# 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.

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### Inverse Bessel Process as continuous local martingale

Let $B$ be a $n$-dimensional brownian motion. This question shows, that $$\Big(\sum_{i=1}^n\int_0^t\frac{B^i_s}{||Bs||}dB^i_s,||B_t||\Big)$$ is a weak solution of $dX=\frac{n-1}{2X}dt+dW$. Now I would ...
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### Solving an Ornstein-Uhlenbeck style SDE

Solve the SDE $dN=a N \log(\frac MN)dt+\sigma NdB_t,$ where $M,a \in \mathbb{R}$. Apply Ito's formula to $f = \log(N)$ gives \begin{align*} df &= \frac{\partial f}{\partial N} dN +\frac{1}...
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### Why is $\int_S^T f dB_t$ (Itô integral) $\mathcal{F}_t$-measurable?

In Oksendal's book, in Theorem 3.2.1, the author states that $\int_S^T f dB_t$ is $\mathcal{F}_T$-measurable. In the proof he says that since it holds for elementary functions, when we take the limits ...
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### Is there any relationship between Doob-Meyer Decomposition and Martingale Representation theorem?

Doob-Meyer Decomposition Theorem decomposes a supermartingale into a martingale and an increasing adapted process. Martingale Representation seeks to decomposes a random variable into Ito's Integral. ...
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Let $(B_t)_{t\geq 0}$ be a brownian motion, $P$ the measure of the probability space which satisfys the usual conditions and $\mathbb E$ the expected value. I like to show $$P\left(\inf_{t\geq0}\int_0^... 0answers 30 views ### Integral of a function of Brownian Motion I am trying to integrate,$$\int_{0}^{t}e^{\alpha s - B(s)}ds$$where, B(s) is the standard Brownian motion. My approach was to use integrate by parts,$$\int_{0}^{t}e^{\alpha s - B(s)}ds = e^{\alpha ...
Ito formula is defined as a process of the form $$X(t)=X(0)+\int_0^t\delta(s)dW_s +\int_0^t\theta(s)ds$$ one obvious way to simulate it is to first generate a Brownian motion and then use the Euler ...