Order in probability of a ratio between two integrals











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Suppose that $mu$ is an adapted bounded stochastic process and suppose that $sigma$ is an adapted bounded left-continuous stochastic process. I want to prove that



$$
frac{int_0^{Delta}mu_s,ds}{int_0^{Delta}sigma_s,dW_s}=O_pleft(sqrt{Delta}right)
$$



The idea is that, since $int_0^{Delta}mu_s,ds=O_p(Delta)$ and $int_0^{Delta}sigma_s,dW_s=O_pleft(sqrt{Delta}right)$ then the ratio should be $O_pleft(frac{Delta}{sqrt{Delta}}right)=O_pleft(sqrt{Delta}right)$.






stochastic-processes asymptotics stochastic-integrals







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    up vote
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    down vote

    favorite












    Suppose that $mu$ is an adapted bounded stochastic process and suppose that $sigma$ is an adapted bounded left-continuous stochastic process. I want to prove that



    $$
    frac{int_0^{Delta}mu_s,ds}{int_0^{Delta}sigma_s,dW_s}=O_pleft(sqrt{Delta}right)
    $$



    The idea is that, since $int_0^{Delta}mu_s,ds=O_p(Delta)$ and $int_0^{Delta}sigma_s,dW_s=O_pleft(sqrt{Delta}right)$ then the ratio should be $O_pleft(frac{Delta}{sqrt{Delta}}right)=O_pleft(sqrt{Delta}right)$.






    stochastic-processes asymptotics stochastic-integrals







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Suppose that $mu$ is an adapted bounded stochastic process and suppose that $sigma$ is an adapted bounded left-continuous stochastic process. I want to prove that



      $$
      frac{int_0^{Delta}mu_s,ds}{int_0^{Delta}sigma_s,dW_s}=O_pleft(sqrt{Delta}right)
      $$



      The idea is that, since $int_0^{Delta}mu_s,ds=O_p(Delta)$ and $int_0^{Delta}sigma_s,dW_s=O_pleft(sqrt{Delta}right)$ then the ratio should be $O_pleft(frac{Delta}{sqrt{Delta}}right)=O_pleft(sqrt{Delta}right)$.






      stochastic-processes asymptotics stochastic-integrals







      share|cite|improve this question













      Suppose that $mu$ is an adapted bounded stochastic process and suppose that $sigma$ is an adapted bounded left-continuous stochastic process. I want to prove that



      $$
      frac{int_0^{Delta}mu_s,ds}{int_0^{Delta}sigma_s,dW_s}=O_pleft(sqrt{Delta}right)
      $$



      The idea is that, since $int_0^{Delta}mu_s,ds=O_p(Delta)$ and $int_0^{Delta}sigma_s,dW_s=O_pleft(sqrt{Delta}right)$ then the ratio should be $O_pleft(frac{Delta}{sqrt{Delta}}right)=O_pleft(sqrt{Delta}right)$.






      stochastic-processes asymptotics stochastic-integrals




      stochastic-processes asymptotics stochastic-integrals






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