continuous dependence estimates for viscosity solutions of integro pdes we present a general framework for deriving continuous depen dence estimates for possibly polynomially growing viscosity solutions of fully nonlinear degenerate parabolic integro pdes we use this framework to pro vide explicit estimates for the continuous dependence on the coecients and the levy measure in the bellman isaacs integro pdes arising in stochas tic control dierenti al games moreover these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems finally we illustrate our results on some integro pdes arising when attempting to price european american options in an incomplete stock mar ket driven by a geometric levy process many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure pde methods

a priori estimates of smoothness of solutions to difference bellman equations with linear and quasi linear operators a priori estimates for finite difference approximations for the first and second order derivatives are obtained for solutions of parabolic equations described in the title

identifying the odds ratio estimated by a two stage instrumental variable analysis with a logistic regression model an adjustment for an uncorrelated covariate in a logistic regression changes the true value of an odds ratio for a unit increase in a risk factor even when there is no variation due to covariates the odds ratio for a unit increase in a risk factor also depends on the distribution of the risk factor we can use an instrumental variable to consistently estimate a causal effect in the presence of arbitrary confounding with a logistic outcome model we show that the simple ratio or two stage instrumental variable estimate is consistent for the odds ratio of an increase in the population distribution of the risk factor equal to the change due to a unit increase in the instrument divided by the average change in the risk factor due to the increase in the instrument this odds ratio is conditional within the strata of the instrumental variable but marginal across all other covariates and is averaged across the population distribution of the risk factor where the proportion of variance in the risk factor explained by the instrument is small this is similar to the odds ratio from a rct without adjustment for any covariates where the intervention corresponds to the effect of a change in the population distribution of the risk factor this implies that the ratio or two stage instrumental variable method is not biased as has been suggested but estimates a different quantity to the conditional odds ratio from an adjusted multiple regression a quantity that has arguably more relevance to an epidemiologist or a policy maker especially in the context of mendelian randomization

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

on the lp formulation in measure spaces of optimal control problems for jump diffusions abstract in this short note we formulate a infinite horizon stochastic optimal control problem for jump diffusions of ito levy type as a lp problem in a measure space and prove that the optimal value functions of both problems coincide the main tools are the dual formulation of the lp primal problem which is strongly connected to the notion of sub solution of the partial integro differential equation of hamilton jacobi bellman type associated with the optimal control problem and the krylov regularization method for viscosity solutions

difference quadrature schemes for nonlinear degenerate parabolic integro pde we derive and analyze monotone difference quadrature schemes for bellman equations of controlled levy jump diffusion processes these equations are fully nonlinear degenerate parabolic integro pdes interpreted in the sense of viscosity solutions we propose new direct discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular levy measures furthermore we develop a new general theory for deriving error estimates for approximate solutions of integro pdes which thereafter is applied to the proposed difference quadrature schemes

duality based a posteriori error estimates for some approximation schemes for optimal investment problems abstract we consider a markov chain approximation scheme for utility maximization problems in continuous time which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments the error estimates previously derived in picarelli and reisinger 2018 are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only in the present paper we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds the theoretical results are confirmed by our numerical tests

asymptotic perron s method and simple markov strategies in stochastic games and control we introduce a modification of perron s method where semisolutions are considered in a carefully defined asymptotic sense with this definition we can show in a rather elementary way that in a zero sum game or a control problem with or without model uncertainty the value function over all strategies coincides with the value function over markov strategies discretized in time therefore there are always discretized markov varepsilon optimal strategies uniform with respect to the bounded initial condition with a minor modification the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies

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

numerical methods for controlled regime switching diffusions and regime switching jump diffusions this work is concerned with numerical methods for controlled regime switching diffusions and regime switching jump diffusions numerical procedures based on markov chain approximation techniques are developed convergence of the algorithms is derived by means of weak convergence methods in addition examples are also provided for demonstration purpose

an error estimate for a new scheme for mean curvature motion in this work we propose a new numerical scheme for the anisotropic mean curvature equation the solution of the scheme is not unique but for all numerical solutions we provide an error estimate between the continuous solution and the numerical approximation our scheme is also applicable to computing the solution to dislocation dynamics equations we also provide some numerical simulations

mean value theorems for stochastic integrals the distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piece wise constant processes the rate of approximation in some negative sobolev spaces is estimated generalizations are given for problems arising in control theory

discretisation of stochastic control problems for continuous time dynamics with delay as a main step in the numerical solution of control problems in continuous time the controlled process is approximated by sequences of controlled markov chains thus discretising time and space a new feature in this context is to allow for delay in the dynamics the existence of an optimal strategy with respect to the cost functional can be guaranteed in the class of relaxed controls weak convergence of the approximating extended markov chains to the original process together with convergence of the associated optimal strategies is established

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

mayer and optimal stopping stochastic control problems with discontinuous cost we study two classes of stochastic control problems with semicontinuous cost the mayer problem and optimal stopping for controlled diffusions the value functions are introduced via linear optimization problems on appropriate sets of probability measures these sets of constraints are described deterministically with respect to the coefficient functions both the lower and upper semicontinuous cases are considered the value function is shown to be a generalized viscosity solution of the associated hjb system respectively of some variational inequality dual formulations are given as well as the relations between the primal and dual value functions under classical convexity assumptions we prove the equivalence between the linearized mayer problem and the standard weak control formulation counter examples are given for the general framework

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

on time inhomogeneous controlled diffusion processes in domains time inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered bellman s principle and its applications to proving the continuity of value functions are investigated

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

collapsibility of relative risk in contingency tables with a response variable we consider the collapsibility of relative risks in contingency tables with a polytomous response variable simple collapsibility of relative risks means that they are consistent over all levels of a collapsed variable and also coincide with the corresponding marginal relative risk strong collapsibility of relative risks means that they retain simple collapsibility no matter how the variable is partially collapsed we present necessary and sufficient conditions for simple and strong collapsibilities of relative risks

characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems abstract the aim of this paper is to study singularly perturbed control systems firstly we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics secondly we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system

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

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

viscosity solutions of fully nonlinear parabolic path dependent pdes part ii the main objective of this paper and the accompanying one viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint is to provide a notion of viscosity solutions for fully nonlinear parabolic path dependent pdes our definition extends our previous work ann probab 2014 42 204 236 focused on the semilinear case and is crucially based on the nonlinear optimal stopping problem analyzed in stochastic process appl 2014 124 3277 3311 we prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions and satisfies a stability property and a partial comparison result the latter is a key step for the well posedness results established in viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint we also show that the value processes of path dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equations

hedging of covered options with linear market impact and gamma constraint within a financial model with linear price impact we study the problem of hedging a covered european option under gamma constraint using stochastic target and partial differential equation smoothing techniques we prove that the super replication price is the viscosity solution of a fully non linear parabolic equation as a by product we show how e optimal strategies can be constructed finally a numerical resolution scheme is proposed

stochastic target games and dynamic programming via regularized viscosity solutions we study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the opponent our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non linear partial differential equation because abstract measurable selection arguments cannot be used in this context the main obstacle is the construction of measurable almost optimal strategies we propose a novel approach where smooth supersolutions are used to define almost optimal strategies of markovian type similarly as in verification arguments for classical solutions of hamilton jacobi bellman equations the smooth supersolutions are constructed by an extension of krylov s method of shaken coefficients we apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

discontinuous control problems with state constraints linear formulations and dynamic programming principles abstract this paper aims at studying a class of discontinuous deterministic control problems under state constraints using a linear programming approach as for classical control problems gaitsgory and quincampoix 2009 16 goreac and serea 2011 19 the primal linear problem is stated on some appropriate space of probability measures naturally the support of these measures is contained in the set of constraints this linearized value function and its dual can alternatively be seen as the limit of standard penalized problems second we provide a semigroup property for this set of probability measures leading to dynamic programming principles for control problems under state constraints an abstract principle is provided for general bounded costs linearized versions are obtained under further semi continuity assumptions

asymptotic control for a class of piecewise deterministic markov processes associated to temperate viruses we aim at characterizing the asymptotic behavior of value functions in the control of piecewise deterministic markov processes pdmp of switch type under nonexpansive assumptions for a particular class of processes inspired by temperate viruses we show that uniform limits of discounted problems as the discount decreases to zero and time averaged problems as the time horizon increases to infinity exist and coincide the arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics the approach strongly relies on viscosity techniques linear programming arguments and coupling via random measures associated to pdmp as an intermediate step in our approach we present the approximation of discounted value functions when using piecewise constant in time open loop policies

fast semi lagrangian schemes for the eikonal equation and applications we introduce and analyze a fast version of the semi lagrangian algorithm for front propagation originally proposed in m falcone the minimum time problem and its applications to front propagation in motion by mean curvature and related topics a visintin and g buttazzo eds de gruyter berlin 1994 pp 70 88 the new algorithm is obtained using the local definition of the approximate solution typical of semi lagrangian schemes and redefining the set of neighboring nodes necessary for fast marching schemes a new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences we show that the new algorithm converges to the viscosity solution of the problem and that its complexity is o n log n nb as it is for the fast marching method based on finite difference n and n nb being respectively the total number of nodes and the number of nodes in the narrow band a new sufficient condition for the convergence of the standard finite difference fast marching method is also given we present several tests comparing the two algorithms and other fast methods e g fast sweeping on a series of benchmarks which include the minimum time problem and the shape from shading problem

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

some regularity and convergence results for parabolic hamilton jacobi bellman equations in bounded domains abstract we study parabolic hamilton jacobi bellman hjb equations in bounded domains with strong dirichlet boundary conditions we work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well posedness and regularity of a related switching system and the convergence of its components to the hjb equation in particular we show existence of a viscosity solution to the switching system by a novel construction of sub and supersolutions and application of perron s method error bounds for monotone schemes for the hjb equation are then derived from estimates near the boundary where the standard regularisation procedure for viscosity solutions is not applicable and are found to be of the same order as known results for the whole space we deduce error bounds for some common finite difference and truncated semi lagrangian schemes

on power integral bases of unramified cyclic extensions of prime degree let p be a prime number and k a number field containing a primitive p th root of unity it is known that an unramified cyclic extension l k of degree p has a power integral basis if it has a normal integral basis we show that for all p the converse is not true in general

a finite element like scheme for integro partial differential hamilton jacobi bellman equations we construct a finite element like scheme for fully nonlinear integro partial differential equations arising in optimal control of jump processes special cases of these equations include optimal portfolio and option pricing equations in finance the schemes are monotone and robust we prove that they converge in very general situations including degenerate equations multiple dimensions relatively low regularity of the data and for most if not all types of jump models used in finance in all cases we provide probably optimal error bounds these bounds apply when grids are unstructured and integral terms are very singular two features that are new or highly unusual in this setting

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

probabilistic error analysis for some approximation schemes to optimal control problems abstract we introduce a class of numerical schemes for optimal stochastic control problems based on a novel markov chain approximation which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments we provide lower error bounds of order arbitrarily close to 1 2 in time and 1 3 in space for lipschitz viscosity solutions coupling probabilistic arguments with regularization techniques as introduced by krylov the corresponding order of the upper bounds is 1 4 in time and 1 5 in space for sufficiently regular solutions the order is 1 in both time and space for both bounds finally we propose techniques for further improving the accuracy of the individual components of the approximation

on the convergence of finite element methods for hamilton jacobi bellman equations we study the convergence of monotone p1 finite element methods on unstructured meshes for fully nonlinear hamilton jacobi bellman equations arising from stochastic optimal control problems with possibly degenerate isotropic diffusions using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by barles and souganidis we obtain strong uniform convergence of the numerical solutions and under nondegeneracy assumptions strong l 2 convergence of the gradients

rate of convergence of finite difference approximations for degenerate ordinary differential equations in this paper we study finite difference approximations for the following linear stationary convection diffusion equations 1 2 2 x u x b x u x u x f x x r where is allowed to be degenerate we first propose a new weighted finite difference scheme motivated by approximating the diffusion process associated with the equation in the strong sense we show that under certain conditions this scheme converges with the first order rate and that such a rate is sharp to the best of our knowledge this is the first sharp result in the literature moreover by using the connection between our scheme and the standard upwind finite difference scheme we get the rate of convergence of the latter which is also new