In this paper, we investigate the law of the logarithm for delayed sums under sublinear expectation. The necessary and sufficient conditions for the law of the logarithm for delayed sums under sublinear expectation are established which differ from those for partial sums. The law of the logarithm for delayed sums can be treated as a tool to prove the convergence rates of logarithm form for partial sums under sublinear expectation. These results extend the results of Lai [13] from the classical probability setting to the sublinear expectation framework.
A theory of autoregressive (AR) sequences under sublinear expectations is developed in this paper. AR sequences play a significant role in linear time series analysis. The traditional AR time series model requires the data to be stationary; however, data in real world often obey different distributions at different time points, which makes the traditional AR model invalid. To characterize the AR property of this kind of time series without stationarity, we introduce AR sequences under sublinear expectations, taking the distribution uncertainty into account by virtue of the sublinear expectation theory. To give the well-definedness of AR sequences under sublinear expectations, we prove the existence and uniqueness of the solution to AR equations under sublinear expectations. Furthermore, we introduce the Yule-Walker equation of our AR sequences, which gives the relationship between the coefficients and the autocovariance functions. At last, we give an example of white noise, which is generated by a G-Brownian motion, and we derive the distribution of the related AR sequences.
In this paper, we examine two types of nonlinear expectations: Choquet expectation and sublinear expectation. Similar to the role played by the normal distribution and Wiener space in classical probability theory, the G-normal distribution and G-expectation to investigate the relationship between moments of variables following a distribution under Choquet expectation and sublinear expectation. We provide bounds for the p-order absolute moment (where p > 0) of a G-normal distribution, specifically under Choquet expectation. Notably, we derive a formula for the Choquet expectation of phi(X), where phi is a strictly monotonic function and X is a G-normally distributed random variable. As a special case, we obtain the explicit expressions for the odd moments of a G-normal distribution corresponding to Choquet expectation. Consequently, we establish that if G-expectation coincides with a Choquet expectation, then the G-expectation space are crucial in the theory of sublinear expectation. Our objective is G-normal is linear.
Scalar invariance is a fundamental property in the theory of risk measures. In this paper, we investigate scalar invariant maps on throughtwomaindirections.First, L-infinity we introduce the notions of scalar closedness and scalar solidity for pseudo-acceptance sets and show that these properties provide a necessary and sufficient characterization of acceptance sets without requiring convexity or . weak*-closedness Second, we develop a generalized duality framework between and the class of L-infinity scalar invariant maps, based on the evaluation pairing (h, phi) -> phi(h) . This leads to a bipolar-type representation for acceptance sets under minimal structural assumptions. Finally, we introduce a hull operator associated with abstract properties on scalar invariant maps, interpret it as the largest minorant satisfying a given property, and study consistency and representation results. Several classical properties (convexity, subadditivity, monotonicity, etc.) are revisited within this framework, and explicit formulas for the associated hulls are provided.
Based ong-expthe ectation of distributions, we obtain the monotonicity and Jensen's inequality for g-expthe ectation of distributions; and for a sequence of distribution functions, we establish a monotone weak convergence theorem, Fatou's lemma, and a convergence theorem with respect to the g-expectation of distributions.
We study the uniqueness of solutions of backward stochastic differential equations (BSDEs), which generator verifies |F(t, y, z)| <= at + beta t|y| + Bt|z| + f (|y|)|z|2, where , , are positive processes and the function is positive, continuous and alpha t beta t theta t f increasing. The uniqueness of solutions of such BSDEs is derived in two situations, when F is locally Lipschitz and when F is jointly convex. As a byproduct: we show the existence of viscosity solutions to the associated semilinear partial differential equations, which can contain nonlinearity that has quadratic growth in the gradient of the solution.
This paper investigates the optimal investment problem for hybrid pension plans in a financial market with jump-diffusion risky assets, where both the contribution and the benefit are adjusted based on the plan's performance, and risks are shared across different generations. The investment in a risk-free asset and two risky assets is carried out by the managers of the pension fund. The model of risky asset is assumed to be modulated by a compound Poisson process, with the two risky asset price processes correlated through a common shock. The objective of this study is to seek the optimal investment strategies and risk-sharing arrangements for plan trustees and participants that minimize the costs associated with unstable contribution risks, unstable benefit risks, and discontinuous risks. By applying the stochastic optimal control approach, the closedform expressions of the optimal strategy and value function are derived. Numerical examples are provided to analyze the effects of parameters on the optimal strategies. In the context of the hybrid pension plan, these strategies effectively facilitate intergenerational risk-sharing.
This paper explores the optimal consumption, life insurance, and investment strategies of an individual under the influence of habit formation. We assume that the individual can invest in a risk-free asset, a stock, and an index bond in the financial market, where the stock price follows the 4/2 stochastic volatility model. The primary aim of this paper is to maximize the expected utility of consumption, total bequests, and terminal wealth before retirement or death; the utility of consumption is derived from actual consumption exceeding the established habitual consumption level. By applying the dynamic programming method, we derive the Hamilton-Jacobi-Bellman (HJB) equation that the value function satisfies, obtain the asymptotic solutions for the optimal consumption, life insurance, and investment strategies using the asymptotic expansion method, and prove the corresponding verification theorem. Furthermore, we provide numerical examples to analyze the influence of consumption habit patterns and model parameters on the individual's optimal strategies.
Nonlinear filtering problems are encountered in many applications, and one solution approach is the extended Kalman filter, which is not always convergent. Therefore, it is crucial to identify conditions under which the extended Kalman filter provides accurate approximations. This paper generalizes two significant results of Picard (1991) on the efficiency of the continuous-time extended Kalman filter for a filtering system with small noise, to a more general setting where the observation noise may be state-dependent but does not allow signal reconstruction from the quadratic variation of the observation process as for example in epidemic models. First, we show that if the drifts of the signal process and the observation process become nearly linear when the parameter epsilon, which scales the diffusion coefficients, approaches zero, and the drift coefficient of the observation process is strongly injective, then the estimation error is of the order of root epsilon. We then establish conditions under which the impact of the initial filtering error decays exponentially fast.
In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player has the same feedback form as the strategy in the social equilibrium, we say that the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations (FBSDEs). By employing elliptic master equations, a class of distribution-dependent elliptic partial differential equations (PDEs), we provide a representation for the Nash equilibrium strategies. We prove the Yamada-Watanabe type theorem and show weak uniqueness for infinite-time FBSDEs. Furthermore, we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.
The G-expectation is a sublinear expectation. It is an important tool for pricing financial products and managing risk thanks to its ability to deal with model uncertainty. The problem is how to efficiently quantify it since the commonly used Monte Carlo method does not work. Fortunately, the expectation of a G-normal random variable can be linked to the viscosity solution of a fully nonlinear G-heat equation. In this paper, we propose a novel numerical scheme for the two-dimensional G-heat equation and pay more attention to the case that there exists uncertainty on the correlationship, especially to the case that the correlationship ranges from negative to positive. The scheme is monotonic, stable, and convergent. The numerical tests show that the scheme is highly efficient.
This paper studies the pricing of contingent claims of American style, using indifference pricing by fully dynamic convex risk measures. We provide a general definition of risk-indifference prices for buyers and sellers in continuous time, in a setting where buyer and seller have potentially different information, and show that these definitions are consistent with no-arbitrage principles. Specifying to stochastic volatility models, we characterize indifference prices via solutions of Backward Stochastic Differential Equations reflected at Backward Stochastic Differential Equations and show that this characterization provides a basis for the implementation of numerical methods using deep learning.
We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to drift coefficients that may also depend on Z, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.
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.
Let be a fractional Brownian motion with Hurst index B-H = {B-t (H) , t >= 0} < H < 1, and B = let {B-t, t >= 0} be an independent Brownian motion. In this study, we investigate the parameter estimation of a mixed fractional Black-Scholes model integral t integral t StH= S 0H + /.L SsHds + sigma SsHd(Bs+ BsH) 0 0 , where sigma > 0 mu ,mu is an element of 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, ... , n}, we give estimations of the parameters mu and sigma h provided = h(n) -> 0 nh , infinity and 1+eta n -> 1 for some gamma > 0 , n ->infinity . We present the asymptotic normality of the estimators based on the velocity of tendingtozeroas tendstoinfinity.Finally, nh(1+gamma) - 1 n we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.
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.
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 tau > 0 . This threshold ensures the stability of the controlled system for any discrete step size tau that is less than 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 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 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.
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.
