Ito integral exponent of Brownian Motion
Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+int_0^tf_sdB_s$.
I have started with
$$e^{B_t}=E[e^{B_t}]+int_0^tf_sdB_s$$
We know that $E[e^{B_t}]=e^{t/2}$ and it gives us
$$e^{B_t}=e^{t/2}+int_0^tf_sdB_s$$
I should probably use Ito Formula at this moment but I don't know how to apply this. Do you have any ideas?
probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-integrals
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
add a comment |
Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+int_0^tf_sdB_s$.
I have started with
$$e^{B_t}=E[e^{B_t}]+int_0^tf_sdB_s$$
We know that $E[e^{B_t}]=e^{t/2}$ and it gives us
$$e^{B_t}=e^{t/2}+int_0^tf_sdB_s$$
I should probably use Ito Formula at this moment but I don't know how to apply this. Do you have any ideas?
probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-integrals
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
add a comment |
Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+int_0^tf_sdB_s$.
I have started with
$$e^{B_t}=E[e^{B_t}]+int_0^tf_sdB_s$$
We know that $E[e^{B_t}]=e^{t/2}$ and it gives us
$$e^{B_t}=e^{t/2}+int_0^tf_sdB_s$$
I should probably use Ito Formula at this moment but I don't know how to apply this. Do you have any ideas?
probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-integrals
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+int_0^tf_sdB_s$.
I have started with
$$e^{B_t}=E[e^{B_t}]+int_0^tf_sdB_s$$
We know that $E[e^{B_t}]=e^{t/2}$ and it gives us
$$e^{B_t}=e^{t/2}+int_0^tf_sdB_s$$
I should probably use Ito Formula at this moment but I don't know how to apply this. Do you have any ideas?
probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-integrals
probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-integrals
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
edited Nov 26 at 22:27
Davide Giraudo
125k16150259
125k16150259
asked Nov 18 at 16:47
Speeedcuber
42
asked Nov 18 at 16:47
Speeedcuber
42
asked Nov 18 at 16:47
Speeedcuber
42
42
add a comment |
add a comment |
1 Answer
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Use ito's lemma on a function $f(x,t) = e^{x-frac{t}{2}}$ where $x(t) = B_t$
answered Dec 2 at 2:44
Makina
1,178215
add a comment |
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Use ito's lemma on a function $f(x,t) = e^{x-frac{t}{2}}$ where $x(t) = B_t$
answered Dec 2 at 2:44
Makina
1,178215
add a comment |
Use ito's lemma on a function $f(x,t) = e^{x-frac{t}{2}}$ where $x(t) = B_t$
answered Dec 2 at 2:44
Makina
1,178215
add a comment |
Use ito's lemma on a function $f(x,t) = e^{x-frac{t}{2}}$ where $x(t) = B_t$
answered Dec 2 at 2:44
Makina
1,178215
Use ito's lemma on a function $f(x,t) = e^{x-frac{t}{2}}$ where $x(t) = B_t$
answered Dec 2 at 2:44
Makina
1,178215
answered Dec 2 at 2:44
Makina
1,178215
answered Dec 2 at 2:44
Makina
1,178215
answered Dec 2 at 2:44
Makina
1,178215
1,178215
add a comment |
add a comment |
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