We will study the following *** X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of *** that,at each fixedtime t≤T,the information available...
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We will study the following *** X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of *** that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before *** at time T,the random value of Y(ω) will become known to this *** question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time *** ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σ containing an unknown parameter θ=θ_*** then consider theso called super evaluation when the agent is a seller of the asset *** will prove that such super evaluation is afiltration consistent nonlinear *** some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a *** also consider the correspondingnonlinear Markovian situation.
We will study the following *** X,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of *** that,at each fixedtime t≤T,the information available t...
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We will study the following *** X,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of *** that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before *** at time T,the random value of Y(ω) will become known to this *** question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε[Y] to define the value of Y given by this agent at time *** ε[·] assigns an (X)0(?)s(?)T-dependent random variable Y to an (X)0(?)s(?)t-dependent random variableε[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ.We then consider theso called super evaluation when the agent is a seller of the asset *** will prove that such super evaluation is afiltration consistent nonlinear *** some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a *** also consider the correspondingnonlinear Markovian situation.
The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect...
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The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect to the Lebesgue measure on a part of the state space. Existence of a value function and a “martingale dynamic programming principle” are established by mapping the problem to a discrete time control problem. Implications for existence of optimal controls are derived.
This article presents an approach to the shape from shading problem which is based upon the notion of viscosity solutions to the shading partial differential equation, in effect a Hamilton-Jacobi equation. The power o...
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(纸本)3540437444
This article presents an approach to the shape from shading problem which is based upon the notion of viscosity solutions to the shading partial differential equation, in effect a Hamilton-Jacobi equation. The power of this approach is twofolds: 1) it allows nonsmooth, i.e. nondifferentiable, solutions which allows to recover objects with sharp troughs and creases and 2) it provides a framework for deriving a numerical scheme for computing approximations on a discrete grid of these solutions as well as for proving its correctness, i.e. the convergence of these approximations to the solution when the grid size vanishes. Our work extends previous work in the area in three aspects. First, it deals with the case of a general illumination in a simpler and a more general way (since they assume that the solutions are continuously differentiable) than in the work of Dupuis and Oliensis [9]. Second, it allows us to prove the existence and uniqueness of "continuous" solutions to the shading equation in a more general setting (general direction of illumination) than in the work of Rouy and Tourin [24], thereby extending the applicability of shape from shading methods to more realistic scenes. Third, it allows us to produce an approximation scheme for computing approximations of the "continuous" solution on a discrete grid as well as a proof of their convergence toward that solution.
We obtain the uniqueness of lower semicontinuous (LSC) viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a "reachable" part of the target in appropriate direc...
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We obtain the uniqueness of lower semicontinuous (LSC) viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a "reachable" part of the target in appropriate directions. We present a counter-example which shows that the uniqueness does not hold without this convergence assumption. It was shown by Soravia that the uniqueness of LSC viscosity solutions having a "subsolution property" on the target holds. In order to verify this subsolution property, we show that the dynamic programming principle (DPP) holds inside for any LSC viscosity solutions. In order to obtain the DPP, we prepare appropriate approximate PDEs derived through Barles' inf-convolution and its variant.
In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Ha...
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In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satis ed, then the value function is the unique lower semicontinuous solution of the Hamilton Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given.
We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equation u(t) + H(u, Du) = g in R-n x R+ with u(x, 0) = u(0)(x). The Hamiltonian H(s,p) is assumed to be convex and positively homogene...
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We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equation u(t) + H(u, Du) = g in R-n x R+ with u(x, 0) = u(0)(x). The Hamiltonian H(s,p) is assumed to be convex and positively homogeneous of degree one inp for each s in R. If H is non increasing in s, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.
We show that the well-known relationship between the dual extremal are in the maximum principle and the optimal value function (of dynamicprogramming), calculated on the optimal trajectory, is valid for the control o...
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We show that the well-known relationship between the dual extremal are in the maximum principle and the optimal value function (of dynamicprogramming), calculated on the optimal trajectory, is valid for the control of parabolic variational inequalities. It follows that every optimal control is given by a feedback law. In the case when the functions defining the performance index are convex also with respect to the state variable, a more specific result is obtained.
We discuss ergodicity properties of a controlled jumps diffusion process reflected from the boundary of a bounded domain. The control parameters act on the drift term and on a first-order-type jump density. The contro...
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We discuss ergodicity properties of a controlled jumps diffusion process reflected from the boundary of a bounded domain. The control parameters act on the drift term and on a first-order-type jump density. The controlled process is generated via a Girsanov change of probability, and a long-run average criterion is optimized. An optimal stationary feedback is constructed by means of the Hamilton-Jacobi-Bellman equation.
This paper studies themonotone follower problemfor a one-dimensional singular diffusion process. The dynamic programming principle is established. It is shown that the value function is continuous and satisfies the Ha...
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This paper studies themonotone follower problemfor a one-dimensional singular diffusion process. The dynamic programming principle is established. It is shown that the value function is continuous and satisfies the Hamilton-Jacobi-Bellman equation in theviscositysense
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