Showing papers in "Banach Center Publications in 2008"
TL;DR: The main results solve the problem of minimizing the expected liquidity costs within a given convex set of predictable trading strategies by reducing it to a deterministic optimization problem.
Abstract: We consider the problem of optimally placing market orders so as to minimize the expected liquidity costs from buying a given amount of shares. The liquidity price impact of market orders is described by an extension of a model for a limit order book with resilience that was proposed by Obizhaeva and Wang (2006). We extend their model by allowing for a time-dependent resilience rate, arbitrary trading times, and general equilibrium dynamics for the una!ected bid and ask prices. Our main results solve the problem of minimizing the expected liquidity costs within a given convex set of predictable trading strategies by reducing it to a deterministic optimization problem. This deterministic problem is explicitly solved for the case in which the convex set of strategies is defined via finitely many linear constraints. A detailed study of optimal portfolio liquidation in markets with opening and closing call auctions is provided as
104 citations
TL;DR: In this paper, the authors consider a model of chemorepulsion and prove global existence and uniqueness of smooth classical solutions in space dimension n = 2, and for n = 3, 4 they prove the global existence of weak solutions.
Abstract: In this paper we consider a model of chemorepulsion. We prove global existence and uniqueness of smooth classical solutions in space dimension n = 2. For n = 3, 4 we prove the global existence of weak solutions. The convergence to steady states is shown in all cases.
89 citations
TL;DR: It is proved that the basic martingale characterization result for locally risk-minimizing strategies still holds true and it is shown how payment streams can be handled and how the self-contained presentation is more streamlined.
Abstract: One of the earliest concepts for hedging and pricing in incomplete flnancial markets has been the quadratic criterion of local risk-minimization. However, deflnitions and theory have so far been established only for the case of a single (one-dimensional) risky asset. We extend the approach to a general multidi- mensional setting and prove that the basic martingale characterization result for locally risk-minimizing strategies still holds true. In comparison with exist- ing literature, the self-contained presentation is more streamlined, and a number of earlier imposed technical conditions are no longer needed. As a minor ex- tension, we show how payment streams (instead of flnal payofis only) can be handled as well.
68 citations
TL;DR: Some of the mathematical challenges arising from modelling structured populations are outlined, primarily focussing on the interplay between forwards in time models for the evolution of the population and backwards in time model for the genealogical trees relating individuals in a sample from that population.
Abstract: Understanding the evolution of individuals which live in a structured and fluctuating environment is of central importance in mathematical population genetics. Here we outline some of the mathematical challenges arising from modelling structured populations, primarily focussing on the interplay between forwards in time models for the evolution of the population and backwards in time models for the genealogical trees relating individuals in a sample from that population. In addition to classical models we describe a special case of a new model introduced in very recent work with Nick Barton. A number of directions for future research are suggested.
59 citations
TL;DR: In this paper, two versions of stochastic population models with mutation and selection are considered, one based on a multitype branching process and the other based on the Moran model with selection.
Abstract: We consider two versions of stochastic population models with mutation andselection. The first approach relies on a multitype branching process; here,individuals reproduce and change type (i.e., mutate) independently of eachother, without restriction on population size. We analyze the equilibriumbehaviour of this model, both in the forward and in the backward direction oftime; the backward point of view emerges if the ancestry of individuals chosenrandomly from the present population is traced back into the past. The second approach is the Moran model with selection. Here, the populationhas constant size N. Individuals reproduce (at rates depending on their types),the offspring inherits the parent's type, and replaces a randomly chosenindividual (to keep population size constant). Independently of thereproduction process, individuals can change type. As in the branching model,we consider the ancestral lines of single individuals chosen from theequilibrium population. We use analytical results of Fearnhead (2002) todetermine the explicit properties, and parameter dependence, of the ancestraldistribution of types, and its relationship with the stationary distribution inforward time.
32 citations
TL;DR: In this paper, the authors propose a class of discrete-time stochastic models for the pricing of inflation-linked assets, based on an axiomatic scheme for asset pricing and interest rate theory in a discrete time setting.
Abstract: We propose a class of discrete-time stochastic models for the pricing of inflation-linked assets. The paper begins with an axiomatic scheme for asset pricing and interest rate theory in a discrete-time setting. The first axiom introduces a "risk-free" asset, and the second axiom determines the intertemporal pricing relations that hold for dividend-paying assets. The nominal and real pricing kernels, in terms of which the price index can be expressed, are then modelled by introducing a Sidrauski-type utility function depending on (a) the aggregate rate of consumption, and (b) the aggregate rate of real liquidity benefit conferred by the money supply. Consumption and money supply policies are chosen such that the expected joint utility obtained over a specified time horizon is maximised subject to a budget constraint that takes into account the "value" of the liquidity benefit associated with the money supply. For any choice of the bivariate utility function, the resulting model determines a relation between the rate of consumption, the price level, and the money supply. The model also produces explicit expressions for the real and nominal pricing kernels, and hence establishes a basis for the valuation of inflation-linked securities.
28 citations
TL;DR: A survey on the theory of caustics and wave front propagations with applications to differential geometry of hypersurfaces in Euclidean space can be found in this article, where the authors give a brief review of the general theory and apply this theory to hypersurface, getting new geometric properties.
Abstract: This is mainly a survey on the theory of caustics and wave front propagations with applications to differential geometry of hypersurfaces in Euclidean space. We give a brief review of the general theory of caustics and wave front propagations, which are well-known now. We also consider a relationship between caustics and wave front propagations which might be new. Moreover, we apply this theory to differential geometry of hypersurfaces, getting new geometric properties.
28 citations
TL;DR: The affine area symmetry set (AASS) as discussed by the authors is a generalization of the euclidean symmetrization of the symmetry set of a plane curve M, which is defined as the closure of the set of centres of circles (resp. spheres) tangent at more than one point to M.
Abstract: For a smooth closed curve in R (resp. a smooth closed surface in R), M , the symmetry set is the closure of the set of centres of circles (resp. spheres) tangent at more than one point to M . The medial axis of M is the subset of the symmetry set for which the circle (resp. sphere) is maximal in the sense that its radius equals the absolute minimum distance from its centre to M . Both these constructions, which are essentially euclidean because of the use of spheres, have been the subject of extensive investigation both in the mathematical and in the computer vision literature. See for example [3, 8, 23, 24] for some different viewpoints. They also have generalizations to higher dimensions, the most obvious being when M is a hypersurface in R. Together with the focal set, the symmetry set forms the ‘full bifurcation set’ of the family of distance-squared functions on M , parametrized by the points of the ambient euclidean space. There are also a number of constructions which depend only on the affine structure of M , that is they depend not on distance and angle but on affine concepts such as parallelness, midpoint or equality of areas. The motivation for seeking and investigating such constructions lies in the wish to detect symmetry when a scene is presented only after distortion by an affine transformation, such as viewing ‘at an angle’. In this article I shall survey these affinely invariant constructions and give some details and examples of each, with special emphasis on interesting lines of investigation which have so far not been followed up. Nearly all the examples will be in R, with occasional mention of R. There are classical constructions in affine differential geometry which can be used to imitate the above metric definition of the symmetry set of a plane curve; these give rise to the ‘affine distance symmetry set’ (ADSS) which is described in §2. It is also possible to imitate a dual construction of the symmetry set as an envelope of lines and this gives rise to the ‘affine envelope symmetry set’ (AESS); see §3. Both the ADSS and the AESS have close connexions with conics, as the natural analogue of circles. A number of constructions use pairs of points for which the tangent lines (or planes) to M are parallel: the ‘centre symmetry set’ (CSS) measures the extent to which M is centrally symmetric, and the singular points of the ‘affine equidistants’ sweep out the CSS in much the same way that the focal set of a curve or surface M is swept out by the singularities of the euclidean parallels, or offsets, of M . An entirely different approach is given by an area-based symmetry set, the affine area symmetry set (AASS), which was introduced for plane curves in [19] as a construction which is robust to small perturbations of the boundary shape: an objection to constructions which depend on the calculation of higher derivatives of a curve or surface is that these quantities are highly sensitive to noise or to perturbations of M . The AASS, like the euclidean symmetry set or the ADSS, is defined for a plane curve M , at least for a strictly convex curve (see §4) by means of a family of functions on M parametrized by an open set of points of the plane. The bifurcation set of this family consists of the AASS together with the midpoints of chords joining parallel tangent pairs, which is one of the affine equidistants mentioned above. This set also has interesting connexions with the boundary of the ‘tricentre set’, that is the boundary of the set of points inside M which are the midpoints of three chords. See §4.
24 citations
TL;DR: In this article, the authors investigated the properties of a rating migration process assuming that it is given by subordination of a discrete time Markov chain and a Cox process, and gave an explicit solution to the pricing problem in a model with short rate and intensity processes given by the solution of a two-dimensional Ornstein-Uhlenbeck equation with a Lévy noise.
Abstract: We investigate the properties of a rating migration process assuming that it is given by subordination of a discrete time Markov chain and a Cox process. The problem of pricing of defaultable bonds with fractional recovery of par value with rating migration and credit default swaps is considered. As an example of applications of our results, we give an explicit solution to the pricing problem in a model with short rate and intensity processes given by the solution of a two-dimensional Ornstein-Uhlenbeck equation with a Lévy noise.
23 citations
TL;DR: The existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane R are studied for the (supercritical) value of mass greater than 8π in this paper.
Abstract: The existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane R are studied for the (supercritical) value of mass greater than 8π.
23 citations
TL;DR: It is shown that any stochastic volatility model is then completed with an arbitrary European type option and that adding path-dependent options such as a variance swap to the set of primary assets, instead of plain vanilla options, also completes the market.
Abstract: Mathematical models for nancial asset prices which include, for example, stochastic volatility or jumps are incomplete in that derivative securities are generally not replicable by trading in the underlying. In earlier work (Proc. R. Soc. London, 2004), the rst author provided a geometric condition under which trading in the underlying and a nite number of vanilla options completes the market. We complement this result in several ways. First, we show that the geometric condition is not necessary and a weaker, necessary and sucient, condition is presented. While this condition is generally not directly veriable, we show that it simplies to matrix non-degeneracy in a single point when the pricing functions are real analytic functions. In particular, any stochastic volatility model is then completed with an arbitrary European type option. Further, we show that adding path-dependent options such as a variance swap to the set of primary assets, instead of plain vanilla options, also completes the market.
TL;DR: In this article, the application of the moving plane method to different questions concerning stationary accumulations of isentropic gases is discussed, and a short proof of the existence of energy-minimizing gas balls is given.
Abstract: Most of the paper deals with the application of the moving plane method to different questions concerning stationary accumulations of isentropic gases. The first part compares the concepts of stationarity arising from the points of view of dynamics and the calculus of variations. Then certain stationary solutions are shown to be unstable. Finally, using the moving plane method, a short proof of the existence of energy-minimizing gas balls is given.
TL;DR: In this article, a continuous time stochastic model of optimal allocation for a defined contribution pension fund with minimum guarantee is proposed and studied, and the authors show that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation.
Abstract: In the paper [6] the authors propose and study a continuous time stochastic model of optimal allocation for a defined contribution pension fund with minimum guarantee. Their target is to maximize the expected utility from current wealth over an infinite horizon, whereas usually portfolio selection models for pension funds maximize the expected utility from final wealth over a finite horizon (the retirement time). In this model the dynamics of wealth takes directly into account the flows of contributions and benefits; moreover the level of wealth is constrained to stay above a solvency level. The fund manager can invest in a riskless asset and in a risky asset but borrowing and short selling are prohibited. The model is formulated as an optimal stochastic control problem with constraint and is treated by the dynamic programming approach, showing that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation. Then they apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties, giving finally a special example with explicit solution. Nevertheless the aim of the authors is to study the problem starting from the time in which the first retirements of contributors occur (when the contribution flow becomes constant and the state equation homogeneous on time), leaving out the accumulation phase and the optimization problem in the first period. In our paper we describe the model and the problem in the accumulation phase, when the state equation is time-dependent. We will show that the value function is continuous and that it solves the associated Hamilton-Jacobi-Bellman equation in a viscosity sense. In the special case when the boundary is absorbing (therefore the value function is explicitally computable on this boundary and so a Dirichlet type condition is available for the boundary differential problem), we will show that it is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
TL;DR: In this paper, the authors considered the single layer potential associated to the fundamental solution of the time-dependent Oseen system and showed that this potential belongs to L(0,∞, H(Ω)) and to H(∂Ω × (0, ∞)).
Abstract: We consider the single layer potential associated to the fundamental solution of the time-dependent Oseen system. It is shown this potential belongs to L(0,∞, H(Ω)) and to H(0,∞, V ′) if the layer function is in L(∂Ω × (0,∞)). (Ω denotes the complement of a bounded Lipschitz set; V denotes the set of smooth solenoidal functions in H 0 (Ω) .) This result means that the usual weak solution of the time-dependent Oseen function with zero initial data and zero body force may be represented by a single layer potential, provided a certain integral equation involving the boundary data may be solved.
TL;DR: In this article, the authors considered a free interface problem for the Navier-Stokes equations and proved the existence and uniqueness of solutions for sufficiently small initial data for any initial data and external forces.
Abstract: We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time Lp-Lq maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.
TL;DR: In this article, the authors discuss similarities and differences between systems of interacting players max-imizing their individual payoffs and particles minimizing their interaction energy and construct an example of a spatial game with three strategies, where stochastic stability of Nash equilibria depends on the number of players and the kind of dynamics.
Abstract: We discuss similarities and differences between systems of interacting players max- imizing their individual payoffs and particles minimizing their interaction energy. Long-run behavior of stochastic dynamics of spatial games with multiple Nash equilibria is analyzed. In particular, we construct an example of a spatial game with three strategies, where stochastic stability of Nash equilibria depends on the number of players and the kind of dynamics.
TL;DR: In this paper, the authors survey a systematic approach starting from a stochastic process discrete both in time and state, leading to a replicator-type equation in zero order, and to a Fokker-Planck-like equation in first order in O(1/n) time.
Abstract: Finite-size fluctuations in coevolutionary dynamics arise in models of biological as well as of social and economic systems. This brief tutorial review surveys a systematic approach starting from a stochastic process discrete both in time and state. The limit $N\to \infty$ of an infinite population can be considered explicitly, generally leading to a replicator-type equation in zero order, and to a Fokker-Planck-type equation in first order in $1/\sqrt{N}$. Consequences and relations to some previous approaches are outlined.
TL;DR: In this article, the case of a dimeric lattice gas, the transport in the presence of pointwise disorder and along coupled tracks has been analyzed and the qualitative and quantitative nonequilibrium properties of these model systems have been discussed.
Abstract: Nonequilibrium collective motion is ubiquitous in nature and often results in a rich collection of intriguing phenomena, such as the formation of shocks or patterns, subdiffusive kinetics, traffic jams, and nonequilibrium phase transitions. These stochastic many-body features characterize transport processes in biology, soft condensed matter and, possibly, also in nanoscience. Inspired by these applications, a wide class of lattice-gas models has recently been considered. Building on the celebrated totally asymmetric simple exclusion process (TASEP) and a generalization accounting for the exchanges with a reservoir, we discuss the qualitative and quantitative nonequilibrium properties of these model systems. We specifically analyze the case of a dimeric lattice gas, the transport in the presence of pointwise disorder and along coupled tracks.
TL;DR: In this article, the authors investigated the influence of individuals' mobility on the spatial structures emerging in rock-paper-scissors games and devised a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution.
Abstract: The formation of out-of-equilibrium patterns is a characteristic feature of spatially-extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the ‘rock-paper-scissors’ game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals' mobility on the spatial structures emerging in rock-paper-scissors games. We have devised a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior - in the proper continuum limit - is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals' wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system. [Nature 448, 1046-1049 (2007)]View Large Image | View Hi-Res Image | Download PowerPoint Slide
TL;DR: In this article, the authors studied the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation ∂tu−∆u = u.
Abstract: We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation ∂tu−∆u = u . The approach is based on suitable iterative fixed point methods in L based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ R for some conservative nonlinear terms with symmetries.
TL;DR: In this paper, the authors consider a two-dimensional Navier-Stokes shear flow with time dependent boundary driving and subject to Tresca law and prove the existence of the pullback attractor for the associated cocycle.
Abstract: We consider a two-dimensional Navier-Stokes shear flow with time dependent boundary driving and subject to Tresca law. We establish the existence of a unique global in time solution and then, using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set, we prove the existence of the pullback attractor for the associated cocycle. This research is motivated by a problem from lubrication theory.
TL;DR: In this paper, a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented, where variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model are employed.
Abstract: Parameter sensitivities of prices for derivative contracts play an important role in model calibration as well as in quantification of model risk. In this paper a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented. Variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model differentiated with respect to the model parameters are employed. Superconvergent approximations to second and higher derivatives of prices with respect to the price process’ state variables are extracted from approximate, computed prices with low, C regularity by postprocessing. The extracted numerical sensitivities are proved to converge with optimal rates as the mesh width tends to zero. Numerical experiments for uniand multivariate models with sparse tensor product discretization confirm the theoretical results.
TL;DR: In this paper, the authors consider the nonstationary Navier-Stokes equations completed by the equation of conservation of internal energy and prove the existence of a weak solution such that: 1) the weak form of the conservation of the internal energy involves a defect measure, and 2) the equality for the total energy is satisfied.
Abstract: Abstract. We consider the non-stationary Navier-Stokes equations completed by the equation of conservation of internal energy. The viscosity of the fluid is assumed to depend on the temperature, and the dissipation term is the only heat source in the conservation of internal energy. For the system of PDE’s under consideration, we prove the existence of a weak solution such that: 1) the weak form of the conservation of internal energy involves a defect measure, and 2) the equality for the total energy is satisfied.
TL;DR: In this article, a boundary value problem for Navier-Stokes equations with prescribed u·n, curlu·n and alternatively (∂u/∂n)·n or curlu n on the boundary is formulated.
Abstract: We formulate a boundary value problem for the Navier–Stokes equations with prescribed u·n, curlu·n and alternatively (∂u/∂n)·n or curlu·n on the boundary. We deal with the question of existence of a steady weak solution. AMS Subject Classification: Primary: 35 Q 30; secondary: 76 D 05