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2 votes
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Showing bounds of Stochastic Process

Consider the equation $$ \mathrm{d}Y_t = \biggl( 2 + 8 \frac{e^{Y_t} - 1}{e^{Y_t} + 1} \biggr) \, \mathrm{d}t + 4 \, \mathrm{d}B_t. \tag{1}\label{e:1} $$ The coefficients $\mu(y, t) = 2+\operatorname{...
Sangchul Lee's user avatar
1 vote

Expressing a continuous local martingale as an integral against a Brownian motion

Since $X_t$ is constant when $A_t = 0$, we have $\int_0^t 1_{A_s = 0}dX_t = 0$. Since you showed $A^{1/2}_t dB_t = 1_{A_t>0}dX_t$, we have \begin{align*} \int_0^t A^{1/2}_s dB_s &= \int_0^t 1_{...
user6247850's user avatar
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1 vote
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Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

Note that we can rewrite the random variable in the following way: $$\begin{aligned} &e^{-\gamma^2t/2+\gamma W_t}E^P[e^{\gamma^2T/2-\gamma W_T}F(Y_T)|\mathscr{F}_t]\\ &=e^{-\gamma^2t/2+\gamma ...
Snoop's user avatar
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How to calculate this easy stochastic integral?

At least, it is easy to compute the law of this stochastic integral. $$\int_{0}^{t} s \,dWs = \lim_{n\to\infty} \sum_{k=1}^{n} t_k(W_{t_{k+1}} - W_{t_k})$$ such that it is a limit of gaussian random ...
Mattttcan's user avatar
0 votes

Integration of Gaussian noise process

The Gaussian noise you describe is not White noise, because you have defined X to have finite variance. If the covariance function of the process you are talking about is a constant times a Kronecker ...
Matthew's user avatar
  • 21
2 votes
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Variance of an Itô Integral

The variance of $$ \int_t^TZ_s-Z_t\ ds $$ is \begin{align}&\textstyle \mathbb E\left[\left(\int_t^TZ_s-Z_t\ ds\right)\left(\int_t^TZ_u-Z_t\ du\right)\right]=\displaystyle\int_t^T\int_t^T\mathbb E\...
Kurt G.'s user avatar
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0 votes

Prove that $(t+1) X_{\frac{t}{t+1}}$ is a Brownian Motion using Levy's characterisation where $X$ is a Brownian Bridge

The following is a partial proof with one assumption needed to get it over the line. This assumption is labelled as such below. Martingale Property By Ito's Product Rule, $$d \beta _t = (t+1) dX_{\...
FD_bfa's user avatar
  • 4,303
1 vote

Solving this SDE $dX_t = aX_tdt + bdW_t$, $X_0 = x$ to find $E[X_t^2]$

This question is old but there are errors in signs and many typos in the accepted answer above. The answer is also has many upvotes (and rightfully so). But still, it posseses an issue for a reader. ...
Mr.Gandalf Sauron's user avatar
0 votes

Prove that $(t+1) X_{\frac{t}{t+1}}$ is a Brownian Motion using Levy's characterisation where $X$ is a Brownian Bridge

This is more a very long comment: Formally all your formulas are correct but the conclusion at the end of Method 2 is implausible to me. From $$ \beta_t=(1+t)X_{\tfrac t{1+t}}\,, $$ we have $$ \beta_{\...
Kurt G.'s user avatar
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