Tagged Questions
Questions on the calculus of stochastic processes, or processes that have a random component.
1
vote
1answer
19 views
Change of measure of conditional expectation
How can I prove that:
$E_π [ (dQ_X/dπ) S (T)| F_t ]= E_{Q_X} [S(T) | F_t]E_π [ dQ_X/dπ | F_t ]$.
Obviously $E_π [(dQ_X/dπ) S(T) ]= E_{Q_X} [S(T)]$ I know that much, but how to prove when it is ...
1
vote
0answers
26 views
Question regarding Ito Process
I am new to Ito Process, so I have a following question.
Consider a standard Ito Process,
$$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$
where W is the m-dimentional Brownian motion and X is a ...
1
vote
0answers
11 views
How to find sigma-algebra over omega3 generated by the log-return ln(S2/S1) and ln(S3/S2)?
I calculated {S2/S1} = {u, u*u/d, d, d*d/u}, and then get ln(S2/S1)= {ln(u),ln(u*u/d), ln(d),ln( d*d/u)}.
I am not sure my way of doing this question is right, because i m confuse about how to get ...
0
votes
0answers
19 views
how to show difference between sigma(S1,S2) and sigma(S1*S2)
Show that in the three-period binomial model, the smallest sigma-algebra respectively generated by the random variable vector (S1, S2, S3) and by the product S1*S2*S3 are not the same. What can you ...
0
votes
0answers
26 views
moving window supremum of a Wiener process
Let $W$ be a Wiener process.
For each fixed $t>\frac12$, is it true that,$$\sup_{s\in[0,\frac12]}|W(t-s)|$$ has the same law as $$\sup_{s\in[0,\frac12]}|W(t)-W(s)| ?$$
0
votes
1answer
39 views
exit time and indicator function
let $D$ open set of $\mathbb{R}^{n}$ and $T_{D}=\inf\{t\geq 0 : X_{t}\notin D\} $ be the first exit time from the $D$ and $1_{A}$ is Indicator function of $A \subseteq \partial D$
$$ ...
5
votes
1answer
46 views
Girsanov: Change of drift, that depends on the process
Known:
If I am looking at an SDE like:
$dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$.
I know that I can change the drift by using Girsanov to
$dX_t = ...
1
vote
0answers
21 views
Variance of a stochastic integral?
Does there exist a variance formula for stochastic integrals?
Suppose we have
$dX = \sigma (X) dW + \mu(X) dt$
Do we have a formula for $Var(X_t)$ or an intergral of $X$ against $B$
More ...
0
votes
1answer
24 views
Mean and variance of a brownian bridge
I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$.
$$X_t = y + ...
0
votes
1answer
21 views
Not using stochasstic integral how to prove $E\int_0^T W^2(t)dt<+\infty$?
Can anyone help me to prove this?
Suppose $W_t$ ~ $N(0,t)$, then not using stochasstic integral (or anything related with Ito) how to prove $E\int_0^T W^2(t)dt<+\infty$?
Thanks.
1
vote
1answer
29 views
Ito vs Stratonovich SDE with irregular time-dependence in coefficients
Suppose I am interested in the Stratonovich SDE
$$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$
If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito ...
2
votes
1answer
33 views
Joint distribution of $W(t)$ Brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$
Let $(W(t))$ be a brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$. Does one know the joint distribution $(W(s),B(s))$ for a given $s$? I know some related theory like Tanaka's ...
2
votes
0answers
27 views
an exetension of Doob's inequality
Doob's inequality gives an estimation of
$$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$
where $X$ is a martingale. Now I wonder how to estimate
$$\mathbb{P}(\sup_{0\leq t,s\leq 1, ...
2
votes
1answer
62 views
Characteristics of stochastic integral?
I need to describe a couple of integrals which are supposed to be evaluated in terms of Ito calculus.
$$
I_1 = \int_0^t e^{-2\tau}dW(\tau); \\
I_2 = \int_0^t e^{-3 W(\tau)} dW(\tau);
$$
Here ...
0
votes
0answers
31 views
Is Brownian motion an adapted process?
In establishing theorems in stochastic calculus, a basic stochastic integral is defined as
$\int^T_0 \Delta(t) dW(t)$, where $\Delta(t)$ is an adapted process, i.e. $F(t)$ measurable at time $t$. ...
