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 ] &lt; ∞ 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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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    ]  &amp;lt;  ∞ 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?

hodowlanyb1rq2 · Accepted Answer

While it&#039;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&#039;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&#039;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  .

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

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