# Tagged Questions

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

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### Yet another application of Ito's formula

Question : Let $dW^4(t)$ be the sum of an ordinary integral with respect to time and an Ito integral. Where $W^4(t)$ are standard Brownian motion. I am trying to apply Ito's formula to this, say ...
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### SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $dr_t = [v_t - ar_t]dt + \sigma dW_t,$ where $a$ is constant and $v,t$ are, potentially, time-...
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### Use of Itô isometry for correlation calculation

When calculating the covariance of the Ornstein-Uhlenbeck process, the Wikipedia article applies implicitly the Itô isometry with the fact of non-overlapping independent increments of the Wiener ...
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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 ...