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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.

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Local martingale and convergence theorem

I have $E[x(t)^2]\leq A\operatorname{exp}(Bt)+C/Bt$ it's clear that for a finite time less than $T$, x(t)^2 is a "local martingale" because $\lim E[x(t)^2]<\infty$. But one can see that if $t$ tends to plus infinity then the limit of $E[x^2]$ is…
fidel
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Is a Wiener-Squared a Chi-Squared?

If $W_0=0$, then $W_t\sim N(0,t)$, so $W_1^2\sim\chi_1^2$. But Ito's lemma for $f(x)=x^2$ states $\mathrm{d}\left(W_t^2\right)=\mathrm{d}t+2W_t\mathrm{d}W_t$, so $W_t^2=t+2\int_0^t{W_s\mathrm{d}W_s}$ and hence…
Junyong Kim
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Question on proof of Ito formula given in the book "Statistics of random processes" by Shiryaev

The main question is: Does the autor consider random function $f$ dependent on $\omega$? It seems that the answer is yes, because at the page 125 (see 4.90) he consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ are random…
Anton Sorokovskiy
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Darboux versus stochastic integral

I don't know if my question is obscure. I'm astonished why there not mention the Darboux sums in the definition of stochastic integral
Zbigniew
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Stochastic integral 2

Let $B_t$ be a standard Brownian motion such that the processes $X_n(t)=e^{-nt}1_{[0,T]}(t)$, and $\mathcal{H}$=$\{$($h_t$): $h_t$ is adapted, $\mathbb{E}\int_{0}^{\infty}h_t^2dt<\infty\}.$ How can we prove that $X_n \in \mathcal{H} $ and…
Sam
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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 predictable process, then $ H \cdot M $ is a true martingale.…
Mad Si
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Simple differential equation and Integral Ito

With stochastic differential equation dx(t) = dW (t), and knowing that all integrals occurring are integral Ito. Witch variable changes y = tx. How I can prove? integral between 0 and t[sdW(s)] = tW(t) - [integral between 0 and t[sdW(s)] W(s)…
user273153
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