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H , we have E [ ∫ 1 0 V$_{z}$ ( Y$_{s}$, Z$_{s}$, s ) dB 2 s ] = 0. Therefore 

Moreover, due to the properties of V we have 

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<img src='content_image/315522.jpg'>

<img src='content_image/315523.jpg'>

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The above inequalities become equalities if and only if the following conditions hold: Δ θ = 0 for equation (3.15); [ θ $^{c}$]$_{1}$ = 0 for equation (3.16); H ( Y ∗ 1 , 1) = f ( Z$_{1}$ ) for equation (3.17). 

Therefore, for any function V satisfying equations (3.12), (3.14) and the final condition given by V ( y, z, 1) ≥ V ( y $^{∗}$( z ) , z, 1) = 0 for every z and any y ̸ = y $^{∗}$( z ) (where y $^{∗}$( z ) is the solution of H ( y $^{∗}$( z ) , 1) = f ( z )), we have that 

<img src='content_image/315521.jpg'>

This expression holds with equality if and only if θ is continuous and condition (3.10) is satisfied. Hence, if θ is such that these conditions are satisfied, then it is optimal. 

(Necessity) Consider the continuous martingale given by 

This martingale is well defined since H is an admissible pricing rule. 

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Consider θ$_{t}$ = ∫ t 0 X$_{s}$ − Y$_{s}$ 1 − s ds . In this case, we can solve the stochastic differential equation for 

