19,187 research outputs found
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
A forward--backward stochastic algorithm for quasi-linear PDEs
We propose a time-space discretization scheme for quasi-linear parabolic
PDEs. The algorithm relies on the theory of fully coupled forward--backward
SDEs, which provides an efficient probabilistic representation of this type of
equation. The derivated algorithm holds for strong solutions defined on any
interval of arbitrary length. As a bypass product, we obtain a discretization
procedure for the underlying FBSDE. In particular, our work provides an
alternative to the method described in [Douglas, Ma and Protter (1996) Ann.
Appl. Probab. 6 940--968] and weakens the regularity assumptions required in
this reference.Comment: Published at http://dx.doi.org/10.1214/105051605000000674 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Quantifying Differential Privacy in Continuous Data Release under Temporal Correlations
Differential Privacy (DP) has received increasing attention as a rigorous
privacy framework. Many existing studies employ traditional DP mechanisms
(e.g., the Laplace mechanism) as primitives to continuously release private
data for protecting privacy at each time point (i.e., event-level privacy),
which assume that the data at different time points are independent, or that
adversaries do not have knowledge of correlation between data. However,
continuously generated data tend to be temporally correlated, and such
correlations can be acquired by adversaries. In this paper, we investigate the
potential privacy loss of a traditional DP mechanism under temporal
correlations. First, we analyze the privacy leakage of a DP mechanism under
temporal correlation that can be modeled using Markov Chain. Our analysis
reveals that, the event-level privacy loss of a DP mechanism may
\textit{increase over time}. We call the unexpected privacy loss
\textit{temporal privacy leakage} (TPL). Although TPL may increase over time,
we find that its supremum may exist in some cases. Second, we design efficient
algorithms for calculating TPL. Third, we propose data releasing mechanisms
that convert any existing DP mechanism into one against TPL. Experiments
confirm that our approach is efficient and effective.Comment: accepted in TKDE special issue "Best of ICDE 2017". arXiv admin note:
substantial text overlap with arXiv:1610.0754
Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints
This paper presents a stochastic model predictive control approach for
nonlinear systems subject to time-invariant probabilistic uncertainties in
model parameters and initial conditions. The stochastic optimal control problem
entails a cost function in terms of expected values and higher moments of the
states, and chance constraints that ensure probabilistic constraint
satisfaction. The generalized polynomial chaos framework is used to propagate
the time-invariant stochastic uncertainties through the nonlinear system
dynamics, and to efficiently sample from the probability densities of the
states to approximate the satisfaction probability of the chance constraints.
To increase computational efficiency by avoiding excessive sampling, a
statistical analysis is proposed to systematically determine a-priori the least
conservative constraint tightening required at a given sample size to guarantee
a desired feasibility probability of the sample-approximated chance constraint
optimization problem. In addition, a method is presented for sample-based
approximation of the analytic gradients of the chance constraints, which
increases the optimization efficiency significantly. The proposed stochastic
nonlinear model predictive control approach is applicable to a broad class of
nonlinear systems with the sufficient condition that each term is analytic with
respect to the states, and separable with respect to the inputs, states and
parameters. The closed-loop performance of the proposed approach is evaluated
using the Williams-Otto reactor with seven states, and ten uncertain parameters
and initial conditions. The results demonstrate the efficiency of the approach
for real-time stochastic model predictive control and its capability to
systematically account for probabilistic uncertainties in contrast to a
nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro
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