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17 votes
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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), ...
Alexander Sokol's user avatar
14 votes
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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 ...
Jay Zha's user avatar
  • 7,832
12 votes
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Some version of Itô isometry with conditional expectations

Let $G \in \mathcal{F}_r$. Using that $$1_G \int_r^t u_s \, dB_s = \int_r^t 1_G u_s \, dB_s \quad \text{a.s.} $$ (see the lemma below) it follows from Itô's isometry that $$\begin{align*} \int_G \...
saz's user avatar
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11 votes
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Integral of Wiener Squared process

That should not be true. The problem is that you're not considering the square of the stochastic integral, i.e. the random variable $J= (\int_0^1 {W_s} ds)^2$, which would indeed be the square of a ...
Pasriv's user avatar
  • 1,070
11 votes

Product Rule for Ito Processes

Note Let $X_t$ and $Y_t$ be two Ito processes. By application of Ito's lemma, we have $$d(X_t\,Y_t)=Y_t\,dX_t+X_t\,dY_t+d[X_t\,,\,Y_t]$$ Now, if $Y_t$ is of finite variation, then covariation $[X_t\...
Behrouz Maleki's user avatar
11 votes
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Conversion between solution to Stratonovich SDE and Itô SDE

You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following hold: $h \colon \mathbb{R}^n \to \mathbb{R}^n$, $\gamma = (\gamma_{ij})_{i,j = 1}^{...
Arthur11's user avatar
  • 1,016
10 votes
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prove that Doléans-Dade exponential is a local martingale

Define $X_t = -\int_0^t\beta_s\,dW_s$ and $Y_t = -\frac{1}{2}\int_0^t\beta_s^2\,ds$. Then $Z_t = e^{X_t+Y_t}$. Even though you did not mention it I am guessing that there is a condition on $(\beta_s)_{...
Calculon's user avatar
  • 5,755
10 votes
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When is a stochastic integral a martingale?

Well, in the form stated above, none of the statements are true, because you're only assuming $f$ to be progressive and not predictable, and you're not assuming that the integrator $X$ has continuous ...
Alexander Sokol's user avatar
10 votes
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Understand better stochastic integral through a.s. convergence

The convergence of the subsequence holds almost surely, i.e. there is an exceptional null set where convergence fails to hold. This null set depends on the partitioning sequence $(t^{(n)})_{n \geq 1}$....
saz's user avatar
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9 votes

Expectation of geometric brownian motion

Even though it's pretty late I think this still may be helpful for a few of you. First of all notice as $B_t$ is a geometric Brownian motion, by definition it is normally distributed with mean $0$ ...
dcutrr's user avatar
  • 91
9 votes

Product Rule for Ito Processes

This can be obtained directly from Ito's product rule: $$ d(X(t)Y(t)) = X(t)dY(t) + Y(t)dX(t) + dX(t)dY(t) $$ For illustration, assume your $dX(t)$ and $dp(t)$ has form: $$ d(X_t) = \mu_1dt + \...
Son Le's user avatar
  • 91
9 votes
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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 ...
saz's user avatar
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9 votes
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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 ...
saz's user avatar
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9 votes
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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 ...
hypernova's user avatar
  • 6,112
9 votes

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 ...
Jose Avilez's user avatar
  • 13.2k
8 votes

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 ...
Pantelis Sopasakis's user avatar
8 votes
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Step in proof of Ito isometry

Due to the settings of a brownian motion $(B_{t_{j+1}} - B_{t_j})$ is independent of $F_{t_j}$ so for every measurable function is holds $$E[f(B_{t_{j+1}} - B_{t_j}) \mid \mathcal{F}_{t_j}] = E[f(B_{...
Gono's user avatar
  • 5,643
8 votes
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Difference between Ito calculus and Malliavin calculus

The Ito calculus extends the methods of classical calculus to stochastic functions of random variables. The Malliavin calculus extends the classical calculus of variations to stochastic functions. ...
user3658307's user avatar
  • 10.4k
8 votes
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Is the Ito integral of a predictable process Gaussian distributed?

No. Let $f_t = 1$ for all $t$ with probability $1/2$ and $2$ with probability $1/2.$ $f_t$ is previsible. Then $W_T$ is a 50-50 mixture of normal with variance $T$ and $4T$, which is not a normal. ...
spaceisdarkgreen's user avatar
8 votes

Expectation of solution to SDE $dX_t=-\tanh(X_t) dt + dW_t$

Step 1: Let's first construct a weak solution to the SDE $$dX_t = - \tanh(X_t) \, dt + dB_t, \qquad X_0 = x_0. \tag{1}$$ For a Brownian motion $(W_t)_{t \geq 0}$ on a probability space $(\Omega,\...
saz's user avatar
  • 121k
8 votes
Accepted

Inequality on Brownian motions and Ito integrals

It can not be true, since the process $M$ defined by $$M_t = \int_0^t (|B_s|-B_s) ~\mathrm{d}B_s$$ is a martingale with expectation $0$. If it was non-negative almost surely, it would be null almost ...
Christophe Leuridan's user avatar
7 votes

Brownian Motion and stochastic integration on the complete real line

There are two notions that are getting mixed up here. When people speak of a Brownian motion on the real line (or more generally of a martingale on the real line) they usually refer to a martingale ...
Ronan's user avatar
  • 301
7 votes

Integral of Wiener Squared process

The distribution of $\int_0^t W^2_s\,ds$ was found by Cameron and Martin in the 1940s. Mark Kac (1949, Transactions of the AMS) applied his method to find the Laplace transform of this integral: $$ \...
John Dawkins's user avatar
  • 26.6k
7 votes
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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 ...
layman's user avatar
  • 20.3k
7 votes

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 ...
baibo's user avatar
  • 619
7 votes

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 ...
John Dawkins's user avatar
  • 26.6k
7 votes
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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 ...
saz's user avatar
  • 121k
7 votes
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Calculation of the quadratic variation of an Itô process.

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 ...
saz's user avatar
  • 121k

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