Mixing
ito vs Stratonovich
$begingroup$
I need to sum up the advantages of ito and stratonovich. I often heard, that
the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future".
Can you explain me why Stratonovich looks into the future?
Thanks a lot.
stochastic-integrals
asked Nov 29 '13 at 15:15
user112260user112260
4613
$endgroup$
add a comment |
$begingroup$
I need to sum up the advantages of ito and stratonovich. I often heard, that
the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future".
Can you explain me why Stratonovich looks into the future?
Thanks a lot.
stochastic-integrals
asked Nov 29 '13 at 15:15
user112260user112260
4613
$endgroup$
add a comment |
$begingroup$
I need to sum up the advantages of ito and stratonovich. I often heard, that
the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future".
Can you explain me why Stratonovich looks into the future?
Thanks a lot.
stochastic-integrals
asked Nov 29 '13 at 15:15
user112260user112260
4613
$endgroup$
I need to sum up the advantages of ito and stratonovich. I often heard, that
the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future".
Can you explain me why Stratonovich looks into the future?
Thanks a lot.
stochastic-integrals
stochastic-integrals
asked Nov 29 '13 at 15:15
user112260user112260
4613
asked Nov 29 '13 at 15:15
user112260user112260
4613
asked Nov 29 '13 at 15:15
user112260user112260
4613
asked Nov 29 '13 at 15:15
user112260user112260
4613
asked Nov 29 '13 at 15:15
user112260user112260
4613
4613
add a comment |
add a comment |
2 Answers
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oldest
votes
$begingroup$
The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.
Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.
edited Apr 13 '17 at 12:58
Community♦
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
$endgroup$
add a comment |
$begingroup$
Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$sumfrac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.
edited Jan 30 at 2:31
danijar
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.
Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.
edited Apr 13 '17 at 12:58
Community♦
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
$endgroup$
add a comment |
$begingroup$
The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.
Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.
edited Apr 13 '17 at 12:58
Community♦
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
$endgroup$
add a comment |
$begingroup$
The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.
Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.
edited Apr 13 '17 at 12:58
Community♦
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
$endgroup$
The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.
Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.
edited Apr 13 '17 at 12:58
Community♦
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
edited Apr 13 '17 at 12:58
Community♦
1
edited Apr 13 '17 at 12:58
Community♦
1
edited Apr 13 '17 at 12:58
Community♦
1
1
answered Feb 5 '14 at 0:28
DimasDimas
46539
answered Feb 5 '14 at 0:28
DimasDimas
46539
answered Feb 5 '14 at 0:28
DimasDimas
46539
46539
add a comment |
add a comment |
$begingroup$
Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$sumfrac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.
edited Jan 30 at 2:31
danijar
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
$endgroup$
add a comment |
$begingroup$
Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$sumfrac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.
edited Jan 30 at 2:31
danijar
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
$endgroup$
add a comment |
$begingroup$
Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$sumfrac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.
edited Jan 30 at 2:31
danijar
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
$endgroup$
Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$sumfrac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.
edited Jan 30 at 2:31
danijar
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
edited Jan 30 at 2:31
danijar
227414
edited Jan 30 at 2:31
danijar
227414
edited Jan 30 at 2:31
danijar
227414
227414
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
answered Feb 27 '14 at 1:04
mookidmookid
25.7k52547
25.7k52547
add a comment |
add a comment |
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