and t ∈$_{R}$ sufficiently close to zero, we have that Proof. Let H$_{t}$ ( o ) ≜ 1 + th$_{1}$ ( o ) + t $^{2}$h$_{2}$ ( o ) and ˙ H$_{t}$ ( o ) ≜ h$_{1}$ ( o ) + 2 th$_{2}$ ( o ). We will pass the derivative inside the integral using (S2) and apply the product rule. The first term we need to consider is The second is Returning to (A.6), this shows that d 2 dt $_{2}$Φ $^{T U}$( P$_{t}$ ) ∣ ∣ t =0 is equal to 46 2$^{∫}$ ∫ [ − 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ˙ T$_{0}$ ( o$_{1}$ ) h$_{1}$ ( o$_{1}$ ) + 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) 2 ˙ T$_{0}$ ( o$_{1}$ ) 2 + h$_{2}$ ( o$_{1}$ ) − ˙ T$_{0}$ ( o$_{1}$ ) 2 − ( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ¨ T$_{0}$ ( o$_{1}$ ) $^{]}$e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) + 2$^{∫}$ ∫ $^{[}$2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ˙ U$_{0}$ ( o$_{2}$ ) h$_{1}$ ( o$_{2}$ ) + 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) 2 ˙ U$_{0}$ ( o$_{2}$ ) 2 + h$_{2}$ ( o$_{2}$ ) − ˙ U$_{0}$ ( o$_{2}$ ) 2 + ( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ¨ U$_{0}$ ( o$_{2}$ ) $^{]}$e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) d dt e − [ T$_{t}$ ( o$_{1}$ ) − U$_{t}$ ( o$_{2}$ )] $_{2}$∣ ∣ ∣ t =0 = − 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ( ˙ T$_{0}$ ( o$_{1}$ ) − ˙ U$_{0}$ ( o$_{2}$ ) $^{)}$e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )] $^{2}$. d dt$^{[}$ H$_{t}$ ( o$_{1}$ ) ˙ H$_{t}$ ( o$_{2}$ ) + ˙ H$_{t}$ ( o$_{1}$ ) H$_{t}$ ( o$_{2}$ ) − 2( T$_{t}$ ( o$_{1}$ ) − U$_{t}$ ( o$_{2}$ )) ( ˙ T$_{t}$ ( o$_{1}$ ) − ˙ U$_{t}$ ( o$_{2}$ ) $^{)}$H$_{t}$ ( o$_{1}$ ) H$_{t}$ ( o$_{2}$ ) ] ∣ ∣ t =0 = 2 [ h$_{2}$ ( o$_{1}$ ) + h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) + h$_{2}$ ( o$_{2}$ )] − 2 ( ˙ T$_{0}$ ( o$_{1}$ ) − ˙ U$_{0}$ ( o$_{2}$ ) 2 − 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) $^{(}$¨ T$_{0}$ ( o$_{1}$ ) − ¨ U$_{0}$ ( o$_{2}$ ) − 2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ( ˙ T$_{0}$ ( o$_{1}$ ) − ˙ U$_{0}$ ( o$_{2}$ ) $^{)}$( h$_{1}$ ( o$_{1}$ ) + h$_{1}$ ( o$_{2}$ )) . d 2 dt $_{2}$Φ $^{T U}$( P$_{t}$ ) ∣ ∣ t =0 = 2$^{∫}$ ∫ Γ T U 0 ( o$_{1}$, o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) + Π( T ) + Π( U ) . d 2 dt $_{2}$Φ $^{T U}$( P$_{t}$ ) ∣ ∣ t =0 = d dt$^{∫}$ ∫ $^{[}$H$_{t}$ ( o$_{1}$ ) ˙ H$_{t}$ ( o$_{2}$ ) + ˙ H$_{t}$ ( o$_{1}$ ) H$_{t}$ ( o$_{2}$ ) − 2( T$_{t}$ ( o$_{1}$ ) − U$_{t}$ ( o$_{2}$ )) ( ˙ T$_{t}$ ( o$_{1}$ ) − ˙ U$_{t}$ ( o$_{2}$ ) $^{)}$H$_{t}$ ( o$_{1}$ ) H$_{t}$ ( o$_{2}$ ) ] × e − [ T$_{t}$ ( o$_{1}$ ) − U$_{t}$ ( o$_{2}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) ∣ ∣ t =0 (A.6)