Question: Exercise 16.8 For the default time epoch defined in (16.17), let Show that the process {M(t)} is a martingale with respect to the filtration

Exercise 16.8 For the default time epoch τ defined in (16.17), let

M(t) = = 1{K(u)=0} [dK (u)-h(u)du]. 0

Show that the process {M(t)} is a martingale with respect to the filtration {Ft}. Hint: Since ∫ t 0 1{K(u−)=0}dK(u) = 1{τ≤t}, we have

image text in transcribed

Apply (16.18) to the case {τ > t}. The case {τ ≤ t} is obvious.

M(t) = = 1{K(u)=0} [dK (u)-h(u)du]. 0

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