The explicit Laplace transform for the Wishart process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.Comment: Accepted on: Journal of Applied Probability 51(3), 201

A Dilated Inception Network for Visual Saliency Prediction

Recently, with the advent of deep convolutional neural networks (DCNN), the improvements in visual saliency prediction research are impressive. One possible direction to approach the next improvement is to fully characterize the multi-scale saliency-influential factors with a computationally-friendly module in DCNN architectures. In this work, we proposed an end-to-end dilated inception network (DINet) for visual saliency prediction. It captures multi-scale contextual features effectively with very limited extra parameters. Instead of utilizing parallel standard convolutions with different kernel sizes as the existing inception module, our proposed dilated inception module (DIM) uses parallel dilated convolutions with different dilation rates which can significantly reduce the computation load while enriching the diversity of receptive fields in feature maps. Moreover, the performance of our saliency model is further improved by using a set of linear normalization-based probability distribution distance metrics as loss functions. As such, we can formulate saliency prediction as a probability distribution prediction task for global saliency inference instead of a typical pixel-wise regression problem. Experimental results on several challenging saliency benchmark datasets demonstrate that our DINet with proposed loss functions can achieve state-of-the-art performance with shorter inference time.Comment: Accepted by IEEE Transactions on Multimedia. The source codes are available at https://github.com/ysyscool/DINe

Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investor's initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.Comment: 26 page

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 35