Moreover, due to the properties of V we have X$_{t}$ = G ( Z$_{t}$, t ) = n ∑ i =1 E [ h − 1 i ( F ( Z$_{t}$$_{i}$, t$_{i}$ )) |F I t ] 1$_{{}$$_{t}$$_{∈}$$_{(}$$_{t}$$_{i}$$_{−}$$_{1}$$_{,t}$$_{i}$$_{]}$$_{}}$ ξ$_{t}$ = X$_{t}$ − ( t$_{i}$ − t ) ( X$_{t}$$_{i}$$_{−}$$_{1}$ − ξ$_{t}$$_{i}$$_{−}$$_{1}$ t$_{i}$ t$_{i}$$_{−}$$_{1}$ + ∫ t t$_{i}$$_{−}$$_{1}$ 1 t$_{i}$ − s dX$_{s}$ − ∫ t t$_{i}$$_{−}$$_{1}$ σ i y t$_{i}$ − s dB 2 s ) . ∑ s ≤ 1 (Δ V ( ξ$_{s}$, Z$_{s}$, s ) − V$_{ξ}$ ( ξ$_{s}$, Z$_{s}$, s )Δ ξ$_{s}$ ) ≤ 0 , (4.5) − ∫ 1 0 V$_{ξξ}$ ( ξ$_{s}$, Z$_{s}$, s ) w $^{2}$( s ) 2 d [ θ $^{c}$]$_{s}$ ≤ 0 , (4.6) − J $^{i}$( t$_{i}$, ξ$_{t}$$_{i}$, Z$_{t}$$_{i}$ ) ≤ 0 . (4.7) 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}$$_{i}$, t$_{i}$ ) for equations (4.7). Therefore, we have that This expression holds with equality if and only if θ is continuous with bounded variation and condition (4.2) is satisfied. Hence, if θ is such that these conditions are satisfied, then it is optimal. E [ X θ 1 ] ≤ V (0 , Z$_{0}$, 0) . (Necessity) Consider the process given by with X (0) = E [ h − 1 1 ( F ( Z$_{t}$$_{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 Notice that ξ$_{t}$ is finite almost surely, therefore θ$_{t}$ has bounded variation almost surely. More- 23