0
votes
0answers
96 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
1
vote
1answer
67 views

How to calculate this easy stochastic integral?

I have a relatively simple homework for stochastic calculus that I recently started to learn. I cannot seem to calculate the following integral: $$ \int_0^t s dW_s $$ In principle, it should be solved ...
0
votes
1answer
37 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
0
votes
1answer
279 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
2
votes
1answer
138 views

Ito differential equation

Define $$X_t := \left( \begin{matrix} \cos W_t \\ \sin W_t \end{matrix} \right).$$ where $W = \left( W_t,\mathcal F_t \right) _{t\ge0}$ is a standard Wiener process. Find the Ito differential of X ...
2
votes
2answers
79 views

stochastic integral equation

For $0 \leq t \leq T$, define $$Z_t:=\exp {\left\lbrace \int_0^t X_sdW_s - \frac 12 \int_0^t X_s^2ds \right\rbrace }$$ Show that this process satisfies the stochastic integral equation ...
2
votes
1answer
149 views

generalized derivative of Wiener process

Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that ...
1
vote
0answers
44 views

Supermartingale Lemma + related problems

Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
1
vote
0answers
179 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
3
votes
1answer
395 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
0
votes
0answers
250 views

Ito process generator and Dynkin's Formula

I have attempted to answer the following multi-part question but am especially having trouble with part iv) of the question. Any feedback would be greatly appreciated!. Consider the Ito process $X(t) ...
0
votes
0answers
100 views

Why is this a martingale?

In our homework assignment, we are supposed to prove: If $ M $ is a countinuous local martingale and if for each $ T > 0, E[\sup_{t \leq T } |M_t|] < + \infty $ and $ H^T $ is a bounded ...
7
votes
2answers
773 views

Is this local martingale a true martingale?

Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where $$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$ $B_t$ - is a standard Brownian motion I ...
4
votes
1answer
279 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
1
vote
1answer
199 views

Integration Order Reversal

I have a question regarding integration order reversal in a stochastic integral. This is a homework problem of the form "Show this is true". My problem is 1) my results are not exactly the same as the ...