to a contingent claim h in an incomplete market a fair hedging price for h should take into account an optimal dynamical hedging plan against h let ct be the cumulative cost and t be the set of events of the history up to time t you can choose the plan at time t such that you minimize i e ct 1 ct 2 t ii e ct ct 2 t or iii e ct c0 2 sufficient conditions on the underlying stochastic process in discrete time are provided such that the fair hedging price does not depend on the choice of i ii or iii which fact should increase its acceptability

on l 2 projections on a space of stochastic integrals let x be an ir d valued continuous semimartingale t a fixed time horizon and the space of all ir d valued predictable x integrable processes such that the stochastic integral g d xis a square integrable semimartingale a re cent paper of delbaen monat schachermayer schweizer stricker 1996 gives necessary and sufficient conditions on x for gt to be closed in l 2 p in this paper we describe the structure of the l 2 projection mapping an ft measurable random variable h l 2 p o ngt and provide the resulting integrand h in feedback form this is related to variance optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on x our proofs use the variance optimal martin gale measure p for x and weighted norm inequalities relating p to the original measure p

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

recursive robust pca or recursive sparse recovery in large but structured noise this paper studies the recursive robust principal components analysis problem if the outlier is the signal of interest this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors st in the presence of large but structured noise lt the structure that we assume on lt is that lt is dense and lies in a low dimensional subspace that is either fixed or changes slowly enough a key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background lt from moving foreground objects st on the fly to solve the above problem in recent work we introduced a novel solution called recursive projected cs reprocs in this paper we develop a simple modification of the original reprocs idea and analyze it this modification assumes knowledge of a subspace change model on the lt s under mild assumptions and a denseness assumption on the unestimated part of the subspace of lt at various times we show that with high probability the proposed approach can exactly recover the support set of st at all times and the reconstruction errors of both st and lt are upper bounded by a time invariant and small value in simulation experiments we observe that the last assumption holds as long as there is some support change of st every few frames

computing greatest common divisors and squarefree decompositions through matrix methods the parametric and approximate cases abstract barnett s method through bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way two different uses of this method in computer algebra are introduced here firstly we describe an algorithm for parameterizing the greatest common divisor of several polynomials in k x y being x a parameter taking values in an real field k secondly we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following corless et al r m corless p m gianni b m trager s watt the singular value decomposition for polynomial systems in acm international symposium on symbolic and algebraic computation 1995 pp 195 207 given a family of polynomials whose coefficients are imperfectly known we describe an algorithm for computing their approximate greatest common divisor by using as main tools barnett s method and singular value decomposition computations furthermore we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients

option pricing with regime switching by trinomial tree method we present a fast and simple tree model to price simple and exotic options in markov regime switching model mrsm with multi regime we modify the trinomial tree model of boyle 1986 12 by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure in mrsm the market might not be complete therefore we provide some ideas and discussions on managing the regime switching risk in support of our results

fast parallel computation of the polynomial remainder sequence via bezout and hankel matrices if u x and v x are polynomials of degree n and m respectively m n all the coefficients of the polynomials generated by the euclidean scheme applied to u x and v x can be computed by using o log 3 n parallel arithmetic steps and n 2 log n processors over any field of characteristic 0 supporting fft fast fourier transform if the field does not support fft the number of processors is increased by a factor of log log n if the field does not allow division by n the number of processors is increased by a factor of n this result is obtained by reducing the euclidean scheme to computing the block triangular factorization of the bezout matrix associated with u x and v x this approach is also extended to the evaluation of polynomial gcd greatest common divisor over any field of constants in o log 2 n steps with the same number of processors

mean variance hedging and stochastic control beyond the brownian setting we show for continuous semimartingales in a general filtration how the mean variance hedging problem can be treated as a linear quadratic stochastic control problem the adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process and we give necessary and sufficient conditions for the solvability of these generalized stochastic riccati equations motivated from mathematical finance this paper takes a first step toward linear quadratic stochastic control in more general than brownian settings

intrinsic grassmann averages for online linear robust and nonlinear subspace learning principal component analysis pca and kernel principal component analysis kpca are fundamental methods in machine learning for dimensionality reduction the former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing kernel hilbert space rkhs in this paper we present a geometric framework for computing the principal linear subspaces in both situations as well as for the robust pca case that amounts to computing the intrinsic average on the space of all subspaces the grassmann manifold points on this manifold are defined as the subspaces spanned by k tuples of observations the intrinsic grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a gaussian distribution we show similar results in the rkhs case and provide an efficient algorithm for computing the projection onto the this average subspace the result is a method akin to kpca which is substantially faster further we present a novel online version of the kpca using our geometric framework competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets