Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

A stochastic integral is an integral of stochastic processes with respect to stochastic processes. This may include Ito's integrals, but also variants such as Stratonovich integrals.

2333 questions
3
votes
1 answer

Ito's Lemma application

$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process. Find $dZ$ ?
vishal Shekhar
  • 67
2
votes
1 answer

How shall I prove this Stochastic integral equation?

I want to prove $$ \int_0^T B_t^2 dB_t = \frac{B_T^3}{3} - \int_0^T B_t dt $$ by the definition of Ito integral. I have tried this so far. Given a partition $0=t_0 < t_1 < ... < t_n=T$, I want to have $$ \sum_i B_{t_i}^2 (B_{t_{i+1}} - B_{t_i}) -…
Tim
  • 47,382
2
votes
2 answers

Stratonovich integral of $\sin(W^2)$

I have to solve the following Stratonovich integral: $$\int_{0}^{t}\sin(W^2_s)\circ{dW_s}$$ First of all I use the conversion from Stratonovich to Ito, obtaining $$\int_{0}^{t}\sin(W^2_s)dW_s+\int_{0}^{t}W_s{\cos(W_s)}ds$$ Is it sufficient? Or can…
Hannah Scott
  • 41
2
votes
2 answers

Question about Ito integral

I was wondering if Ito integral: $\int_0^T B(t)dB(t) $ is Gaussian (in which B(t) is Brownian Motion)?? Thank you so much, I appreciate any help ^^
user94464
  • 51
2
votes
0 answers

Why progressively measurable is important for stochastic integral

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, $(\mathcal F_t)$ a filtration and $B$ a Brownian motion adapted to the filtration. Let denote $\mathcal V(0,T)$ the space of function $f: [0,T]\times \Omega \to \mathbb R$ s.t. $(i)$…
Todd
  • 574
2
votes
0 answers

Stochastic differential equation

Using stochastic(!) methods find explicit solution to each of the two ($i = 1, 2$) initial value problems $$\partial_t u(t, x) = \frac{1}{2} \beta^2 \partial_x^ 2 u(t, x) + (−\alpha x + \gamma )\partial_x u(t, x)$$ with $u(0, x) = f_i(x)$ where…
lisa
  • 79
2
votes
1 answer

integral of square of Brownian motion

What is expectation of $$\int_0^t B(s)^2ds$$ where $B(s) is standard Brownian motion. Is the integral a well known random variable?
Sida
  • 31
1
vote
1 answer

Calculate Stochastic Integral

I found the following integral $\int_{0}^1 B_t t^{-1}dt,$ where $B_t$ is a standard Brownian motion. Using Ito formula with $f(t,x)=x\log(t)$ I achieved $$0=\log(1)B_1=\int_{0}^1 B_s s^{-1}ds +\int_{0}^1 \log(s)dB_s,$$ and further by properties of…
Seneca
  • 235
  • 2
  • 8
1
vote
1 answer

How to calculate the Multiple Stratonovich Integral?

My question is about multiple Stratonovich-Integrals. I have the following Stratonovich-Integral $ \int \limits_{t_n}^{t_{n+1}} \int \limits_{t_n}^{s_1}1\,dW(s)dW(s_1).$ How can I calculate it? Is it right, that $\int…
user112260
  • 161
1
vote
0 answers

quantile of Ito integral when integration limit goes to zero.

I woud like to calculate the Value at Risk of an Ito Integral in the following form in the limit! $$\lim_{\Delta t\to 0}\frac{1}{\Delta t}VaR_{q,t}\left[\int_t^{t+\Delta t}b(s,y(s))\pi_y^c(s,y(s))d…
Ahmad.S
  • 11
  • 1
1
vote
1 answer

Verify Brownian motion stochastic integral identity only using definition

I have to verify, using only the definition of stochastic integral, that: $$ (B \cdot B)_t := \int_0^t B_s \text{d}B_s = \frac{1}{2} B_t^2 - \frac{1}{2}t. $$ Below what I've tried so far. Given $M \in H^2$, $K \in L^2(M)$ the stochastic integral $K…
user515933
  • 563
1
vote
1 answer

Can the derivative of an integral of a Brownian motion be calculated?

For an equation given below: $$ \int_0^t\phi(s)dW(s) $$ where $W(s)$ is a Brownian Motion. Can I calculate the derivate w.r.t $t$ as follows: $$ \frac{d}{dt} \left( \int_0^t\phi(s)dW(s)\right) = (\phi(t)-\phi(0))(W(t)-W(0)) $$ ? Or is this wrong?
coffee-raid
  • 333
1
vote
0 answers

What's wrong with this stochastic integral calculation?

To compute $E[(\int_0^1 W(s) ds)^2]$, I know that using integration by parts and Ito isometry, we have: \begin{align} E\left[\left(\int_0^1 W(s) ds\right)^2\right] &= E\left[\left(W(1) - \int_0^1 sdW(s)\right)^2\right]\\ & =…
Michael
  • 111
1
vote
2 answers

Does $\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$ holds?

I was wondering if the inequality $$\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$$ holds for stochastic integral. In fact, I don't see such a property in any book, neither on Google, so I have some doubt. What do you think…
John
  • 858
1
vote
1 answer

How to compute ito's calculus without knowing its solution already?

I'm having trouble computing Itô's calculus. Take $\int_0^tB_sdB_s$ for example, can I solve it base solely on Itô-Doeblin Formula, instead of assuming $f(x)=\frac{x^2}{2}$? Please help!
Bubblethan
  • 261
  • 2
  • 10
1
2
3 4