# Questions tagged [stochastic-approximation]

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11
questions

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### Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...

1
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1
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### Phase space Brownian bridge

I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$
s.t. $x_0=0$ and $x_1=0$
where $dw_t$ is a Wiener process.
I am thinking about ...

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1
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### Understanding the approximation of a random sum of random processes

I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...

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2
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### A double sum with complex numbers having stochastic variables

I am very confused by a sum I have been trying to solve analytically/ numerically for a long time. It comes from the idea of a physical problem where the observation is made that has a combined ...

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0
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### Adiabatic elimination of "fast"/"velocity" variable

My question comes from section IV, part A of the paper titled Stochastic resonance. Specifically, their equation (4.1) states that, if we start with a Langevin equation of the form $$m\ddot{x} = -m\...

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### A semimartingale interpolation problem

This question is a direct extension of this one.
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...

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### A martingale extension/interpolation problem

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a partition of $[0,T]$ with $t_0=0,t_n<t_{n+1},t_N=...

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### The stochastic approximation algorithm of Robbins-Monro

I am reading A Stochastic Approximation Method by Herbert Robbins and Sutton Monro and have a question concerning their algorithm. In below, I will basically follow their construction but will change ...

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### Applications of products of random matrices

I'm studying the paper "Matrix concentration for products" and I'm trying to find simple applications of the inequalities for the expected value of the spectral norm of products of random ...

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### Minimizing $\int\lambda({\rm d}y)\frac{\left|g(y)-\frac{p(y)}c\lambda g\right|^2}{r((i,x),y)}$ with respect to discrete parameter $i$

Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to(...

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### Reference for the positive probability of convergence to a stable point of a stochastic approximation algorithm

Consider a stochastic approximation process with
$$x_{t+1} = x_t + \frac{1}{t} (g(x_t)+u_t)$$
where $(u_s)_s$ is a sequence of i.i.d. shocks.
Assume $g$ is Lipschitz, $u_t$ has finite variance, and ...