# Tagged Questions

Questions on the calculus of stochastic processes, or processes that have a random component.

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### Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
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### Lifetime of a spaceship run by three computers

A spaceship is controlled by three independent computers. The ship can function as long as at least two of the three computers are functioning. Suppose the lifetimes of the computers are i.i.d. ...
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### Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
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### Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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### Fourier transform of n-th power of autocorrelation of a random process

I'm having troubles in understanding how Fourier transform of the n-th power of a time function is obtained. In particular I came across to a particular result with respect to the calculation of the ...
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### Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$dV_t = -\beta V_t dt + \sigma dB_t$$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
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### Running average of Brownian motion

Question : Let us define the cumulative sum (Brownian motion): $$x_k = \sum_{i=1}^k y_i$$ and the running average : $$\overline{x_k} =\frac{1}{W}\sum_{i=k-W+1}^k x_i$$ for $k>W$, $W$ ...
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### Relation between two stochastics

I got stuck trying to figuring out how to show the following question in probabilistic theory: "We say that A and B (with P(A), P(B) > 0) attract each other when P(A|B) > P(A)." I've shown that ...
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### Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
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### Poisson integral and discontinuous martingale (Ito-Levy formula)

Consider compounded Poisson process $P$ given by $P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz)$ where $N$ is a Poisson random measure of intensity $dt \otimes \nu$ and $\nu$ is a Levy measure. Why ...
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### Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
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### Distribution of Levy driven O-U process

Is there a way to find an analytical expression for $E\left[\exp\left(-\int_0^T \gamma_s ds\right)\right]$, where $d\gamma_t=k(\theta-\gamma_t)dt+\sigma dL_t$, and $L_t$ is a symmetric alpha ...
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### Relation between autocorrelation and mean of a stochastic process

It is said that if the autocorrelation approaches zero as $\tau$ tends to zero, then the mean of the stochastic process is also zero. I am having trouble understanding the above concept. Say we have ...
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### What does it mean by “mean of a process”?

Say $X(t)$ is a stochastic process. Now when it says that the mean of the process, does it mean that the mean of $X(t)$. Elaborating further, a process is an collection of random variables - ...
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### $\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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### Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
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### An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
I have the following: Let $(X_n)$ be a sequence of i.i.d. random variables. (a) Assume $\frac{1}{n} S_n=\frac{1}{n} \sum_{i=1}^n X_i$ converges a.s. to a real-valued random variable $Y$. Show that ...