135,112 research outputs found
An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise
We study a linear quadratic problem for a system governed by the heat
equation on a halfline with Dirichlet boundary control and Dirichlet boundary
noise. We show that this problem can be reformulated as a stochastic evolution
equation in a certain weighted L2 space. An appropriate choice of weight allows
us to prove a stronger regularity for the boundary terms appearing in the
infinite dimensional state equation. The direct solution of the Riccati
equation related to the associated non-stochastic problem is used to find the
solution of the problem in feedback form and to write the value function of the
problem.Comment: 16 pages. Many misprints have been correcte
Synchronization of a large number of continuous one-dimensional stochastic elements with time delayed mean field coupling
We study synchronization as a means of control of collective behavior of an ensemble
of coupled stochastic units in which oscillations are induced merely by external noise.
We determine the boundary of the synchronization domain of a large number of onedimensional
continuous stochastic elements with time delayed non-homogeneous
mean-field coupling. Exact location of the synchronization threshold is shown to
be a solution of the boundary value problem (BVP) which was derived from the
linearized Fokker-Planck equation. Here the synchronization threshold is found by
solving this BVP numerically. Approximate analytics is obtained by expanding the
solution of the linearized Fokker-Planck equation into a series of eigenfunctions of
the stationary Fokker-Planck operator. Bistable systems with a polynomial and
piece-wise linear potential are considered as examples. Multistability and hysteresis
is observed in the Langevin equations for finite noise intensity. In the limit of small
noise intensities the critical coupling strength was shown to remain finite
The Optimal Consumption Function in a Brownian Model of Accumulation Part A: The Consumption Function as Solution of a Boundary Value Problem
We consider a neo-classical model of optimal economic growth with c.r.r.a. utility in which the traditional deterministic trends representing population growth, technological progress, depreciation and impatience are replaced by Brownian motions with drift. When transformed to 'intensive' units, this is equivalent to a stochastic model of optimal saving with diminishing returns to capital. For the intensive model, we give sufficient conditions for optimality of a consumption plan (open-loop control) comprising a finite welfare condition, a martingale condition for shadow prices and a transversality condition as t ? ?. We then replace these by conditions of optimality of a plan generated by a consumption function (closed-loop control), i.e. a function H(z) expressing log-consumption as a time-invariant, deterministic function of log-capital z. Making use of the exponential martingale formula we replace the martingale condition by a non-linear, non-autonomous second order o.d.e. which an optimal consumption function must satisfy; this has the form H"(z) = F[H'(z),?(z),z], where ?(z) = exp{H(z)-z}. Economic considerations suggest certain limiting values which H'(z) and ?(z) should satisfy as z ? ? ?, thus defining a two-point boundary value problem (b.v.p.) - or rather, a family of problems, depending on the values of parameters. We prove two theorems showing that a consumption function which solves the appropriate b.v.p. generates an optimal plan. Proofs that a unique solution of each b.v.p. exists will be given in a separate paper (Part B).Consumption, capital accumution, Brownian motion, optimisation, orderinary differential equation, boundary value problems.
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
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Dynamic pricing of general insurance in a competitive market
A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer’s objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation.
In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer’s risk aversion which leads to reduced market exposure and lower overall wealth generation
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