# Tagged Questions

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

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### Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
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### Mean Value of a Random Process

Consider a random process $X(t) = Z(t)\sin(wt-Q)$. Here $Q$ is a random variable taking values $q$ in $[-\pi/2,\pi/2]$ with PDF given by $$p_1^Q(q) = \frac{\cos(q)}{2}$$ $Z(t)$ is some random ...
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### The Stratonovich Integral and its meaning as the limit in mean square of a sum?

I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as: $$\int_0^T X_t \circ dW_t$$ ...
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### Construct a martingale with a given distribution?

Given a random variable Y, is it possible to construct a martingale M such that $$M_1 \stackrel{D}{=} Y$$ I'm not sure how to go about proving that such an M exists under such general conditions, but ...
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### How many types of martingale related stochastic processes are there?

Previously I had thought that the only concepts in this direction were martingales, submartingales, and supermartingales. However, at least when discussing quadratic variation and stochastic ...
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### Rephrase a multiparameter SDE indexed by time and space as an infinite dimensional SDE indexed by time

Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle ...
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### Stochastic Different Equation

Consider the stochastic differential equation $\frac{dX_t}{X_t}=adt+bdW_t$ for the diffusion $X_t$ . The parameters $a,b$ are constant.Using Ito's lemma and suitable integration over $[0,T]$, show ...
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### Generating a list of numbers

A set of numbers is generated starting from $0$ in the following way: Add the current number to the resultset In a chance of 50:50, do Either add $2$ to the current number Or subtract $1$ from the ...
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### Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: $$X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1},$$...
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### Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
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### Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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Say we have a Gausian process $X_s=\int_0^sh(x)\,dW(x)$ where $W(x)$ is a Wiener process. Now define $$Z=\frac{\int_0^1\operatorname{sign}(X_s) \, dW(s)}{\int_0^1|X_s| \, ds}$$ Intuitively we have $EZ=... 1answer 50 views ### How can we identify$\omega\in\Omega$with a path of Brownian motion$t\rightarrow B_t(\omega)$? In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ... 0answers 25 views ### PDE for Brownian Bridge Expectation? Let$\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where$B(t)$is the standard Brownian motion and$v(t)$a deterministic function. Compute$m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} Y(s)\big|Y(t)=y\...
Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
### $X_t=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{n}})dB(u)$ is a Brownian motion for suitable non-zero constants $a_0,\ldots,a_n$
Let $B(t)$ be brownian motion. Show that for any integer $n \geq 1$, there exist nonzero constants $a_{0},\ldots,a_{n}$ such that \$X_{t}=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{...