442 research outputs found
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Small Noise Asymptotics for Invariant Densities for a Class of Diffusions: A Control Theoretic View (with Erratum)
The uniqueness argument in the proof of Theorem 5, p. 483, of "Small noise
asymptotics for invariant densities for a class of diffusions: a control
theoretic view, J. Math. Anal. and Appl. (2009) " is flawed. We give here a
corrected proof.Comment: 23 pages; Journal of Mathematical Analysis and Applications, 200
Large deviations for some fast stochastic volatility models by viscosity methods
We consider the short time behaviour of stochastic systems affected by a
stochastic volatility evolving at a faster time scale. We study the asymptotics
of a logarithmic functional of the process by methods of the theory of
homogenisation and singular perturbations for fully nonlinear PDEs. We point
out three regimes depending on how fast the volatility oscillates relative to
the horizon length. We prove a large deviation principle for each regime and
apply it to the asymptotics of option prices near maturity
Random conformal dynamical systems
We consider random dynamical systems such as groups of conformal
transformations with a probability measure, or transversaly conformal
foliations with a Laplace operator along the leaves, in which case we consider
the holonomy pseudo-group. We prove that either there exists a measure
invariant under all the elements of the group (or the pseudo-group), or almost
surely a long composition of maps contracts exponentially a ball. We deduce
some results about the unique ergodicity.Comment: 61 page
Relative Value Iteration for Stochastic Differential Games
We study zero-sum stochastic differential games with player dynamics governed
by a nondegenerate controlled diffusion process. Under the assumption of
uniform stability, we establish the existence of a solution to the Isaac's
equation for the ergodic game and characterize the optimal stationary
strategies. The data is not assumed to be bounded, nor do we assume geometric
ergodicity. Thus our results extend previous work in the literature. We also
study a relative value iteration scheme that takes the form of a parabolic
Isaac's equation. Under the hypothesis of geometric ergodicity we show that the
relative value iteration converges to the elliptic Isaac's equation as time
goes to infinity. We use these results to establish convergence of the relative
value iteration for risk-sensitive control problems under an asymptotic
flatness assumption
Importance Sampling for Multiscale Diffusions
We construct importance sampling schemes for stochastic differential
equations with small noise and fast oscillating coefficients. Standard Monte
Carlo methods perform poorly for these problems in the small noise limit. With
multiscale processes there are additional complications, and indeed the
straightforward adaptation of methods for standard small noise diffusions will
not produce efficient schemes. Using the subsolution approach we construct
schemes and identify conditions under which the schemes will be asymptotically
optimal. Examples and simulation results are provided
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