# 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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### Stochastic Process

I would like to know if anyone here could help me with this exercise. Y(t) = X(t +d) - X(t), where X(t) is a Gaussian Stochastic process. (A) Calculate the mean and covariance of Y(t) (B) Calculate ...
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### Couple/Compare two stochastic processes and prove an intuitive proposition

Consider a stochastic process (denoted $X$) $X_0, X_1, X_2, \ldots$ (not necessarily a Markov Chain) over state space $\{0, 1, \cdots, n \}$. The transition probabilities are ($n$ is the sink state) ...
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### Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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### Expected value and variance of a stochastic process

Having trouble finding expected value and variance of a stochastic process defined by SDE: $dX_{t} = a X_{t} dt + b dB_{t}$ $X_0 = x$, $a$ and $b$ are constant values, $B_t$~$N(0,t)$ Thank you for ...
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### Solving inverse problem related to Iterated function systems?

I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : How do we justify the shape generated from those equations? Is it possible to ...
### Find a process $f=f(t,W_t)$ such that another process is a martingale
Find a process $f=f(t,W_t)$ such that process: $$X_t=\exp(W_t^2-2tW_t^2)+\int_0^tf(s,W_s)ds$$ is a martingale. Justify the fact that $X_t$ is martingale. I think I should find a process such that ...