Ito Lemma and identifying martingale parts
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Suppose that $X_t$ is a càdlàg semi-martingale with decomposition
$$
X_t= X_0+ B_t + M_t.
$$
I know that using the Ito lemma for any $C^2$-function $f$,
$$
f(X_t)= f(X_0)\
+ int_{0^+}^tf_x(X_{s-})dB_s + int_{0^+}^t frac{f_{xx}}{2}(X_{s-})[M]_t\
+int_{0^+}^tf_x(X_{s-})dM_s\
\
+ sum_{0<sleq t}left(f(X_{s-}) - f(X_s)right) +f_x(X_{s-})Delta B_s +frac1{2}f_x(X_{s-})(Delta M)^2_s
\
+ sum_{0<sleq t} f_x(X_{s-})Delta M_s
$$
My partial Solution
Suppose that $X_t = X_0 +int_0^t mu(t,X_t)dt + int_0^t sigma(t,X_t)dW_t$, then
$$
int_0^tleft(f_x(X_t)mu(t,X_t) + f_{xx}frac{sigma(t,X_t)}{2}right)dt
$$
is the drift part and
$$
int_0^tf_{x}sigma(t,X_t)dW_t,
$$
is the local martingale part of $f(X_t)$. However, I am having trouble identifying which part is which in the general setting. I know that from the continuous case that the second line of the Ito Lemma is finite-variation and that the third is a martingale, but I am not sure which portions are local martingale or fv in the last 3 lines...
stochastic-processes stochastic-calculus martingales stochastic-integrals stochastic-analysis
edited Nov 19 at 7:19
TheBridge
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
add a comment |
up vote
1
down vote
favorite
Suppose that $X_t$ is a càdlàg semi-martingale with decomposition
$$
X_t= X_0+ B_t + M_t.
$$
I know that using the Ito lemma for any $C^2$-function $f$,
$$
f(X_t)= f(X_0)\
+ int_{0^+}^tf_x(X_{s-})dB_s + int_{0^+}^t frac{f_{xx}}{2}(X_{s-})[M]_t\
+int_{0^+}^tf_x(X_{s-})dM_s\
\
+ sum_{0<sleq t}left(f(X_{s-}) - f(X_s)right) +f_x(X_{s-})Delta B_s +frac1{2}f_x(X_{s-})(Delta M)^2_s
\
+ sum_{0<sleq t} f_x(X_{s-})Delta M_s
$$
My partial Solution
Suppose that $X_t = X_0 +int_0^t mu(t,X_t)dt + int_0^t sigma(t,X_t)dW_t$, then
$$
int_0^tleft(f_x(X_t)mu(t,X_t) + f_{xx}frac{sigma(t,X_t)}{2}right)dt
$$
is the drift part and
$$
int_0^tf_{x}sigma(t,X_t)dW_t,
$$
is the local martingale part of $f(X_t)$. However, I am having trouble identifying which part is which in the general setting. I know that from the continuous case that the second line of the Ito Lemma is finite-variation and that the third is a martingale, but I am not sure which portions are local martingale or fv in the last 3 lines...
stochastic-processes stochastic-calculus martingales stochastic-integrals stochastic-analysis
edited Nov 19 at 7:19
TheBridge
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
1
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose that $X_t$ is a càdlàg semi-martingale with decomposition
$$
X_t= X_0+ B_t + M_t.
$$
I know that using the Ito lemma for any $C^2$-function $f$,
$$
f(X_t)= f(X_0)\
+ int_{0^+}^tf_x(X_{s-})dB_s + int_{0^+}^t frac{f_{xx}}{2}(X_{s-})[M]_t\
+int_{0^+}^tf_x(X_{s-})dM_s\
\
+ sum_{0<sleq t}left(f(X_{s-}) - f(X_s)right) +f_x(X_{s-})Delta B_s +frac1{2}f_x(X_{s-})(Delta M)^2_s
\
+ sum_{0<sleq t} f_x(X_{s-})Delta M_s
$$
My partial Solution
Suppose that $X_t = X_0 +int_0^t mu(t,X_t)dt + int_0^t sigma(t,X_t)dW_t$, then
$$
int_0^tleft(f_x(X_t)mu(t,X_t) + f_{xx}frac{sigma(t,X_t)}{2}right)dt
$$
is the drift part and
$$
int_0^tf_{x}sigma(t,X_t)dW_t,
$$
is the local martingale part of $f(X_t)$. However, I am having trouble identifying which part is which in the general setting. I know that from the continuous case that the second line of the Ito Lemma is finite-variation and that the third is a martingale, but I am not sure which portions are local martingale or fv in the last 3 lines...
stochastic-processes stochastic-calculus martingales stochastic-integrals stochastic-analysis
edited Nov 19 at 7:19
TheBridge
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
Suppose that $X_t$ is a càdlàg semi-martingale with decomposition
$$
X_t= X_0+ B_t + M_t.
$$
I know that using the Ito lemma for any $C^2$-function $f$,
$$
f(X_t)= f(X_0)\
+ int_{0^+}^tf_x(X_{s-})dB_s + int_{0^+}^t frac{f_{xx}}{2}(X_{s-})[M]_t\
+int_{0^+}^tf_x(X_{s-})dM_s\
\
+ sum_{0<sleq t}left(f(X_{s-}) - f(X_s)right) +f_x(X_{s-})Delta B_s +frac1{2}f_x(X_{s-})(Delta M)^2_s
\
+ sum_{0<sleq t} f_x(X_{s-})Delta M_s
$$
My partial Solution
Suppose that $X_t = X_0 +int_0^t mu(t,X_t)dt + int_0^t sigma(t,X_t)dW_t$, then
$$
int_0^tleft(f_x(X_t)mu(t,X_t) + f_{xx}frac{sigma(t,X_t)}{2}right)dt
$$
is the drift part and
$$
int_0^tf_{x}sigma(t,X_t)dW_t,
$$
is the local martingale part of $f(X_t)$. However, I am having trouble identifying which part is which in the general setting. I know that from the continuous case that the second line of the Ito Lemma is finite-variation and that the third is a martingale, but I am not sure which portions are local martingale or fv in the last 3 lines...
stochastic-processes stochastic-calculus martingales stochastic-integrals stochastic-analysis
stochastic-processes stochastic-calculus martingales stochastic-integrals stochastic-analysis
edited Nov 19 at 7:19
TheBridge
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
edited Nov 19 at 7:19
TheBridge
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
edited Nov 19 at 7:19
TheBridge
3,74611424
edited Nov 19 at 7:19
TheBridge
3,74611424
edited Nov 19 at 7:19
TheBridge
3,74611424
3,74611424
asked Nov 17 at 19:58
AIM_BLB
2,3522718
asked Nov 17 at 19:58
AIM_BLB
2,3522718
asked Nov 17 at 19:58
AIM_BLB
2,3522718
2,3522718
1
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41
add a comment |
1
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41
1
1
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41
add a comment |
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1
You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards
– TheBridge
Nov 19 at 7:24
Yes, is that possible?
– AIM_BLB
Nov 30 at 8:41