y , we can use the monotone convergence theorem to obtain Thus, due to the definition of an admissible pricing rule, we have As a consequence, we can express E [ X θ 1 ] in terms of V as (notice that ∫ t 0 B 2 $_{t}$dF ( Z$_{t}$, t ) is a martingale) On the other hand, by applying the Itˆ o formula for semimartingales to V directly ([14], Theorem II.6.33, p. 81) we get Notice that, due to the definition of the fundamental value F and of admissible pricing rule 14 E [ V ( Y$_{1}$, Z$_{1}$, 1)] = E [ V (0 , Z$_{0}$, 0) − X θ 1 + ∫ 1 0 V$_{y}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) dB 2 s − 1 2 ∫ 1 0 V$_{yy}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) d [ θ $^{c}$]$_{s}$ + ∑ s ≤ 1 [Δ V ( Y$_{s}$, Z$_{s}$, s ) − V$_{y}$ ( Y$_{s}$, Z$_{s}$, s )Δ Y$_{s}$ ]   . E [ X θ 1 ] = E [ − ∫ 1 0 V$_{y}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) dθ$_{s}$ − ∫ 1 0 V$_{z}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) dZ$_{s}$ − ∫ 1 0 V$_{zy}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) d [ θ $^{c}$, Z ]$_{s}$ − ∫ 1 0 V$_{yy}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) d [ θ $^{c}$]$_{s}$ − ∫ 1 0 V$_{yy}$ ( Y$_{s}$$_{−}$, Z$_{s}$, s ) d [ θ $^{c}$, B $^{2}$] s − ∑ s ≤ 1 Δ V$_{y}$ ( Y$_{s}$, Z$_{s}$, s )Δ θ$_{s}$   . V$_{y}$ ( y, z, t ) + F ( z, t ) − H ( y, t ) = 0 . (3.14) lim Δ → 0+ V ( y + Δ , z, t ) − V ( y, z, t ) Δ + F ( z, t ) − H ( y, t ) = 0 . (3.13) lim Δ → 0+ V ( y + Δ , z, t ) − V ( y, z, t ) Δ = lim Δ → 0+ E [ ∫ y + B 2 $_{1}$− B 2 t y +Δ+ B 2 $_{1}$− B 2 t ( f ( z + ∫ 1 t σ$_{z}$ ( s ) dB 1 $_{s}$) − H ( x, 1) ) dx ] Δ = − F ( z, t ) − E   lim Δ → 0+ ∫ y + B 2 $_{1}$− B 2 t y +Δ+ B 2 $_{1}$− B 2 t H ( x, 1) dx Δ   = E [ H ( y + B 2 1 − B 2 $_{t}$, 1) ] − F ( z, t ) . The same argument can be applied to the left derivative of V with respect to y to obtain