62.11. HITTING THIS BEFORE THAT 2109

It follows

aP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa])≤

0≤ bP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa]) (62.11.43)

Note that [τb < τa] , [τa < τb]⊆ [M∗ > 0] and so

[τb < τa]∪ [τa < τb]∪ ([τa = τb]∩ [M∗ > 0]) = [M∗ > 0] (62.11.44)

The following diagram may help in keeping track of the various substitutions.

[τa < τb] [τb < τa] [τb = τa]∩ [M∗ > 0]

[M∗ > 0]

Left side of 62.11.43

From 62.11.44, this yields on substituting for P([τa < τb])

0 ≥ aP([τa = τb]∩ [M∗ > 0])+a [P([M∗ > 0])−P([τa ≥ τb]∩ [M∗ > 0])]+bP([τb < τa])

and so since [τa ̸= τb]⊆ [M∗ > 0] ,

0≥ a [P([M∗ > 0])−P([τa > τb])]+bP([τb < τa])

−aP([M∗ > 0])≥ (b−a)P([τb < τa]) (62.11.45)

Next use 62.11.44 to substitute for P([τb < τa])

0≥ aP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa])

= aP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])

+b [P([M∗ > 0])−P([τa ≤ τb]∩ [M∗ > 0])]

= aP([τa ≤ τb]∩ [M∗ > 0])+b [P([M∗ > 0])−P([τa ≤ τb]∩ [M∗ > 0])]

and so(b−a)P([τa ≤ τb])≥ bP([M∗ > 0]) (62.11.46)

Right side of 62.11.43

From 62.11.44, used to substitute for P([τa < τb]) this yields

0≤ bP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa])

= bP([τa = τb]∩ [M∗ > 0])+a [P([M∗ > 0])−P([τa ≥ τb]∩ [M∗ > 0])]+bP([τb < τa])