Small deviations for admixture additive & multiplicative processes

Abstract

Abstract Define the admixture additive processes Xγ,H,αa1,a2,a3,a4(t)≜a1B(t1)+a2Wγ(t2)+a3BH(t3)+a4Sα(t4)∈R, $$\mathbb{X}^{a_{1}, a_{2}, a_{3}, a_{4}}_{\gamma,H,\alpha}(\mathrm{t})\triangleq a_{1}B(t_{1})+a_{2}W_{\gamma}(t_{2})+a_{3}B_{H}(t_{3})+a_{4}S_{\alpha}(t_{4}) \in\mathbb{R}, $$ and the admixture multiplicative processes Yγ,H,α(t)≜B(t1)⋅Wγ(t2)⋅BH(t3)⋅Sα(t4)∈R, $$\mathbb{Y}_{\gamma,H,\alpha}(\mathrm{t})\triangleq B(t_{1})\cdot W_{\gamma}(t_{2})\cdot B_{H}(t_{3})\cdot S_{\alpha}(t_{4})\in\mathbb{R}, $$ where t=(t1,t2,t3,t4)∈R+4,a1,a2,a3,a4 $\mathrm{t}=(t_{1},t_{2},t_{3},t_{4})\in\mathbb{R}_{+}^{\mathrm{4}},a_{1},a_{2},a_{3},a_{4}$ are finite constants, B(t1) $B(t_{1})$ is the standard Brownian motion, Wγ(t2) $W_{\gamma}(t_{2})$ is the fractional integrated Brownian motion with index parameter γ>−1/2 $\gamma>-1/2$, BH(t3) $B_{H}(t_{3})$ is the fractional Brownian motion with Hurst parameter H∈(0,1) $H\in(0,1)$, Sα(t4) $S_{\alpha}(t_{4})$ is the stable process with index α∈(0,2] $\alpha\in(0,2]$, and they are independent of each other. The small deviation for Xγ,H,αa1,a2,a3,a4(t) $\mathbb{X}^{a_{1}, a_{2}, a_{3}, a_{4}}_{\gamma,H,\alpha}(\mathrm{t})$ and the lower bound of small deviation for Yγ,H,α(t) $\mathbb{Y}_{\gamma,H,\alpha}(\mathrm{t})$ are obtained. As an application, limit inf type LIL is given for Xγ,H,αa1,a2,a3,a4(t) $\mathbb{X}^{a_{1}, a_{2}, a_{3}, a_{4}}_{\gamma,H,\alpha}(\mathrm{t})$.

Keywords

• Small deviation
• Admixture additive process
• Admixture multiplicative process
• Limit theorem