Deng Ding

Gaussian mixture models (GMMs) are powerful tools specifically suited for problems where data distributions are multi-modal. Inspired by Fourier expansion, we propose a concept called GMM expansion to approximate arbitrary densities. A simple learning method is introduced under this framework. Theoretical and numerical analyses are provided to show that densities under GMM expansion are able to approximate arbitrary densities with certain accuracy. The proposed learning algorithm demonstrates better accuracy and time efficiency compared to classic density approximation methods such as the Expectation Maximization (EM) algorithm and Bayesian variational inference (BVI) for Gaussian mixture models. The proposed framework also shows better accommodation of neural networks. Three neural network applications are built to demonstrate its usability. The accuracy of density estimation in neural networks is reported to be significantly improved in one of the applications. Another application shows that latent variables can be easily turned into random variables. This user-friendly method opens up more possibilities in terms of latent manipulation, embedding techniques, and many other potential uses.

In this work, we present a novel machine learning approach for pricing high-dimensional American options based on the modified Gaussian process regression (GPR). We incorporate deep kernel learning and sparse variational Gaussian processes to address the challenges traditionally associated with GPR. These challenges include its diminished reliability in high-dimensional scenarios and the excessive computational costs associated with processing extensive numbers of simulated paths Our findings indicate that the proposed method surpasses the performance of the least squares Monte Carlo method in high-dimensional scenarios, particularly when the underlying assets are modeled by Merton's jump diffusion model. Moreover, our approach does not exhibit a significant increase in computational time as the number of dimensions grows. Consequently, this method emerges as a potential tool for alleviating the challenges posed by the curse of dimensionality.

We propose an Gaussian Mixture Model (GMM) learning algorithm, based on our previous work of GMM expansion idea. The new algorithm brings more robustness and simplicity than classic Expectation Maximization (EM) algorithm. It also improves the accuracy and only take 1 iteration for learning. We theoretically proof that this new algorithm is guarantee to converge regardless the parameters initialisation. We compare our GMM expansion method with classic probability layers in neural network leads to demonstrably better capability to overcome data uncertainty and inverse problem. Finally, we test GMM based generator which shows a potential to build further application that able to utilized distribution random sampling for stochastic variation as well as variation control.

Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today’s context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives.

The problem of domain aiming control is formulated for controlled stochastic nonlinear systems. This issue involves regularity of the solution to the resulting closed-loop stochastic system. To begin with, an extended existence and uniqueness theorem for stochastic differential equation with local Lipschitz coefficients is proven by using a Lyapunov-type function. A Lyapunov-based sufficient condition is also given under which there is no regularity of the solution for a class of stochastic differential equations. The notions of domain recurrence and residence time for stochastic nonlinear systems are introduced, and various criteria for the recurrence and non-recurrence relative to a bounded open domain or an unbounded domain are provided. Furthermore, upper bounds of either the expectation or the moment-generating function of the residence time are derived. In particular, a connection between the mean residence time and a Dirichlet problem is investigated and illustrated with a numerical example. Finally, the problem of domain aiming control is considered for certain types of nonlinear and linear stochastic systems. Several examples are provided to illustrate the theoretical results.