# Tag Info

## Hot answers tagged stochastic-integrals

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### Why predictable processes?

Being predictable (as in being measurable with respect to the predictable $\sigma$-algebra) always implies being progressive (as in being measurable with respect to the progressive $\sigma$-algebra), ...
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### Stochastic Differential Equation solution for Geometric Brownian Motion

Your integration step is wrong. $Z$ here refers to Brownian motion, and thus you need to apply Ito Integration. One way I like to see the extra drift is needed, is that we take derivative of the ...
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### Ito representation unique up to indistinguishability? Proof?

Since $$0 = \int_0^t F(s) \, ds + \int_0^t G(s) \, dW_s$$ we have $$M_t := \int_0^t G(s) \, dW_s = - \int_0^t F(s) \, ds.$$ In particular, $(M_t)_{t \geq 0}$ is a martingale with continuous ...
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### 4th moment of a Wiener stochastic integral?

Although you don't mention it explicitly, your calculations suggest that $g$ is a deterministic function; I'll assume this throughout my answer. If $g$ is not deterministic, then things are getting ...
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### Stochastic differential equation with respect to general stochastic process

You can if you want. Soon you will learn Ito's lemma, which states that $${\rm d}f(X_t)=f'(X_t)\,{\rm d}X_t+\frac{1}{2}f''(X_t)\,{\rm d}\left<X\right>_t$$ for all Ito processes $X_t$ and twice ...
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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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### Table of Ito Integrals

$\newcommand{\d}{\mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - \tfrac{1}{2}t}$: \begin{array} {|r|r|} \hline X_t & \d ...
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### Solution of SDE $dX_t = \mu(t)X_tdt + \sigma X_t dW_t$

The reason $\ln(x)$ is chosen is this: (Note: where I say $B_{t}$ I mean Brownian motion, which you denoted in your question as $W_{t}$) The SDE you provided is one of the few we can explicitly ...
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### Stochastic Differential Equation solution for Geometric Brownian Motion

I believe the answer by @Yujie Zha can be simplified substantially. Thanks to @Dr. Lutz Lehmann for providing a link to this, my solution is the same as the solution on page 15, but with more ...
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### Is the Ito integral of a predictable process Gaussian distributed?

To follow up on spaceisdarkgreen's answer: For any distribution on the real line there is an $\mathcal F_T$-measurable random variable with that distribution. (Even of the form $g(W_T)$ for an ...
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### Solve the SDE $dX_t=\sqrt t(X_t+\sin t)dW_t$

The SDE is a particular example of a so-called linear SDE $$dX_t = (\alpha(t)+\beta(t) X_t) \, dt + (\gamma(t)+\delta(t) X_t) \, dW_t \tag{1}$$ where $\alpha, \beta,\gamma,\delta$ are deterministic ...
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Since you only impose mild assumptions on $f$, $g$, the proof is somewhat technical, e.g. we cannot work with Riemann sums because $f^2$ might not be Riemann integrable. Without loss of generality, I ...