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Are differentials on their own in stochastic calculus just an abuse of notation?

Differentials in stochastic calculus have very precise interpretations. The stochastic differential equation in differential form $$dX_t = \mu (t, X_t) dt + \sigma (t, X_t) dB_t$$ translates precisely ...
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Stochastic integration by parts of geometric brownian motion

You can't go any further than $\int e^{B_{t}+at}dt$. This is called the integrated Geometric Brownian motion. For references on its law see "The Integral of Geometric Brownian Motion" &...
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$M_t=\int_0^t e^{-3W_s}dW_s$ properties

Is $M_t$ well defined when $\mathbb EM_t<\infty$? Is this enough or do I need to check any other conditions? Any random variable is well-defined when its expectation is bounded Does finite ...
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What is the expectation of the positive part of a stochastic integral?

Being a square-integrable martingale, its expectation is zero as pointed out by other users. Going further, consider the function $f(x) = \frac{1}{2}\max\{0, x\}^2$. Although this is not a $C^2$-...
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Here we simply apply Ito isometry. So as explained Quadratic Variations and the Ito Isometry we have $$EX_{t}X_{r}=E[\int^t b ds\int^r b ds]+E[\int^t b ds \int^r \sigma dB_{s}]+E[\int^r b ds \int^t \... • 3,716 2 votes Apply Ito to exp((\mu-0.5*\sigma^2)t+\sigma*W_t) for Brownian Motion W_t The problem here is how you have written Ito's formula, the way you have it written is actually incorrect if we are considering a general stochastic process X_t, the \sigma^2 term appears from the ... • 495 2 votes Differentiating Stochastic Integrals (Ito Integrals) X_{t} is not in general differentiable in the sense of ordinary calculus. The notation$$ dX_{t}=B_{t}^{2}dB_{t} $$is equivalent to$$ X_{t_{1}}-X_{t_{0}}=\int_{t_{0}}^{t_{1}}B_{t}^{2}dB_{t}\qquad\...
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