Search Results (33)

This paper reveals a simple methodology to create local-correlation models suitable for the closed-form pricing of two-asset financial derivatives. The multivariate models are built to ensure two conditions. First, marginals follow desirable processes, e.g., we choose the Geometric Brownian Motion (GBM), popular for stock prices. Second, the payoff of the derivative should follow a desired one-dimensional process. These conditions lead to a specific choice of the dependence structure in the form of a local-correlation model. Two popular multi-asset options are entertained: a spread option and a basket option. Full article

The stochastic correlation for Brownian motions is the integrand in the formula of their quadratic covariation. The estimation of this stochastic process becomes available from the temporally localized correlation of latent price driving Brownian motions in stochastic volatility models for asset prices. By analyzing this process for Apple and Microsoft stock prices traded minute-wise, we give statistical evidence for the roughness of its paths. Moment scaling indicates fractal behavior, and both fractal dimensions (approx. 1.95) and Hurst exponent estimates (around 0.05) point to rough paths. We model this rough stochastic correlation by a suitably transformed fractional Ornstein–Uhlenbeck process and simulate artificial stock prices, which allows computing tail dependence and the Herding Behavior Index (HIX) as functions in time. The computed HIX is hardly variable in time (e.g., standard deviation of 0.003–0.006); on the contrary, tail dependence fluctuates more heavily (e.g., standard deviation approx. 0.04). This results in a higher correlation risk, i.e., more frequent sudden coincident appearance of extreme prices than a steady HIX value indicates. Full article

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results. Full article

This paper examines the impacts of observed versus uncertain (stochastic) institutional quality of corporate debt financing. This paper compares the impacts of two distinct levels of institutional quality in developed and developing economies. World governance indicators (WGIs) are used as proxies for institutional quality. Stochastic Geometric Brownian Motion (GBM) is used to quantify the institutional uncertainty of WGIs. The results of GLS estimates using a sample of 309 nonfinancial listed firms in G8 countries and 373 nonfinancial listed firms in MENA countries covering the years 2016–2022 show (a) positive (negative) stochastic impacts of voice and accountability (government effectiveness and political stability) on debt financing in the G8 and MENA regions; (b) although potential improvements in institutional quality are shared concerns among G8 and MENA countries, the former outperforms the latter in terms of creditors’ contract protection and enforcement, paving the way for public policy makers in the MENA region to enhance regulations that protect debt contractual obligations; (c) macroeconomic variables have sporadic impacts; GDP growth is significant in G8 but not in MENA countries; (d) the negative impacts of inflation rates are consistent in both regions; and (e) unemployment plays a negative signaling role in the G8 region only. This paper contributes to the related literature by examining the uncertain impact of institutional quality on corporate debt financing. This paper offers implications for policy makers, directing them to focus on institutional endeavors in a way that helps companies secure the debt financing required to support investment growth. Full article

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM): $d\xi \left(t\right)=f(t,\xi (t-{\kappa }_1\left(t\right)))dt+g(t,\xi (t-{\kappa }_2\left(t\right)))d\mathcal{Z}\left(t\right),$ where ${\kappa }_1(·),{\kappa }_2(·):{\mathbb{R}}_{+}\to [0,\tau ]$ denote variable delays, and $\mathcal{Z}\left(t\right)$ denotes scalar G-Brownian motion, which has a symmetry distribution. It is shown that the SDDE-GBM is quasi-surely exponentially stable for each $\tau >0$ bounded by ${\tau }^{*}$, where the corresponding (non-delay) stochastic differential equation driven by G-Bronwian motion (SDE-GBM), $d\eta \left(t\right)=f(t,\eta (t\left)\right)dt+g(t,\eta (t\left)\right)d\mathcal{Z}\left(t\right)$, is quasi-surely exponentially stable. Moreover, by solving the non-linear equation on $\tau$, we can obtain the implicit lower bound ${\tau }^{*}$. Finally, illustrating examples are provided. Full article

In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear matrix inequalities (LMI) and Lyapunov–Krasovskii functions, we derive a sufficient condition for stabilization. Our findings demonstrate that an unstable system can be stabilized with a control interval within $({\theta }^{*},1)$. Some numerical examples are provided at the end to validate the correctness of our theoretical results. Full article

In crowded fluids, polymer segments can exhibit anomalous subdiffusion due to the viscoelasticity of the surrounding environment. Previous single-particle tracking experiments revealed that such anomalous diffusion in complex fluids (e.g., in bacterial cytoplasm) can be described by fractional Brownian motion (fBm). To investigate how the viscoelastic media affects the diffusive behaviors of polymer segments without resolving single crowders, we developed a novel fractional Brownian dynamics method to simulate the dynamics of polymers under confinement. In this work, instead of using Gaussian random numbers (“white Gaussian noise”) to model the Brownian force as in the standard Brownian dynamics simulations, we introduce fractional Gaussian noise (fGn) in our homemade fractional Brownian dynamics simulation code to investigate the anomalous diffusion of polymer segments by using a simple “bottle-brush”-type polymer model. The experimental results of the velocity autocorrelation function and the exponent that characterizes the subdiffusion of the confined polymer segments can be reproduced by this simple polymer model in combination with fractional Gaussian noise (fGn), which mimics the viscoelastic media. Full article

Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space ${\mathbb{C}}_r((-\infty ,0];R^n)$. The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both ${\mathbb{L}}_G^2$ and exponential estimates. Full article

We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference. Full article

Due to CO${}_2$ emissions, humans are encountering grave environmental crises (e.g., rising sea levels and the grim future of submerged cities). Governments have begun to offset emissions by constructing emission-trading schemes (carbon-offset markets). Investors naturally crave carbon-offset options to effectively control risk. However, the research and practice for these options are relatively limited. This paper contributes to the literature in this area. Specifically, according to carbon-emission allowances’ empirical distributions, we implement fractal Brownian motions and jump diffusions instead of traditional geometric Brownian motions. We contribute to extending the theoretical model based on carbon-offset option-pricing methods. We innovate the carbon-offset options of Asian styles. We authenticate the options’ stochastic differential equations and analytically price the options in the form of theorems. We verify the parameter sensitivity of pricing formulas by illustrations. We also elucidate the practical implications of an emission-trading scheme. Full article

Biological and financial models are examples of dynamical systems where both stochastic and historical behavior are important to be considered. The fractional Brownian motion (fBM) is commonly used, sometimes with fractional-order derivatives, to model the combined stochastic and fractional effects. Recently, spectral techniques are used to analyze models with fBM using, e.g., iterated Itô fractional integrals such as the fractional Wiener-Hermite (FWHE). In the current work, FWHE is generalized and adapted to be consistent with the Malliavin calculus approach. The conditions for existence and uniqueness are outlined in addition to the proof of convergence. The solution algorithm is described in detail. Using FWHE, the stochastic fractional model is replaced by a deterministic fractional-order system that can be handled using well-known mathematical tools to evaluate the solution statistics. Analytical solutions can be obtained for many important models such as the fractional stochastic Black–Scholes model. The convergence is studied and compared with the exact solution and high convergence is noticed compared with other techniques. A general numerical algorithm is described to analyze the resultant deterministic system in the case of no feasible analytical solutions. The algorithm is applied to study and simulate the population model with nonlinear losses for different values of the Hurst parameter. The results show the efficiency of FWHE in analyzing practical linear and nonlinear models. Full article

This paper investigates the mean-square stability of uncertain time-delay stochastic systems driven by G-Brownian motion, which are commonly referred to as G-SDDEs. To derive a new set of sufficient stability conditions, we employ the linear matrix inequality (LMI) method and construct a Lyapunov–Krasovskii function under the constraint of uncertainty bounds. The resulting sufficient condition does not require any specific assumptions on the G-function, making it more practical. Additionally, we provide numerical examples to demonstrate the validity and effectiveness of the proposed approach. Full article

Metal–organic frameworks (MOFs) with hierarchical porous structures have been attracting intense interest currently due to their promising applications in catalysis, energy storage, drug delivery, and photocatalysis. Current fabrication methods usually employ template-assisted synthesis or thermal annealing at high temperatures. However, large-scale production of hierarchical porous metal–organic framework (MOF) particles with a simple procedure and mild condition is still a challenge, which hampers their application. To address this issue, we proposed a gelation-based production method and achieved hierarchical porous zeolitic imidazolate framework-67 (called HP-ZIF67-G thereafter) particles conveniently. This method is based on a metal–organic gelation process through a mechanically stimulated wet chemical reaction of metal ions and ligands. The interior of the gel system is composed of small nano and submicron ZIF-67 particles as well as the employed solvent. The relatively large pore size of the graded pore channels spontaneously formed during the growth process is conducive to the increased transfer rate of substances within the particles. It is proposed that the Brownian motion amplitude of the solute is greatly reduced in the gel state, which leads to porous defects inside the nanoparticles. Furthermore, HP-ZIF67-G nanoparticles interwoven with polyaniline (PANI) exhibited an exceptional electrochemical charge storage performance with an areal capacitance of 2500 mF cm⁻², surpassing those of many MOF materials. This stimulates new studies on MOF-based gel systems to obtain hierarchical porous metal–organic frameworks which should benefit further applications in a wide spectrum of fields ranging from fundamental research to industrial applications. Full article

We studied the random variable $V_t={\mathrm{vol}}_{S^2}(g_{t}B\cap B)$, where B is a disc on the sphere $S^2$ centered at the north pole and ${\left(g_t\right)}_{t\ge 0}$ is the Brownian motion on the special orthogonal group $SO\left(3\right)$ starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of $V_t$ for $0\le t\le \tau$, where $\tau$ is the first time when $V_t$ vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ${\mathsf{\Gamma }}_B$ seen as the support of the function $f\left(g\right)={\mathrm{vol}}_{S^2}(gB\cap B)$ immersed in $SO\left(3\right)$. The integral formula depends on the mean curvature of ${\mathsf{\Gamma }}_B$ and the diameter of B. Full article

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values. Full article