Proving alternative notation of quadratic variation of Brownian motion












0












$begingroup$


Setting: Let ${W_t,tgeq 0}$ be a Brownian with respect to the standard filtration ${mathcal{F}_t,tgeq 0}$.



Problem: We fix $t>0$ and must prove that:




$left langle W,W rightrangle^{(n)}_{t}=sum^n_{j=1}(W_{frac{tj}{n}}-W_{frac{t(j-1)}{n}})^2$




Can be written as:




$left langle W,W right rangle^{(n)}_{t}=frac{t}{n}sum^n_{i=1}X_i^2$




Where $X_i$ are independent identically distributed and have a standard normal distribution.



Question: I've attempted to solve this exercise, obviously, but kept getting stuck because I think my approach is wrong. I tried writing out the sum and using telescoping but I can't seem to get factors to drop out. Writing out the first sum does yield some similar factors but the square prohibits them from dropping off against eachother. Writing out the square and then applying telescoping also provides no luck. If I can't use telescoping for this problem I'm not sure how to approach it. Any tips or advice on solving this problem?



Any help is appreciated. Thanks!






probability-theory brownian-motion







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    0












    $begingroup$


    Setting: Let ${W_t,tgeq 0}$ be a Brownian with respect to the standard filtration ${mathcal{F}_t,tgeq 0}$.



    Problem: We fix $t>0$ and must prove that:




    $left langle W,W rightrangle^{(n)}_{t}=sum^n_{j=1}(W_{frac{tj}{n}}-W_{frac{t(j-1)}{n}})^2$




    Can be written as:




    $left langle W,W right rangle^{(n)}_{t}=frac{t}{n}sum^n_{i=1}X_i^2$




    Where $X_i$ are independent identically distributed and have a standard normal distribution.



    Question: I've attempted to solve this exercise, obviously, but kept getting stuck because I think my approach is wrong. I tried writing out the sum and using telescoping but I can't seem to get factors to drop out. Writing out the first sum does yield some similar factors but the square prohibits them from dropping off against eachother. Writing out the square and then applying telescoping also provides no luck. If I can't use telescoping for this problem I'm not sure how to approach it. Any tips or advice on solving this problem?



    Any help is appreciated. Thanks!






    probability-theory brownian-motion







    share|cite|improve this question









    $endgroup$















      0












      0








      0


      1



      $begingroup$


      Setting: Let ${W_t,tgeq 0}$ be a Brownian with respect to the standard filtration ${mathcal{F}_t,tgeq 0}$.



      Problem: We fix $t>0$ and must prove that:




      $left langle W,W rightrangle^{(n)}_{t}=sum^n_{j=1}(W_{frac{tj}{n}}-W_{frac{t(j-1)}{n}})^2$




      Can be written as:




      $left langle W,W right rangle^{(n)}_{t}=frac{t}{n}sum^n_{i=1}X_i^2$




      Where $X_i$ are independent identically distributed and have a standard normal distribution.



      Question: I've attempted to solve this exercise, obviously, but kept getting stuck because I think my approach is wrong. I tried writing out the sum and using telescoping but I can't seem to get factors to drop out. Writing out the first sum does yield some similar factors but the square prohibits them from dropping off against eachother. Writing out the square and then applying telescoping also provides no luck. If I can't use telescoping for this problem I'm not sure how to approach it. Any tips or advice on solving this problem?



      Any help is appreciated. Thanks!






      probability-theory brownian-motion







      share|cite|improve this question









      $endgroup$




      Setting: Let ${W_t,tgeq 0}$ be a Brownian with respect to the standard filtration ${mathcal{F}_t,tgeq 0}$.



      Problem: We fix $t>0$ and must prove that:




      $left langle W,W rightrangle^{(n)}_{t}=sum^n_{j=1}(W_{frac{tj}{n}}-W_{frac{t(j-1)}{n}})^2$




      Can be written as:




      $left langle W,W right rangle^{(n)}_{t}=frac{t}{n}sum^n_{i=1}X_i^2$




      Where $X_i$ are independent identically distributed and have a standard normal distribution.



      Question: I've attempted to solve this exercise, obviously, but kept getting stuck because I think my approach is wrong. I tried writing out the sum and using telescoping but I can't seem to get factors to drop out. Writing out the first sum does yield some similar factors but the square prohibits them from dropping off against eachother. Writing out the square and then applying telescoping also provides no luck. If I can't use telescoping for this problem I'm not sure how to approach it. Any tips or advice on solving this problem?



      Any help is appreciated. Thanks!






      probability-theory brownian-motion




      probability-theory brownian-motion






      share|cite|improve this question













      share|cite|improve this question











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      asked Dec 11 '18 at 7:25









      S. CrimS. Crim

      363112




      363112






















          2 Answers
          2






          active

          oldest

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          1












          $begingroup$

          Hints:




          1. Show that $$langle W,W rangle_t^{(n)} = sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $langle W,W rangle_t^{(n)}$ ...)

          2. Show that $mathbb{E}(Y_j)=0$ and $mathbb{E}(Y_j^2) = t/n$ for all $j=1,ldots,n$.

          3. Define $$X_j := sqrt{frac{n}{t}} Y_j, qquad j=1,ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.

          4. Use Step 1 to show that $$langle W,Wrangle_t^{(n)} = frac{t}{n} sum_{j=1}^n X_j^2.$$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
            $endgroup$
            – S. Crim
            Dec 11 '18 at 7:50










          • $begingroup$
            @S.Crim You are welcome.
            $endgroup$
            – saz
            Dec 11 '18 at 8:10



















          1












          $begingroup$

          Just define $X_i= {sqrt {frac n t}} (W_{frac {tj} n}-W_{frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.






          share|cite|improve this answer









          $endgroup$













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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Hints:




            1. Show that $$langle W,W rangle_t^{(n)} = sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $langle W,W rangle_t^{(n)}$ ...)

            2. Show that $mathbb{E}(Y_j)=0$ and $mathbb{E}(Y_j^2) = t/n$ for all $j=1,ldots,n$.

            3. Define $$X_j := sqrt{frac{n}{t}} Y_j, qquad j=1,ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.

            4. Use Step 1 to show that $$langle W,Wrangle_t^{(n)} = frac{t}{n} sum_{j=1}^n X_j^2.$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
              $endgroup$
              – S. Crim
              Dec 11 '18 at 7:50










            • $begingroup$
              @S.Crim You are welcome.
              $endgroup$
              – saz
              Dec 11 '18 at 8:10
















            1












            $begingroup$

            Hints:




            1. Show that $$langle W,W rangle_t^{(n)} = sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $langle W,W rangle_t^{(n)}$ ...)

            2. Show that $mathbb{E}(Y_j)=0$ and $mathbb{E}(Y_j^2) = t/n$ for all $j=1,ldots,n$.

            3. Define $$X_j := sqrt{frac{n}{t}} Y_j, qquad j=1,ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.

            4. Use Step 1 to show that $$langle W,Wrangle_t^{(n)} = frac{t}{n} sum_{j=1}^n X_j^2.$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
              $endgroup$
              – S. Crim
              Dec 11 '18 at 7:50










            • $begingroup$
              @S.Crim You are welcome.
              $endgroup$
              – saz
              Dec 11 '18 at 8:10














            1












            1








            1





            $begingroup$

            Hints:




            1. Show that $$langle W,W rangle_t^{(n)} = sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $langle W,W rangle_t^{(n)}$ ...)

            2. Show that $mathbb{E}(Y_j)=0$ and $mathbb{E}(Y_j^2) = t/n$ for all $j=1,ldots,n$.

            3. Define $$X_j := sqrt{frac{n}{t}} Y_j, qquad j=1,ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.

            4. Use Step 1 to show that $$langle W,Wrangle_t^{(n)} = frac{t}{n} sum_{j=1}^n X_j^2.$$






            share|cite|improve this answer











            $endgroup$



            Hints:




            1. Show that $$langle W,W rangle_t^{(n)} = sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $langle W,W rangle_t^{(n)}$ ...)

            2. Show that $mathbb{E}(Y_j)=0$ and $mathbb{E}(Y_j^2) = t/n$ for all $j=1,ldots,n$.

            3. Define $$X_j := sqrt{frac{n}{t}} Y_j, qquad j=1,ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.

            4. Use Step 1 to show that $$langle W,Wrangle_t^{(n)} = frac{t}{n} sum_{j=1}^n X_j^2.$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 11 '18 at 10:19

























            answered Dec 11 '18 at 7:38









            sazsaz

            80.5k860125




            80.5k860125












            • $begingroup$
              Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
              $endgroup$
              – S. Crim
              Dec 11 '18 at 7:50










            • $begingroup$
              @S.Crim You are welcome.
              $endgroup$
              – saz
              Dec 11 '18 at 8:10


















            • $begingroup$
              Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
              $endgroup$
              – S. Crim
              Dec 11 '18 at 7:50










            • $begingroup$
              @S.Crim You are welcome.
              $endgroup$
              – saz
              Dec 11 '18 at 8:10
















            $begingroup$
            Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
            $endgroup$
            – S. Crim
            Dec 11 '18 at 7:50




            $begingroup$
            Thank you for the detailed hints. I reckon I should be able to solve the problem with these. If not, I'll be back. Thanks again!
            $endgroup$
            – S. Crim
            Dec 11 '18 at 7:50












            $begingroup$
            @S.Crim You are welcome.
            $endgroup$
            – saz
            Dec 11 '18 at 8:10




            $begingroup$
            @S.Crim You are welcome.
            $endgroup$
            – saz
            Dec 11 '18 at 8:10











            1












            $begingroup$

            Just define $X_i= {sqrt {frac n t}} (W_{frac {tj} n}-W_{frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Just define $X_i= {sqrt {frac n t}} (W_{frac {tj} n}-W_{frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Just define $X_i= {sqrt {frac n t}} (W_{frac {tj} n}-W_{frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.






                share|cite|improve this answer









                $endgroup$



                Just define $X_i= {sqrt {frac n t}} (W_{frac {tj} n}-W_{frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 11 '18 at 7:38









                Kavi Rama MurthyKavi Rama Murthy

                59k42161




                59k42161






























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