content_image/325521.jpg content_image/325520.jpg content_image/325518.jpg content_image/325519.jpg content_image/325517.jpg
the fact that H ( y, t ) satisfies equation (3.7), we have that 

<img src='content_image/325521.jpg'>

Therefore, by Theorem 26.6 of [8], and Theorem II.6.29 of [14], we have (notice that Z$_{s}$ and B 2 t are continuous) 

<img src='content_image/325520.jpg'>

On the other hand, consider a function 

where y $^{∗}$( z ) is the solution of H ( y $^{∗}$( z ) , 1) = f ( z ). Let 

<img src='content_image/325518.jpg'>

<img src='content_image/325519.jpg'>

This function is well defined (it is easy to check that E [ | J ( B 2 $_{1}$, Z$_{1}$ ) | ] < ∞ ) and satisfies the partial differential equation 

<img src='content_image/325517.jpg'>

with terminal condition V ( y, z, 1) = J ( y, z ). Therefore V ( y, z, 1) ≥ V ( y $^{∗}$( z ) , z, 1) = 0 for any fixed z and any y ̸ = y $^{∗}$( z ). Moreover, since H ( y, 1) is a nondecreasing continuous function of 

