content_image/320625.jpg content_image/320624.jpg content_image/320623.jpg content_image/320622.jpg content_image/320621.jpg content_image/320620.jpg
On the other hand, if lim$_{t}$$_{→}$$_{t}$$_{1}$ ∫ t 0 α 2 $_{1}$+ σ 2 $_{z}$( s ) λ $^{2}$( s ) ds = ∞ , consider the process 

and a change of time τ ( t ) given by 

<img src='content_image/320625.jpg'>

<img src='content_image/320624.jpg'>

Then, W$_{s}$ = X$_{τ}$$_{(}$$_{s}$$_{)}$ is a Brownian motion. Hence, we can use the law of iterated logarithm to get 

or, in the original time, 

<img src='content_image/320623.jpg'>

<img src='content_image/320622.jpg'>

where Ξ( t ) = ∫ t 0 α 2 $_{1}$+ σ 2 $_{z}$( s ) λ $^{2}$( s ) ds . Due to the conditions (4.8)-(4.11) in this case we have 

therefore it follows that lim$_{t}$$_{→}$$_{t}$$_{1}$ λ ( t ) X$_{t}$ = 0, thus ξ$_{t}$$_{1}$ = α$_{1}$Y$_{t}$$_{1}$ = Z$_{t}$$_{1}$ . 

<img src='content_image/320621.jpg'>

II) Suppose α$_{j}$ ( Y$_{t}$$_{j}$ − Y$_{t}$$_{j}$$_{−}$$_{1}$ ) = Z$_{t}$$_{j}$ − Z$_{t}$$_{j}$$_{−}$$_{1}$ for any j ≤ i . Consider the interval [ t$_{i}$, t$_{i}$$_{+1}$ ]. At t = t$_{i}$ we have ξ$_{t}$$_{i}$ = Z$_{t}$$_{i}$ , and ˜ Y$_{t}$ = Y$_{t}$ − Y$_{t}$$_{i}$ satisfies the following stochastic differential equation on [ t$_{i}$, t$_{i}$$_{+1}$ ]: 

<img src='content_image/320620.jpg'>

