# Tagged Questions

This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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### Ito's Integral's definition: Importance of isometry

I'm reading Oksendal's Stochastic Differential Equations (5th edition). He defines the Ito integral of $f$ as the limit $$\lim_{n\to\infty} \int^T_S \phi_n(t,\omega) dB_t(\omega)$$ Where $\{\phi_n\}$ ...
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### What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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### Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
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### Predictable Processes in Brownian Setting

Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable. But from what I can recall, in the traditional ...
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### The Itō Integral

In stochastic calculus and specifically for mathematical finance Ito's lemma is used for time varying processes I need to know intuitively why the Ito Integral is stochastic?
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I need an help with the following exercise. I would like to know if what I've done is correct. Let $(X_t)_{t\geq 0}$ be the process define as $$X_t=e^{\lambda t} X_0-\sigma \Big(\lambda \int_0^t e^{-... 1answer 17 views ### Conditional expectations one more time Please someone verifies my results: 1) E \Big( \int_0^3W_t^2dt|F_1\Big)=(editing in progress) 2) E \Big( \int_0^2 (tW_t+t^2)dt|F_1\Big)=E \Big( \int_0^2 tW_tdt|F_1\Big)+E \Big( \int_0^2 t^2dt\... 1answer 77 views ### Integral on interval [-\infty,W_t], W_t is Brownian motion Basicaly I have an expectation of an integral on the interval which contains Brownian motion and it look like this.$$ E\left[e^{W_t}\cdot\int_{-\infty}^{W_t} e^{-z^2}dz\right] $$W_t is Brownian ... 1answer 59 views ### How to decompose X_t^2 as an Itô process? I am given the stochastic process X_t to be the unique process starting at X_0 and solution of the following SDE:$$dX_t = (a-bX_t)\,dt + \sigma \sqrt{X_t} \, dW_t,$$where W_t is a real ... 0answers 97 views ### Sufficient condition for martingale property Let (\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P}) be a filtered probability space and M=(M_t)_{t\geq 0} an \mathcal{F}_t-adapted stochastic process. If$$ \forall t<s, \ \mathbb{...
I need an help with the following exercise. Let $(W_t)_{t\geq 0}$ a Wiener process on $(\Omega, \mathcal E, \mathbb P)$ and let $I=[0,T]$ be an interval. We want to prove that the Stratonovich ...
I am trying to compute the following expectation: $$M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right]$$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...