We prove probabilistic error estimates for high-index saddle dynamics with or without constraints to account for the inaccurate values of the model, which could be encountered in various scenarios such as model uncertainties or surrogate model algorithms via machine learning methods. The main contribution lies in incorporating the probabilistic error bound of the model values with the conventional error estimate methods for high-index saddle dynamics. The derived results generalize the error analysis of deterministic saddle dynamics and characterize the affect of the inaccuracy of the model on the convergence rate.

In this research, we summarize the results of a practical study of index options based on the option valuation model which was proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388), where an EMM kernel is integrated which takes into account all risk components of a regime-switching model. Further, the regime-switching risk of an economy in the options is priced using a hidden Markov regime-switching model with the risky underlying asset being modulated by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We apply such a model to the pricing of Hang Seng Index options based on the real-world financial data from October 2009 to October 2010 (i.e., for the year in which the model was proposed). We employed the entropy martingale measure (EMM) approach proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388) to determine the optimal martingale measure for the Markov-modulated GBM. In addition, we have proposed a numerical technique called the weighted difference method to compliment the EMM approach. We have also verified the extended jump-diffusion model under regime-switching that we proposed recently (Int. J. Finan. Eng., 6(4) (2019), 1950038) using the 50ETF options which are obtained from Shanghai Stock Exchange covering a time span from January 2018 to December 2022. Further, we have highlighted the challenges for the EMM kernel-based Markov regime-switching model for pricing the out-of-the-money index options in the real world.

In this paper we investigate the nonconforming $P_1$ finite element approximation to the sequential regularization method for unsteady Navier-Stokes equations. We provide error estimates for a full discretization scheme. Typically, conforming $P_1$ finite element methods lead to error bounds that depend inversely on the penalty parameter $\epsilon.$ We obtain an $\epsilon$-uniform error bound by utilizing the nonconforming $P_1$ finite element method in this paper. Numerical examples are given to verify theoretical results.

The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.