29 research outputs found
Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation
We analyse a class of numerical schemes for solving the HJB equation for stochastic control problems, that generalizes the usual finite difference method. The latter is known to be monotonous, and hence valid, only if the scaled covariance matrix is diagonal dominant. We generalize this result by, given the set of neighbouring points allowed to enter in the scheme, showing how to compute the class of covariance matrices that is consistent with this set of points. We perform this computation for several cases in dimension 2 to 4
Inverse stochastic optimal controls
We study an inverse problem of the stochastic optimal control of general
diffusions with performance index having the quadratic penalty term of the
control process. Under mild conditions on the drift, the volatility, the cost
functions of the state, and under the assumption that the optimal control
belongs to the interior of the control set, we show that our inverse problem is
well-posed using a stochastic maximum principle. Then, with the well-posedness,
we reduce the inverse problem to some root finding problem of the expectation
of a random variable involved with the value function, which has a unique
solution. Based on this result, we propose a numerical method for our inverse
problem by replacing the expectation above with arithmetic mean of observed
optimal control processes and the corresponding state processes. The recent
progress of numerical analyses of Hamilton-Jacobi-Bellman equations enables the
proposed method to be implementable for multi-dimensional cases. In particular,
with the help of the kernel-based collocation method for
Hamilton-Jacobi-Bellman equations, our method for the inverse problems still
works well even when an explicit form of the value function is unavailable.
Several numerical experiments show that the numerical method recover the
unknown weight parameter with high accuracy
The non-locality of Markov chain approximations to two-dimensional diffusions
In this short paper, we consider discrete-time Markov chains on lattices as
approximations to continuous-time diffusion processes. The approximations can
be interpreted as finite difference schemes for the generator of the process.
We derive conditions on the diffusion coefficients which permit transition
probabilities to match locally first and second moments. We derive a novel
formula which expresses how the matching becomes more difficult for larger
(absolute) correlations and strongly anisotropic processes, such that
instantaneous moves to more distant neighbours on the lattice have to be
allowed. Roughly speaking, for non-zero correlations, the distance covered in
one timestep is proportional to the ratio of volatilities in the two
directions. We discuss the implications to Markov decision processes and the
convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in
the Barles-Souganidis framework.Comment: Corrected two errata from previous and journal version: definition of
R in (5) and summations in (7
Consistency of a simple multidimensional scheme for Hamilton–Jacobi–Bellman equations
International audienceThis Note presents an approximation scheme for second-order Hamilton-Jacobi-Bellman equations arising in stochastic optimal control. The scheme is based on a Markov chain approximation method. It is easy to implement in any dimension. The consistency of the scheme is proved, which guarantees its convergence. To cite this article: R. Munos, H. Zidani, C. R. Acad. Sci. Paris, Ser. I 340 (2005)