content_image/297638.jpg content_image/297641.jpg content_image/297636.jpg content_image/297637.jpg
Moreover, due to the properties of V we have 

<img src='content_image/297638.jpg'>

The above inequalities become equalities if and only if the following conditions hold: Δ θ = 0 for equation (4.5); [ θ $^{c}$]$_{1}$ = 0 for equation (4.6); H ( ξ ∗ $_{t$_{i}$}$, t$_{i}$ ) = F ( Z$_{t}$$_{i}$, t$_{i}$ ) for equations (4.7). Therefore, we have that 

<img src='content_image/297641.jpg'>

This expression holds with equality if and only if θ is continuous with bounded variation and condition (4.2) is satisfied. Hence, if θ is such that these conditions are satisfied, then it is optimal. 

(Necessity) Consider the process given by 

<img src='content_image/297636.jpg'>

with X (0) = E [ h − 1 1 ( F ( Z$_{t}$$_{1}$, t$_{1}$ )) |F I 0 ] where h − 1 i is inverse of H ( y, t$_{i}$ ). This process is well defined since H is an admissible pricing rule. 

Consider the trading strategy given by θ$_{0}$ = 0 and dθ$_{t}$ = ∑ n i =1 X$_{t}$ − ξ$_{t}$ σ i $_{y}$( t$_{i}$ − s $_{)}$1$_{{}$$_{t}$$_{∈}$$_{(}$$_{t}$$_{i}$$_{−}$$_{1}$$_{,t}$$_{i}$$_{]}$$_{}}$ dt . In this case, we can solve the stochastic differential equation for ξ on each interval [ t$_{i}$$_{−}$$_{1}$, t$_{i}$ ] to get 

<img src='content_image/297637.jpg'>

Notice that ξ$_{t}$ is finite almost surely, therefore θ$_{t}$ has bounded variation almost surely. More- 

