论文标题
连续时间随机建模中的红噪声
Red noise in continuous-time stochastic modelling
论文作者
论文摘要
时间相关噪声的概念对于应用随机建模很重要。然而，在连续时间随机建模设置中，对红色噪声一词尚无一致的定义。我们在这里提出了针对时间（$ u_t \ mathrm {d} t $）集成的Ornstein-uhlenbeck过程的严格论证，作为独特的红色噪声实现。我们还将一词$ \ mathrm {d} u_t $确定为应用文献中常见的红噪声的错误公式。为此，我们证明了幂谱密度（PSD）与ITô-差异类别的定理链接属性。 PSD衰减的常见红噪声属性为$ s（ω）\simΩ^{ - 2} $限制了可能的Itô-Differentials $ \ Mathrm {d} y_t =α__T=α__T\ Mathrm {d} t+β_t\β_T\ mathrm {D} t+β_t\ Mathrm {d} w_t $ {特别是，任何具有连续的，正方形的综合集合的差异都必须具有消失的martingale部分，即$ \ mathrm {d} y_t =α_t=α_t\ mathrm {d} t $几乎所有$ t \ geq 0 $。我们进一步指出，作为Ornstein-uhlenbeck过程，服用$（α_T）_ {t \ geq 0} $，由于其高斯 - 马尔科夫（Gauss-Markov）属性，构成了独特的模型选择。在文献中的两个示例中讨论了噪声项$ \ mathrm {d} u_t $作为红噪声及其后果的错误使用。
The concept of time-correlated noise is important to applied stochastic modelling. Nevertheless, there is no generally agreed-upon definition of the term red noise in continuous-time stochastic modelling settings. We present here a rigorous argumentation for the Ornstein-Uhlenbeck process integrated against time ($U_t \mathrm{d} t$) as a uniquely appropriate red noise implementation. We also identify the term $\mathrm{d}U_t$ as an erroneous formulation of red noise commonly found in the applied literature. To this end, we prove a theorem linking properties of the power spectral density (PSD) to classes of Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S(ω)\simω^{-2}$ restricts the range of possible Itô-differentials $\mathrm{d}Y_t=α_t\mathrm{d} t+β_t\mathrm{d} W_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}Y_t=α_t\mathrm{d} t$ for almost all $t\geq 0$. We further point out that taking $(α_t)_{t\geq 0}$ to be an Ornstein-Uhlenbeck process constitutes a uniquely relevant model choice due to its Gauss-Markov property. The erroneous use of the noise term $\mathrm{d} U_t$ as red noise and its consequences are discussed in two examples from the literature.
