Adjoint Processes Research Articles

Box splines revisited: Convergence and acceleration methods for the subdivision and the cascade algorithms

A general convergence analysis of the cascade algorithm, for the determination of a refinable function from its mask, is applied to box splines in which case certain difficulties adherent to the general case can be resolved completely and even elegantly. In the process, the understanding of the convergence of the adjoint process, subdivision, is also enhanced.

The Relaxed Stochastic Maximum Principle in Singular Optimal Control of Diffusions

This paper studies optimal control of systems driven by stochastic differential equations, where the control variable has two components, the first being absolutely continuous and the second singular. Our main result is a stochastic maximum principle for relaxed controls, where the first part of the control is a measure valued process. To achieve this result, we establish first order optimality necessary conditions for strict controls by using strong perturbation on the absolutely continuous component of the control and a convex perturbation on the singular one. The proof of the main result is based on the strict maximum principle, Ekeland’s variational principle, and some stability properties of the trajectories and adjoint processes with respect to the control variable.

Explicit Fourier wavefield operators

SUMMARY Explicit wavefield extrapolators are based on direct analytic mathematical formulae that express the output as an extrapolation operator acting on the input, while implicit methods usually require the calculation of the numerical inverse of a matrix to obtain the output. Typically, explicit methods are faster than implicit methods, and they often give more insight into the physics of the wave propagation, but they often suffer from instability. Four different explicit extrapolators based on Fourier theory are presented and analysed. They are: PS (ordinary phase shift), GPSPI (generalized phase shift plus interpolation), NSPS (non-stationary phase shift) and SNPS (symmetric non-stationary phase shift). A formal proof is given that NSPS in a direction orthogonal to the velocity gradient is the mathematical adjoint process to GPSPI in the opposite direction. This motivates the construction of SNPS that combines NSPS and GPSPI in a symmetric fashion. This symmetry (under interchange of input and output lateral coordinates) is required by reciprocity arguments. PS and SNPS are symmetric while NSPS and GPSPI are not. A numerical stability study using SVD (singular value decomposition) shows that all of these extrapolators can become unstable for strong lateral velocity gradients. Unstable operators allow amplitudes to grow non-physically in a recursion. Stability is enhanced by introducing a small (∼3 per cent) imaginary component to the velocities. This causes a numerical attenuation that tends to stabilize the operators but does not address the cause of the instability. For the velocity model studied (a very challenging case) GPSPI and NSPS have exactly the same instability while SNPS is always more stable. Instability manifests in a complicated way as a function of extrapolation step size, frequency, velocity gradient, and strength of numerical attenuation. The SNPS operator can be stabilized over a wide range of conditions with considerably less attenuation than is required for NSPS or GPSPI.

Pricing of American style options with an adjoint process correction method

Pricing of American options is a more complicated problem than pricing of European options. In this work a formula is derived that allows the computation of the early exercise premium, i.e. the price difference between these two option types in terms of an adjoint process evolving in the reversed time direction of the original process determining the evolution of the European price. We show how this equation can be utilised to improve option price estimates from numerical schemes like finite difference or Monte Carlo methods.

Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance

We give a verification theorem by employing Arrow's generalization of the Mangasarian sufficient condition to a general jump diffusion setting and show the connections of adjoint processes to dynamic programming. The result is applied to financial optimization problems.

An Algorithm for Infinite Dimensional Stochastic Control Problems

An algorithm of Bonnans is generalized to the case of optimal control of a stochastic evolution equation. Properties of the adjoint process are also discussed.

Optimal Control of Point Processes with Noisy Observations: The Maximum Principle

This paper studies the optimal control problem for point processes with Gaussian white-noised observations. A general maximum principle is proved for the partially observed optimal control of point processes, without using the associated filtering equation . Adjoint flows—the adjoint processes of the stochastic flows of the optimal system—are introduced, and their relations are established. Adjoint vector fields , which are observation-predictable, are introduced as the solutions of associated backward stochastic integral-partial differential equtions driven by the observation process. In a heuristic way, their relations are explained, and the adjoint processes are expressed in terms of the adjoint vector fields, their gradients and Hessians, along the optimal state process. In this way the adjoint processes are naturally connected to the adjoint equation of the associated filtering equation . This shows that the conditional expectation in the maximum condition is computable through filtering the optimal state, as usually expected. Some variants of the partially observed stochastic maximum principle are derived, and the corresponding maximum conditions are quite different from the counterpart for the diffusion case. Finally, as an example, a quadratic optimal control problem with a free Poisson process and a Gaussian white-noised observation is explicitly solved using the partially observed maximum principle.

Necessary conditions for optimality in relaxed stochastic control problems

In this paper, we are concerned with optimal control problems where the system is driven by a stochastic differential equation of the Ito type. We study the relaxed model for which an optimal solution exists. This is an extension of the initial control problem, where admissible controls are measure valued processes. Using Ekeland's variational principle and some stability properties of the corresponding state equation and adjoint processes, we establish necessary conditions for optimality satisfied by an optimal relaxed control. This is the first version of the stochastic maximum principle that covers relaxed controls.

The adjoint process of reflected brownian motion in a cone

We provide two characterizations of the adjoint of reflected Brownian motion in a cone with radially homogeneous reflection. Roughly speaking, the adjoint is another reflected Brownian motion with a drift given by the logarithmic derivative of the invariant measure density and the reflection field is the reflection of the original field across the inward normal

The Adjoint Process of Killed Reflected Brownian Motion in a Cone and Applications

Let $X_t$ be reflected Brownian motion (RBM) in a cone with radially homogeneous reflection, killed upon reaching the vertex of the cone. We determine the adjoint process and use it to find the Martin boundary of the killed RBM together with all the corresponding positive harmonic functions. Then we can identify and prove uniqueness (up to positive scalar multiples) of the invariant measure for killed RBM and RBM without killing. Along the way, we prove the strong Feller property of the resolvent of RBM (no killing).

Necessary Conditions of Optimization for Partially Observed Controlled Diffusions

Necessary conditions are derived for stochastic partially observed control problems when the control enters the drift coefficient and correlation between signal and observation noise is allowed. The problem is formulated as one of complete information, but instead of considering directly the equation satisfied by the unnormalized conditional density of nonlinear filtering, measure-valued decompositions are used to decompose it into two processes. The minimum principle and the stochastic partial differential equation satisfied by the adjoint process are then derived, and the optimality conditions are shown to be the exact necessary conditions derived by Bensoussan [ Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), pp. 169--222; Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, UK, 1992] when the correlation is zero.

The Maximum Principle for Partially Observed Optimal Control of Stochastic Differential Equations

This paper concerns partially observed optimal control of possibly degenerate stochastic differential equations, with correlated noises between the system and the observation. The control is allowed to enter into all the coefficients. A general maximum principle is proved for the partially observed optimal control, and the relations among the adjoint processes are established. Adjoint vector fields, which are adapted to the past and present observations, are introduced as the solutions to some backward stochastic partial differential equations (BSPDEs), and their relations are established. Under suitable conditions, the adjoint processes are characterized in terms of the adjoint vector fields, their differentials and Hessians, along the optimal state process. Some other formulations of the partially observed stochastic maximum principle are then derived.

Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by the Ito stochastic differential equations (SDEs), where both the drift and diffusion terms are allowed to depend on controls and the systems are allowed to be degenerate. Necessary and sufficient conditions for a control to be near-optimal are studied. It is shown that any near-optimal control nearly maximizes the "$\HH$-function" (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients) in some integral sense, and vice versa if certain additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximum of the $\HH$-function are obtained, based on some delicate estimates of the adjoint processes. Examples are presented to demonstrate the results.

On consistent regularities of control and value functions

In this paper we deal with nonsmooth optimal control problems in the case when the control is allowed to be a discontinuous function. We analyse smoothness assumptions on an adjoint process in deterministic and stochastic cases. Possibilities of steep generalized space-gradients of the adjoint function imply the necessity of an approximation of the Hamiltonian. The key question of such an approximation is a relationship between the control and the value function. Under quite general assumptions it is proved that the performance measure for the original process is determined by the control function with possible discontinuities.

The maximum principle for optimal control of diffusions with non-smooth coefficients

In this paper, we study the necessary conditions for optimality of a control problem where the state process is governed by a stochastic differential equation, with non differentiable coefficients. We obtain a stochastic maximum principle for this model. This is the first version of the stochastic maximum principle that covers the non smooth case. The proof is based on the approximation of the non smooth coefficients by smooth ones. Then we apply Ekeland's variational principle in order to establish some necessary conditions satisfied by ϵ-optimal controls. The result is obtained by passing at the limit by using Krylov's estimate. The adjoint process is defined in terms of almost everywhere derivatives of the coefficients, and is the unique solution of a linear backward stochastic differential equation

Minimum principle for partially observable nonlinear risk-sensitive control problems using measure-valued decompositions

This paper is concerned with the derivation of a minimum principle for partially observed controlled diffusions with correlation between signals and observation noises, when the sample cost criterion is an exponential of integral. Instead of considering the information state which satisfies a version of Zakai's equation, measure-valued decompositions are employed to convert this equation into a P-a.s. deterministic equation, thus avoiding the basic finite dimensional approximation approach. Using weak control variations the minimum principle is derived. It consists of a modified version of Zakai's equation, an adjoint process satisfying a stochastic partial differential equation with terminal condition, and a Hamiltonian functional. This minimum principle is then applied to solve the linear-exponential-quadratic-Gaussian trackmgjproblem. The results of this paper are important in relating H ∞ or robust control problems and risk-sensitive control problems

On the application of minimum principle for solving partially observable risk-sensitive control problems

This paper is concerned with the application of a minimum principle derived for general nonlinear partially observable exponential-of-integral control problems, to solve linear-exponential-quadratic-Gaussian problems. This minimum principle is the stochastic analog of Pontryagin's minimum principle for deterministic systems. It consists of an information state equation, an adjoint process governed by a stochastic partial differential equation with terminal condition, and a Hamiltonian functional. Two methods are employed to obtain the optimal control law. The first method appeals to the well-known approach of completing the squares, by first determining the optimal control law that minimizes the Hamiltonian functional. The second method provides significant insight into relations with the HamiltoniJacobi approach associated with completely observable exponential-of-integral control problems. These methods of solution are particularly attractive because they do not assume a certainty equivalence principle, hence they can be used to solve nonlinear problems as well.

General necessary conditions for partially observed optimal stochastic controls

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

General necessary conditions for partially observed optimal stochastic controls

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

The second order minimum principle and adjoint process

When the control parameter appears in both the drift and diffusion coefficients, Peng has derived a “second order” minimum principle in terms of an adjoint process . Peng has also characterized this process as the unique, adapted solution of a reverse time stochastic differential equation. Using stochastic flows this paper gives an explicit expression for .