The model of partially observed nonlinear system, called extended Kalman filter (EKF), and depending on some unknown parameters is considered. An approximation of the unobserved component is proposed. This approximation is realized in two steps. First a the method of moments estimator of unknown parameter is constructed and then this estimator is substituted in the equations of extended Kalman filter. The obtained equations describe the adaptive extended Kalman filter. The properties of estimator of the unknown parameter and of the unknown state are described in the asymptotic of small noise in observations.

In this paper, we study the path-regularity and martingale properties of the set-valued stochastic integrals defined in our previous work [4]. Such integrals have some fundamental differences from the well-known Aumann-Itô stochastic integrals, and are much better suitable for representing set-valued martingales, whence potentially useful in the study of set-valued backward stochastic differential equations. However, similar to the Aumann-Itô integral, the new integral is only a set-valued submartingale in general, and there is very limited knowledge about the path regularity of the related indefinite integral, much less the sufficient conditions under which the integral is a true martingale. In this paper, we first establish the existence of right- and left-continuous modifications of set-valued submartingales in continuous time, and apply the results to set-valued stochastic integrals. Moreover, we show that a set-valued stochastic integral yields a martingale if and only if the set of terminal values of the stochastic integrals associated to the integrand is closed and decomposable. Finally, as a particular example, we study the set-valued martingale in the form of the conditional expectation of a set-valued random variable. We show that when the random variable is a convex random polytope, the conditional expectation of a vertex stays as a vertex of the set-valued conditional expectation if and only if the random polytope has a deterministic normal fan.

This study investigates adaptive equilibrium strategies for multiple risk-averse informed traders in Almgren-Chriss framework. Dynamic information and transaction costs are taken into account. Using a convex analytic approach, we characterize the open-loop Nash equilibrium in terms of a system of linear forward-backward stochastic differential equations, and further provide an explicit feedback expression of the unique equilibrium. The results show how risk-averse informed traders exploit long-lived information and manage positions in the face of information volatility and inaccuracy.

Let $B^H=\{B_t^H, t \geq 0\}$ be a fractional Brownian motion with Hurst index $0<H<1$, and let $B=\{B_t, t \geq 0\}$ be an independent Brownian motion. In this study, we investigate the parameter estimation of a mixed fractional Black-Scholes model $ S_t^H=S_0^H+\mu\displaystyle \int_0^tS_s^H{\mathrm{d}}s+\sigma\displaystyle \int_{0}^{t}S_s^H{\mathrm{d}}(B_s+B_s^H)$,where $\sigma>0$, $\mu\in {\mathbb R}$ are two unknown parameters. Using quasi-likelihood estimation, when the system is observed at some discrete time instants $\{t_i=ih,i=0,1,2,\ldots,n\}$, we give estimations of the parameters $\mu$ and $\sigma$ provided $h=h(n)\to 0$, $nh\to \infty$ and $h^{1+\gamma}n\to 1$ for some $\gamma>0$, as $n\to \infty$. We present the asymptotic normality of the estimators based on the velocity of $nh^{1+\gamma}-1$ tending to zero as $n$ tends to infinity. Finally, we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.

In this paper, we investigate the mean square and quasi-sure exponential stabilization of stochastic differential equations driven by G-Brownian motion, leveraging discrete-time feedback observations. We introduce a discrete-time feedback control mechanism within the drift part and demonstrate the existence of a threshold $\bar{\tau}>0$. This threshold ensures the stability of the controlled system for any discrete step size 𝜏 that is less than $\bar{\tau}$. To validate our control strategy, we present an illustrative example.

Pricing barrier options pose a significant challenge in financial derivative valuation because they are activated only when the underlying asset reaches a predetermined barrier. The first-hitting time model was employed to characterize the activation process. In addition, the pricing of American barrier options with a floating interest rate is dynamically represented by an uncertain fractional differential equation. The study derives price formulas for various American barrier options, including up-and-in call, down-and-in put, up-and-output, and down-and-out call options. The proposed model enhances the accuracy of capturing the long-tail distribution and tail risk of financial markets, thereby addressing their complexity and nonlinearity. Furthermore, the predictor-corrector method is utilized to compute the numerical prices for the barrier options with floating interest rates, supplemented by illustrative numerical examples.

This paper presents the necessary and sufficient conditions for a kind of discrete-time robust stochastic optimal control problem with convex control domains. The classical variational method is invalid in this context because it is an “inf sup problem”. We obtain the variational inequality with a common reference probability by systematically using weak convergence approach and the minimax theorem. Moreover, a discrete-time robust investment problem is also studied where the explicit optimal control is given.

This paper presents a closed-loop numerical algorithm for the quadratic optimal control problem of a linear mixed system that combines both deterministic and stochastic controls. The core idea is to numerically solve two Riccati equations by using linear quadratic theory. Based on these numerical solutions, a feedback-type discretization method for the original problem is developed, along with an analysis of its convergence rate. A significant advantage of the proposed method is that it avoids the computation of backward stochastic differential equations associated with Pontryagin’s maximum principle, leading to notable improvements in computational efficiency. This work builds upon the theoretical framework established by Hu and Tang (Probab. Uncertain. Quant. Risk, 4 (2019), Paper No. 1) and focuses on its numerical implementation.