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The fashion apparel industry is facing an increasingly growing demand, compounded by the short sales lifecycle and strong seasonality of clothing, posing significant challenges to inventory management in the retail sector. Despite some retailers like Uniqlo and Zara implementing inventory management and dynamic pricing strategies, challenges persist due to the dynamic nature of fashion trends and the stochastic factors affecting inventory. To address these issues, we construct a mathematical model based on the mathematical expression of the deterministic fashion level function, where the geometric Brownian motion, widely applied in finance, is initially utilized in the stochastic fashion level function. Drawing on research findings from deteriorating inventory management and stochastic optimization, we investigate the fluctuation of inventory levels, optimal dynamic pricing, optimal production rates, and profits—four crucial indicators—via Pontryagin’s maximum principle. Analytical solutions are derived, and the numerical simulation is provided to verify and compare the proposed model with deterministic fashion level function models. The model emphasizes the importance of considering stochastic factors in decision-making processes and provides insights to enhance profitability, inventory management, and sustainable consumption in the fashion product industry.

Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u ( t , x ) $u(t,x)$ . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0,where θ > 0 and ν∈R are two parameters. When θ < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.

Let SH be a sub-fractional Brownian motion with index 12

Abstract In this paper, we investigate the pth mean almost periodic solution to a neutral stochastic evolution equation with infinite delay and Poisson jumps. We give a sufficient condition for the existence and uniqueness of pth mean almost periodic solution and the condition depends on the continuity of coefficients and the power of a fractional operator. We give an example to illustrate the abstract results.

In this paper, we study a class of time-fractal-fractional stochastic differential equations with the fractal–fractional differential operator of Atangana under the meaning of Caputo and with a kernel of the power law type. We first establish the Hölder continuity of the solution of the equation. Then, under certain averaging conditions, we show that the solutions of original equations can be approximated by the solutions of the associated averaged equations in the sense of the mean square convergence. As an application, we provide an example with numerical simulations to explore the established averaging principle.

Let BH={BtH,t≥0} be a fractional Brownian motion with Hurst index 12≤H<1. In this paper, we consider the linear self-attracting diffusion: dXtH=dBtH+σXtHdt−θ∫0tXsH−XuHdsdt+νdt with X0H=0, where θ>0 and σ,ν∈R are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, Math. Ann.303 (1995), 87–93). Our main aim is to study the large time behaviors. We show that the solution t−σθHXtH−X∞H converges in distribution to a normal random variable, as t tends to infinity, and obtain two strong laws of large numbers associated with the solution XH.

Abstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 00 $t > 0$, where log+x=max{1,logx} $\log^{+}x=\max{\{1, \log x\}}$ for x≥0 $x\geq0$. As an application, we introduce the ΦH $\Phi_{H}$-variation of SH $S^{H}$ driven by ΦH(x):=[x/2log+log+(1/x)]1/H $\Phi_{H}(x):= [x/\sqrt{2\log^{+}\log ^{+}(1/x)} ]^{1/H}$ (x>0) $(x>0)$ with ΦH(0)=0 $\Phi_{H}(0)=0$.

Abstract Let U, V and W be three Hilbert spaces and let BH $B^{H}$ be a W-valued fractional Brownian motion with Hurst index H∈(12,1) $H\in(\frac{1}{2},1)$. In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation {Dtαc[Lx(t)]=Ax(t)+f(t,xt)+Bu(t)+σ(t)ddtBH(t),t∈(0,T],x(t)=ϕ(t),t∈(−∞,0], $$\textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), &t\in(0,T], \\ x(t)=\phi(t), &t\in(-\infty,0], \end{cases} $$ where Dαc ${}^{\mathrm{c}}D^{\alpha}$ is the Caputo fractional derivative of order α∈(1−H,1) $\alpha\in(1-H,1)$, the time history xt:(−∞,0]→xt(θ)=x(t+θ) $x_{t}:(-\infty,0]\rightarrow x_{t}(\theta)=x(t+\theta)$ with t>0 $t>0$ belonging to the phase space Bh $\mathscr{B}_{h}$, the control function u(⋅) $u(\cdot)$ is given in L2([0,T],V) $L^{2}([0,T],V)$, B is a bounded linear operator from V into U. Under some suitable conditions, we show that the system is approximately controllable on [0,T] $[0,T]$ and we give an example to illustrate the theory.

Abstract In this paper, we consider the stochastic heat equation of the form ∂ u ∂ t = Δ α u + ∂ 2 B ∂ t ∂ x , $$\frac{\partial u}{\partial t}=\Delta_{\alpha}u+\frac{\partial ^{2}B}{\partial t\,\partial x}, $$ where ∂ 2 B ∂ t ∂ x $\frac{\partial^{2}B}{\partial t\,\partial x}$ is a fractional Brownian sheet with Hurst indices H 1 , H 2 ∈ ( 1 2 , 1 ) $H_{1},H_{2}\in(\frac{1}{2},1)$ and Δ α = − ( − Δ ) α / 2 $\Delta _{\alpha}=-(-\Delta)^{\alpha/2}$ is a fractional Laplacian operator with 1 < α ≤ 2 $1

Abstract Let W = λ B + ν B H ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index 0 < H < 1 2 $0< H

In this paper, existence, uniqueness and continuity of the adapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition, continuous dependence on the initial date is also investigated. Finally, stochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.

We consider a class of neutral stochastic partial differential equations with infinite delay in real separable Hilbert spaces. We derive the existence and uniqueness of mild solutions under some local Carathéodory-type conditions and also exponential stability in mean square of mild solutions as well as its sample paths. Some known results are generalized and improved.

We consider the stochastic functional differential equations with finite delay driven by G-Brownian motion. Under the global Carathéodory conditions we prove the existence and uniqueness, and as an application, we price the European call option when the underlying asset's price follows such an equation.

Let SH be a subfractional Brownian motion with index 0

For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameter H>1/2, we prove an existence and uniqueness result for this equation under suitable assumptions.

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both the pth moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

We first present two convergence results about the second-order quadratic variations of the subfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self-similarity parameter associated with one-dimensional subfractional Brownian motion.

Being base on coupling by change of measure and an approximation technique, the Harnack inequalities for a class of stochastic functional differential equations driven by subordinate fractional Brownian motion with Hurst parameter 0 < H < 1/2 are established. By using a transformation formulas for fractional Brownian motion, the Harnack inequalities for stochastic functional differential equations driven by subordinate fractional Brownian motion with Hurst parameter 1/2 < H < 1 are established. [ABSTRACT FROM AUTHOR]