# Tagged Questions

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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### Changing domain of the CGF of a stochastic process

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search ...
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### Ito's formula and Taylor expansions for jumps processes.

Consider some model $$dX_t = \mu d t + \sigma dW_t$$ where $\mu, \sigma$ are some constants. Now let $f \in C^{1,2}$ and consider $$Y_t = f(t,X_t).$$ Say we (informally) consider a second order ...
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### Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that ...
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### If M is a transition matrix, 1 is one of its eigenvalues?

If $A$ is an $n\times n$ Markov, Transition or Stochastic matrix (i.e. $A_{ij} \geq 0$ and $\sum_{i=1}^n A_{ij}=1$) one of its eigenvalues is equal to 1?
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### Expectation of product of Geometric Brownian Motion

If I have two GBMs where $s < t$ $A_t = A_0 \exp((a-\frac{1}{2}\sigma_A^2)t+\sigma_AW_t^A)$ $B_s = B_0 \exp((a-\frac{1}{2}\sigma_B^2)s+\sigma_BW_t^B)$ How would I calculate $E(A_tB_s)$ My ...
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### Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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### How do you find the probability of a brownian motion?

If $B(t)$ is a brownian motion what do these two questions mean? 1. What is the probability of $B(2)$ 2. What is the probability of $B(2) \gt B(1)$ I know this is also called a Wiener Process and ...
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### Problem with compound Poisson process

Let $X_k$ for $k=1,2,...$ be a sequence of i.i.d. random variables with $\mu_k=0$ and $\sigma_k^2=1$ for all $k$. Consider de random process $$S(t)=\sum\limits_{k=1}^{N(t)}X_k$$ where $N(t)$ is a ...
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### Function evaluated in Brownian motion vanishes implies that the function itself vanishes?

tl;dr, here's my question: Question. Let $f(t,x)$ be a measurable function such that $f(t,B_t)=0$ almost everywhere on $[0,T]\times\Omega$ for a Brownian motion $B_t$. Does this imply that ...
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### Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
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### Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$
The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...