Stochastic Differential Equation with Random Coefficients

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Let $b, sigma : [0,T] times Omega times mathbbR rightarrow mathbbR,$ be the drift and diffusion coefficients for a SDE.The typical assumptions for the existence of a solution are the following.



  1. For all $ x in mathbbR$ the processes $(t,omega) mapsto b(t,omega, x), (t,omega) mapsto sigma(t,omega,x)$ are progressively measurable and Lebesgue square integrable.


  2. There exits $ C > 0 $ such that for all $omega, t, x_1, x_2 $ we have
    $$ | b(t,omega, x_1) - b(t,omega, x_2) | + | sigma(t, omega, x_1) - sigma( t, omega, x_2 ) | leq K | x_1 - x_2 |. $$


My question is as follows. For the integrals to be well defined we need that the processes
$$ (t, omega) mapsto b(t,omega, X_t(omega) ) $$
and
$$ (t, omega) mapsto sigma(t,omega, X_t(omega) ) $$
are progressively measurable. This should follow from 1. and 2. but I was not able to prove it. I hope someone can help me out.




measure-theory stochastic-calculus sde




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    up vote
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    down vote

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    Let $b, sigma : [0,T] times Omega times mathbbR rightarrow mathbbR,$ be the drift and diffusion coefficients for a SDE.The typical assumptions for the existence of a solution are the following.



    1. For all $ x in mathbbR$ the processes $(t,omega) mapsto b(t,omega, x), (t,omega) mapsto sigma(t,omega,x)$ are progressively measurable and Lebesgue square integrable.


    2. There exits $ C > 0 $ such that for all $omega, t, x_1, x_2 $ we have
      $$ | b(t,omega, x_1) - b(t,omega, x_2) | + | sigma(t, omega, x_1) - sigma( t, omega, x_2 ) | leq K | x_1 - x_2 |. $$


    My question is as follows. For the integrals to be well defined we need that the processes
    $$ (t, omega) mapsto b(t,omega, X_t(omega) ) $$
    and
    $$ (t, omega) mapsto sigma(t,omega, X_t(omega) ) $$
    are progressively measurable. This should follow from 1. and 2. but I was not able to prove it. I hope someone can help me out.




    measure-theory stochastic-calculus sde




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $b, sigma : [0,T] times Omega times mathbbR rightarrow mathbbR,$ be the drift and diffusion coefficients for a SDE.The typical assumptions for the existence of a solution are the following.



      1. For all $ x in mathbbR$ the processes $(t,omega) mapsto b(t,omega, x), (t,omega) mapsto sigma(t,omega,x)$ are progressively measurable and Lebesgue square integrable.


      2. There exits $ C > 0 $ such that for all $omega, t, x_1, x_2 $ we have
        $$ | b(t,omega, x_1) - b(t,omega, x_2) | + | sigma(t, omega, x_1) - sigma( t, omega, x_2 ) | leq K | x_1 - x_2 |. $$


      My question is as follows. For the integrals to be well defined we need that the processes
      $$ (t, omega) mapsto b(t,omega, X_t(omega) ) $$
      and
      $$ (t, omega) mapsto sigma(t,omega, X_t(omega) ) $$
      are progressively measurable. This should follow from 1. and 2. but I was not able to prove it. I hope someone can help me out.




      measure-theory stochastic-calculus sde




      share|cite|improve this question











      Let $b, sigma : [0,T] times Omega times mathbbR rightarrow mathbbR,$ be the drift and diffusion coefficients for a SDE.The typical assumptions for the existence of a solution are the following.



      1. For all $ x in mathbbR$ the processes $(t,omega) mapsto b(t,omega, x), (t,omega) mapsto sigma(t,omega,x)$ are progressively measurable and Lebesgue square integrable.


      2. There exits $ C > 0 $ such that for all $omega, t, x_1, x_2 $ we have
        $$ | b(t,omega, x_1) - b(t,omega, x_2) | + | sigma(t, omega, x_1) - sigma( t, omega, x_2 ) | leq K | x_1 - x_2 |. $$


      My question is as follows. For the integrals to be well defined we need that the processes
      $$ (t, omega) mapsto b(t,omega, X_t(omega) ) $$
      and
      $$ (t, omega) mapsto sigma(t,omega, X_t(omega) ) $$
      are progressively measurable. This should follow from 1. and 2. but I was not able to prove it. I hope someone can help me out.





      measure-theory stochastic-calculus sde





      share|cite|improve this question










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