content_image/316010.jpg content_image/316009.jpg content_image/316008.jpg content_image/316007.jpg content_image/316006.jpg
Define the function V as 

<img src='content_image/316010.jpg'>

Notice that, due to the properties of the functions V $^{i}$, we have that V is well defined and satisfies the partial differential equation 

<img src='content_image/316009.jpg'>

with conditions V ( ξ, z, t$_{i}$ ) = ( 1 σ i y − 1 σ i +1 y ) J $^{i}$( ξ, z ) + V ( ξ, z, t$_{i}$ +) if i < n and V ( ξ, z, t$_{n}$ ) = 1 σ n y J $^{n}$( ξ, z ). Moreover, we have 

<img src='content_image/316008.jpg'>

As a consequence, we can express E [ X θ 1 ] in terms of V as (notice that ∫ t 0 ∫ u 0 w ( s ) dB 2 $_{s}$dF ( Z$_{u}$, u ) is a martingale) 

<img src='content_image/316007.jpg'>

On the other hand, by applying the Itˆ o formula for semimartingales to V directly ([14], 

Theorem II.6.33, p. 81) and removing martingale terms we get 

<img src='content_image/316006.jpg'>

