Consider the following stochastic differential equation: d Y t </msub> =

Jaylene Duarte 2022-05-15 Answered
Consider the following stochastic differential equation:
d Y t = Z t d W t
and terminal condition Y T = b , for which holds: E [ | b | 2 ] < Furthermore b is adapted to the filtration generated by the Browian motion only at terminal time T.
Z t is a predictable square integrable process. So the right hand side Z t d W t is martingale.
Why is the solution Y t adapted to the underlying filtration If I rewrite the equation, I get:
Y t = Y T t T Z s d W s = b t T Z s d W s
Since b is only mb w.r.t to terminal time T, Y t cannot be adapted.
Where did I do a mistake?
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Answers (1)

hodowlanyb1rq2
Answered 2022-05-16 Author has 12 answers
While it's true that b and t T Z s d W s are not F t measurable, b t T Z s d W s is F t measurable. That's because
b t T Z s d W s = Y T t T Z s d W s = 0 T Z s d W s t T Z s d W s = 0 t Z s d W s .
If you're confused about why there is a solution to this backwards stochastic differential equation, note that Y t := E [ b | F t ] is a martingale satisfying Y T = b, so by the martingale representation theorem there exists an adapted process Z such that d Y t = Z t d W t .

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