85 research outputs found
Quasi-Monte Carlo methods for Choquet integrals
We propose numerical integration methods for Choquet integrals where the
capacities are given by distortion functions of an underlying probability
measure. It relies on the explicit representation of the integrals for step
functions and can be seen as quasi-Monte Carlo methods in this framework. We
give bounds on the approximation errors in terms of the modulus of continuity
of the integrand and the star discrepancy.Comment: 6 page
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
Optimal long term investment model with memory
We consider a financial market model driven by an R^n-valued Gaussian process
with stationary increments which is different from Brownian motion. This
driving noise process consists of independent components, and each
component has memory described by two parameters. For this market model, we
explicitly solve optimal investment problems. These include (i) Merton's
portfolio optimization problem; (ii) the maximization of growth rate of
expected utility of wealth over the infinite horizon; (iii) the maximization of
the large deviation probability that the wealth grows at a higher rate than a
given benchmark. The estimation of paremeters is also considered.Comment: 25 pages, 3 figures. To appear in Applied Mathematics and
Optimizatio
A kernel-based method for Schr\"odinger bridges
We characterize the Schr\"odinger bridge problems by a family of
Mckean-Vlasov stochastic control problems with no terminal time distribution
constraint. In doing so, we use the theory of Hilbert space embeddings of
probability measures and then describe the constraint as penalty terms defined
by the maximum mean discrepancy in the control problems. A sequence of the
probability laws of the state processes resulting from -optimal
controls converges to a unique solution of the Schr\"odinger's problem under
mild conditions on given initial and terminal time distributions and an
underlying diffusion process. We propose a neural SDE based deep learning
algorithm for the Mckean-Vlasov stochastic control problems. Several numerical
experiments validate our methods
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