Integral over conditional expectation

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Suppose that $(t,x)mapsto g(t, x)$ is continuous in $tin [0,1]$ and $mathsfEsup_t |g(t,X)|<infty$, where $X$ is some random variable living on $(Omega,mathcalH,mathsfP)$. Is the following integral well-defined?
$$
int_0^1 mathsfE[g(t,X)mid mathcalF]dt,
$$
where $mathcalFsubsetmathcalH$. If $Y(t,omega)=mathsfE[g(t,X)mid mathcalF](omega)$, then $Y_t:0le tle 1 $ is a stochastic process. So I need to use Kolmogorov's continuity criterion to show that there exists an everywhere continuous modification $tildeY_t$. Then the integral under question is well defined if I replace $Y_t$ with $tildeY_t$. Is this reasoning correct?




probability-theory stochastic-processes conditional-expectation




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    Suppose that $(t,x)mapsto g(t, x)$ is continuous in $tin [0,1]$ and $mathsfEsup_t |g(t,X)|<infty$, where $X$ is some random variable living on $(Omega,mathcalH,mathsfP)$. Is the following integral well-defined?
    $$
    int_0^1 mathsfE[g(t,X)mid mathcalF]dt,
    $$
    where $mathcalFsubsetmathcalH$. If $Y(t,omega)=mathsfE[g(t,X)mid mathcalF](omega)$, then $Y_t:0le tle 1 $ is a stochastic process. So I need to use Kolmogorov's continuity criterion to show that there exists an everywhere continuous modification $tildeY_t$. Then the integral under question is well defined if I replace $Y_t$ with $tildeY_t$. Is this reasoning correct?




    probability-theory stochastic-processes conditional-expectation




    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Suppose that $(t,x)mapsto g(t, x)$ is continuous in $tin [0,1]$ and $mathsfEsup_t |g(t,X)|<infty$, where $X$ is some random variable living on $(Omega,mathcalH,mathsfP)$. Is the following integral well-defined?
      $$
      int_0^1 mathsfE[g(t,X)mid mathcalF]dt,
      $$
      where $mathcalFsubsetmathcalH$. If $Y(t,omega)=mathsfE[g(t,X)mid mathcalF](omega)$, then $Y_t:0le tle 1 $ is a stochastic process. So I need to use Kolmogorov's continuity criterion to show that there exists an everywhere continuous modification $tildeY_t$. Then the integral under question is well defined if I replace $Y_t$ with $tildeY_t$. Is this reasoning correct?




      probability-theory stochastic-processes conditional-expectation




      share|cite|improve this question













      Suppose that $(t,x)mapsto g(t, x)$ is continuous in $tin [0,1]$ and $mathsfEsup_t |g(t,X)|<infty$, where $X$ is some random variable living on $(Omega,mathcalH,mathsfP)$. Is the following integral well-defined?
      $$
      int_0^1 mathsfE[g(t,X)mid mathcalF]dt,
      $$
      where $mathcalFsubsetmathcalH$. If $Y(t,omega)=mathsfE[g(t,X)mid mathcalF](omega)$, then $Y_t:0le tle 1 $ is a stochastic process. So I need to use Kolmogorov's continuity criterion to show that there exists an everywhere continuous modification $tildeY_t$. Then the integral under question is well defined if I replace $Y_t$ with $tildeY_t$. Is this reasoning correct?





      probability-theory stochastic-processes conditional-expectation





      share|cite|improve this question












      share|cite|improve this question




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      edited Jul 17 at 4:46
























      asked Jul 17 at 4:23









      Robert W.

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