# Tagged Questions

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

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### Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
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### what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
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### A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that ...
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### Poisson process and Heaviside function

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside's function: ...
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### Moments of random variable distributed as the product of normal cdf and pdf [on hold]

How do I find the first and second moments of a random variable whose pdf is $c\Phi(-Y-a)\phi(Y-a)$ Where $\Phi$ is the standard normal cdf, $\phi$ is the standard normal pdf, c is a normalising ...
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### stochastic calculus, stopping time, ito integral vector brownian motion

I'm referring to chapter 4, question 7 in Harrison's book 'Brownian Motion and Stochastic Flow Systems.' Problem In the setting of (9) let $f_{n}(x)=E_{x}[\int_{0}^{T}X_{t}^{n}dt]$. Use Ito's ...
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### Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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### Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as ...
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### Conditional probability of geometric brownian motion [on hold]

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
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### Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
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### Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
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### Should I ask about math books? I.e. scan of index page or references page? [on hold]

I mean it's not harmful to anyone if I ask about certain page of certain book? I need page 263 from this book ...
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### Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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### Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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### How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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### Equivalence of two Ito formulae

Let $X$ and $Y$ be two $1$-dimensional Ito processes. There are two Ito formulae for the product $X_tY_t$ given by $d\left(X_tY_t\right)=X_tdY_t+Y_tdX_t+d\left[X_t,Y_t\right]$ ...
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### What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...