# Tagged Questions

Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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### Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as I'...
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### Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
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### càdlàg adapted process of finite variation

$X$ is a semimartingale with $X_0=0$. I have to show, that $S_t:=\prod^{}_{s\le t}(1+\Delta X_s)\exp(-\Delta X_s)$ is a càdlàg adapted process of finite variation. Could you please help me?
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### Estimate for average probability of Ito diffusion falls into an interval

Denote $E^x(X_t)$ be the solution to a Ito diffusion starting with $X_0=x$. Let $K\subset \mathbb{R}$ be a compact subset. I also assume $X^x_t$ has transition probability $p(t,y,x)$. Currently I am ...
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### Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
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### Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
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### Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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### Models for Probability Density Functions with unknown parameters and given mean and variance

The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not ...
In "Mitter SK, Newton NJ. A Variational Approach to Nonlinear Estimation. SIAM J Control Optim. 2003 Jan;42(5):1813–33", it is shown that the path estimation measure $P_{X|Y}(\cdot,y)$ for the ...