# Tag Info

## New answers tagged stochastic-processes

### Penalty shootout

Of course, one can write an expression for the probability in question as a certain sum over the set $\{-1,0,1\}^N$. What kind of expression for this probability do you want? Given the three ...
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### Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function

It is sufficient that $\inf_x \sigma(x)>0$ and $\sup_x \sigma(x)<\infty$, and $f$ does not have to be monotone. In this case, denoting $\mathcal L f(x) = \frac{1}{2}\sigma(x)f''(x)+\mu(x)f'(x)$, ...
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• 4,861
Accepted

### Does $X_t$ with $t>0$ admit a density?

I believe the answer is yes. By independence of $X_0$ from the Brownian increments, by the claim here we can write $X_t = F(X_0, \omega)$, where $F(x, \omega) := x + \int_0^t\sigma(s, x) \, dB_s$. By ...
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### The equivalence of stochastic quantization and path integral quantization

Some older sources than https://arxiv.org/pdf/hep-th/0312315 that prove the equivalence between stochastic quantisation and regular QFT (Euclidean path integrals) include: Equivalence of stochastic ...
• 180k
1 vote
Accepted

### Conditional expectation w.r.t filtration of Brownian motion as a continuous map of its paths

The answer to the question as stated is no. Take the SDE coefficients $\alpha$ and $\beta$ to be deterministic, then $X_t$ is $\mathcal F_t$ measurable, so that $\mathbb E[X_t | \mathcal F_t] = X_t$ ...
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$\newcommand\om\omega\newcommand\Om\Omega\newcommand\F{\mathcal F}$Let $(\Om,\F,P)$ be a probability space. A necessary and sufficient condition for $(\Om,\F,P)$ to support a sequence of independent ...