M$_{1}$ = κ ∫ ∞ µ exp { ty } Φ ( y ∗ α − αp$_{α}$$_{−}$ ) exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 ( αp$_{α}$$_{−}$ ) 2 } dy, M$_{2}$ = − κ ∫ ∞ µ exp { ty } Φ( − αp$_{α}$$_{−}$ ) exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 ( αp$_{α}$$_{−}$ ) 2 } dy, (A.19) M$_{3}$ = κ ∫ ∞ µ exp { ty } Φ ( αp$_{α}$$_{+}$ − y ∗ α ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 ( αp$_{α}$$_{+}$ ) 2 } dy, M$_{4}$ = κ ∫ µ $_{−∞}$exp { ty } Φ ( αp$_{α}$$_{+}$ ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 ( αp$_{α}$$_{+}$ ) 2 } dy. (A.20) and when α > 0 and y ∗ ≤ 0 (i.e., y ≤ µ ) we have, We first consider M$_{1}$ , substitute z = y ∗ α − αp$_{α}$$_{−}$ , change the limits of integration and integrate- by-parts as follows: M$_{1}$ = ∫ ∞ − αp$_{α}$$_{−}$ κασ exp { µt + ασ ( z + αp$_{α}$$_{−}$ ) t } Φ( z ) exp { − αp$_{α}$$_{−}$ ( z + αp$_{α}$$_{−}$ ) + 1 2 α $^{2}$p2 α$_{−}$ } dz = κασ exp { µt + α $^{2}$p$_{α}$$_{−}$ σt − 1 2 α $^{2}$p2 α$_{−}$ } ∫ ∞ − αp$_{α}$$_{−}$ Φ( z ) exp { − α ( p$_{α}$$_{−}$ − σt ) z } dz = κασ exp { µt + α $^{2}$p$_{α}$$_{−}$ σt − 1 2 α $^{2}$p2 α$_{−}$ }   Φ( z ) exp {− α ( p$_{α}$$_{−}$ − σt ) z } − α ( p$_{α}$$_{−}$ σt ) ∣ ∣$^{∞}$ − αp$_{α}$$_{−}$ − ∫ ∞ − αp$_{α}$$_{−}$ φ ( z ) exp {− α ( p$_{α}$$_{−}$ − σt ) z } − α ( p$_{α}$$_{−}$ σt ) dz   (limit exists only if t < p/σ ) = κσ exp { µt + α $^{2}$p$_{α}$$_{−}$ σt − 1 $_{2}$α$^{2}$p 2 α$_{−}$ } ( p$_{α}$$_{−}$ − σt )   exp { α $^{2}$p$_{α}$$_{−}$ ( p$_{α}$$_{−}$ − σt ) } Φ( − αp$_{α}$$_{−}$ ) + exp { α 2 2 ( p$_{α}$$_{−}$ − σt ) 2 } ∫ ∞ − αp$_{α}$$_{−}$ 1 √$_{2}$$_{π}$ exp { − 1 2 ( z + α ( p$_{α}$$_{−}$ − σt ) 2 ) } dz   = κσ exp { µt + 1 $_{2}$α$^{2}$p 2 α$_{−}$ } ( p$_{α}$$_{−}$ − σt ) Φ( − αp$_{α}$$_{−}$ ) + κσ exp { µt + 1 $_{2}$α$^{2}$σ$^{2}$t 2 } ( p$_{α}$$_{−}$ − σt ) Φ ( z + α ( p$_{α}$$_{−}$ − σt ) ) ∣ ∣$^{∞}$ − αp$_{α}$$_{−}$ = κσ exp { µt + 1 $_{2}$α$^{2}$p 2 α$_{−}$ } ( p$_{α}$$_{−}$ − σt ) Φ( − αp$_{α}$$_{−}$ ) + κσ exp { µt + 1 $_{2}$α$^{2}$σ$^{2}$t 2 } ( p$_{α}$$_{−}$ − σt ) Φ( ασt ) . (A.21) 34 depending on y ∗ > 0 (i.e., y > µ ) or y ∗ ≤ 0 (i.e., y ≤ µ ). Case (i) : When α > 0 and y ∗ > 0 (i.e., y > µ ), we have the following three components,