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Issues with notation of Ito Integral
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How should I interoperate $X_t_j$, does usually when I previously have seen double subscript it refer to having multidimensions.
In this case it looks like we are splitting up t into sections.
In other words we could have that t goes from 0 to T, in steps of t.
however, is the second subscript referring to splitting up [0,t] into furter increments?
stochastic-processes stochastic-calculus stochastic-analysis
asked Jul 16 at 23:12
ALEXANDER
850818
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up vote
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down vote
favorite
How should I interoperate $X_t_j$, does usually when I previously have seen double subscript it refer to having multidimensions.
In this case it looks like we are splitting up t into sections.
In other words we could have that t goes from 0 to T, in steps of t.
however, is the second subscript referring to splitting up [0,t] into furter increments?
stochastic-processes stochastic-calculus stochastic-analysis
asked Jul 16 at 23:12
ALEXANDER
850818
1
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
1
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
1
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55
 |Â
show 7 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How should I interoperate $X_t_j$, does usually when I previously have seen double subscript it refer to having multidimensions.
In this case it looks like we are splitting up t into sections.
In other words we could have that t goes from 0 to T, in steps of t.
however, is the second subscript referring to splitting up [0,t] into furter increments?
stochastic-processes stochastic-calculus stochastic-analysis
asked Jul 16 at 23:12
ALEXANDER
850818
How should I interoperate $X_t_j$, does usually when I previously have seen double subscript it refer to having multidimensions.
In this case it looks like we are splitting up t into sections.
In other words we could have that t goes from 0 to T, in steps of t.
however, is the second subscript referring to splitting up [0,t] into furter increments?
stochastic-processes stochastic-calculus stochastic-analysis
asked Jul 16 at 23:12
ALEXANDER
850818
asked Jul 16 at 23:12
ALEXANDER
850818
asked Jul 16 at 23:12
ALEXANDER
850818
asked Jul 16 at 23:12
ALEXANDER
850818
850818
1
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
1
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
1
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55
 |Â
show 7 more comments
1
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
1
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
1
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55
1
1
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
1
1
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
1
1
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55
 |Â
show 7 more comments
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1
$X_t$ is a (random) function of $t.$ $X_t_i$ is that function evaluated at $t=t_i.$ The $t_i$ are an evenly spaced sequence of sample points in the interval... compare to the evaluation points for a Riemann integral or something like that.
– spaceisdarkgreen
Jul 16 at 23:51
@spaceisdarkgreen So I could consider it being the ith element of $X_t$ ?
– ALEXANDER
Jul 17 at 0:49
1
No... that's not what I said.
– spaceisdarkgreen
Jul 17 at 1:06
@spaceisdarkgreen Would you be able to write more in depth answer to the solution provided to this answer, where he uses the term ith element.: math.stackexchange.com/questions/2710012/…
– ALEXANDER
Jul 17 at 2:06
1
In this example in this question, the $i$ refers to the $i$-th time step in a discretization of the time axis. In other words we're taking a function defined on some interval, but only looking at a finite number of points (and then taking a limit as we go to finer and finer spaced points in order to recover information about the continuous time function... as we do in, say, the Riemann or Stieljes integral)
– spaceisdarkgreen
Jul 17 at 2:55