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For any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_pi$ = 0?

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$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$
and defined the quadratic variation of g by
$triangle^2(g)=sum_k=0^n-1(g(t_k+1)-g(t_k))^2$
, $[g, g]_pi=limlimits_ntoinftytriangle_n^2(g)$

if the limit exist
then for any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_pi$ = 0 ?

real-analysis probability-theory stochastic-calculus

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asked Mar 17 at 11:46

mathgogomathgogo

1

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• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to MSE. Please read this text about how to ask a good question.
  $endgroup$
  – José Carlos Santos
  Mar 17 at 11:54
  

  
  
  
  

add a comment | 

0

$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$
and defined the quadratic variation of g by
$triangle^2(g)=sum_k=0^n-1(g(t_k+1)-g(t_k))^2$
, $[g, g]_pi=limlimits_ntoinftytriangle_n^2(g)$

if the limit exist
then for any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_pi$ = 0 ?

real-analysis probability-theory stochastic-calculus

share|cite|improve this question

asked Mar 17 at 11:46

mathgogomathgogo

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to MSE. Please read this text about how to ask a good question.
  $endgroup$
  – José Carlos Santos
  Mar 17 at 11:54
  

  
  
  
  

add a comment | 

$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$
and defined the quadratic variation of g by
$triangle^2(g)=sum_k=0^n-1(g(t_k+1)-g(t_k))^2$
, $[g, g]_pi=limlimits_ntoinftytriangle_n^2(g)$

if the limit exist
then for any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_pi$ = 0 ?

real-analysis probability-theory stochastic-calculus

share|cite|improve this question

asked Mar 17 at 11:46

mathgogomathgogo

1

$endgroup$

share|cite|improve this question

asked Mar 17 at 11:46

mathgogomathgogo

1

share|cite|improve this question

asked Mar 17 at 11:46

mathgogomathgogo

1

asked Mar 17 at 11:46

mathgogomathgogo

1

asked Mar 17 at 11:46

mathgogomathgogo

1