Define the function V as Notice that, due to the properties of the functions V $^{i}$, we have that V is well defined and satisfies the partial differential equation with conditions V ( ξ, z, t$_{i}$ ) = ( 1 σ i y − 1 σ i +1 y ) J $^{i}$( ξ, z ) + V ( ξ, z, t$_{i}$ +) if i < n and V ( ξ, z, t$_{n}$ ) = 1 σ n y J $^{n}$( ξ, z ). Moreover, we have As a consequence, we can express E [ X θ 1 ] in terms of V as (notice that ∫ t 0 ∫ u 0 w ( s ) dB 2 $_{s}$dF ( Z$_{u}$, u ) 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) and removing martingale terms we get 22 E [ X θ 1 ] = E [ V (0 , Z$_{0}$, 0) − n − 1 ∑ i =1 ( 1 σ i y − 1 σ i +1 y ) J $^{i}$( ξ$_{t}$$_{i}$, Z$_{t}$$_{i}$, t$_{i}$ ) − 1 σ n y J $^{n}$( ξ$_{t}$$_{i}$, Z$_{t}$$_{i}$, t$_{i}$ ) − 1 2 ∫ 1 0 V$_{ξξ}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) w $^{2}$( s ) d [ θ $^{c}$]$_{s}$ + ∑ s ≤ 1 [Δ V ( ξ$_{s}$, Z$_{s}$, s ) − V$_{ξ}$ ( ξ$_{s}$, Z$_{s}$, s )Δ ξ$_{s}$ ]   . E [ X θ 1 ] = E [ − ∫ 1 0 V$_{ξ}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) w ( s ) dθ$_{s}$ − ∫ 1 0 V$_{z}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) dZ$_{s}$ − ∫ 1 0 V$_{zξ}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) w ( s ) d [ θ $^{c}$, Z ]$_{s}$ − 1 0 V$_{ξξ}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) w $^{2}$( s ) d [ θ $^{c}$]$_{s}$ − ∫ 1 0 V$_{ξξ}$ ( ξ$_{s}$$_{−}$, Z$_{s}$, s ) w $^{2}$( s ) d [ θ $^{c}$, B $^{2}$] s − ∑ s ≤ 1 Δ( w ( s ) V$_{ξ}$ ( ξ$_{s}$, Z$_{s}$, s ))Δ θ$_{s}$   . V$_{ξ}$ ( ξ, z, t ) + F ( z, t ) − H ( ξ, t ) w ( t ) = 0 . (4.4) V$_{t}$ ( ξ, z, t ) + w $^{2}$( s ) 2 V$_{ξξ}$ ( ξ, z, t ) + σ 2 $_{z}$( t ) 2 V$_{zz}$ ( ξ, z, t ) = 0 , (4.3) V ( ξ, z, t ) = ∑ i<n : t ≤ t$_{i}$ ( 1 σ i y − 1 σ i +1 y ) V $^{i}$( ξ, z, t ) + 1 σ n y V $^{n}$( ξ, z, t )