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 $n$ 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