On the stochastic differentiability of noncausal processes with respect to the process with quadratic variation

Abstract

Let $(V_t{)}_{t\in [0,L]}$ be a stochastic process with quadratic variation on a probability space $(\mathrm{\Omega },\mathcal{F},P)$ and $Q(\ni 0)$ a dense subset of $[0,L]$, where $[0,L]$ is regarded as the infinite interval $[0,\mathrm{\infty })$ when $L=\mathrm{\infty }$. First, we introduce the $L^0\left(\mathrm{\Omega }\right)$-module ${\mathcal{D}}_Q\left(V\right)$ of V-differentiable noncausal processes on Q and V-derivative operator ${\mathcal{D}}_{V,Q}=\frac{d_Q}{d_{Q}V}$ defined on ${\mathcal{D}}_Q\left(V\right)$, which enjoys the modularity: ${\mathcal{D}}_{V,Q}(\alpha X+\text{β}Y)=\alpha {\mathcal{D}}_{V,Q}X+\text{β}{\mathcal{D}}_{V,Q}Y$ for any $X,Y\in {\mathcal{D}}_Q\left(V\right)$ and $\alpha ,\text{β}\in L^0\left(\mathrm{\Omega }\right)$. Second, we show that the class ${\mathcal{Q}}_{V,Q}=\{X\in {\mathcal{D}}_Q\left(V\right)|\frac{\mathrm{d}[X{]}_{Q,t}}{\mathrm{d}[V{]}_{Q,t}}={\left|\frac{d_Q{X}_t}{d_Q{V}_t}\right|}^2\}$ forms an $L^0\left(\mathrm{\Omega }\right)$-module, where $[{]}_{Q,t}$ stands for the quadratic variation on Q. As a result, we have the isometry: ${〈X,Y〉}_{Q,t}={〈{\mathcal{D}}_{V,Q}X,{\mathcal{D}}_{V,Q}Y〉}_{L^2\left(\right[0,t],[V\left]\right)}$ for any $X,Y\in {\mathcal{Q}}_{V,Q}$, where ${〈,〉}_{Q,t}$ stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with ${\mathcal{D}}_{V,Q}\circ I={\mathrm{i}\mathrm{d}}_{D\left(I\right)}$. This result is essentially used for solving the identification problem from the stochastic Fourier coefficients.

Keywords:

• Stochastic derivative
• stochastic integral
• quadratic variation
• quadratic covariation
• noncausal
• anticipative

MSC2020 subject classifications:

• 60H05
• 60H07

Acknowledgments

The author would like to express his grate gratitude to Professor Tetsuya Kazumi for supervising to draw this paper and to Professor Shigeyoshi Ogawa and Professor Hideaki Uemura for paying kind attention and giving me some advice and comments to this study on several occasions and to Professor Masanori Hino for giving some comments and advice at an opportunity to present this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).