Questions tagged [stochastic-approximation]
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13 questions
0 votes
1 answer
114 views
Optimal probing problem
We are provided with a set of $n$ targets. Each target is characterized by a utility value. We know the distribution of the utility value for each target, but do not know its current value. Therefore, ...
- 327
2 votes
0 answers
74 views
Approximate the adjoint generator of the discretization of an SDE
Let $d\in\mathbb N$; $\sigma\in\mathbb R^{d\times d}$; $p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$ $(X_t)_{t\ge0}$ denote ...
- 249
2 votes
0 answers
231 views
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 ...
- 121
1 vote
1 answer
150 views
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 ...
- 13
0 votes
1 answer
231 views
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 ...
- 135
1 vote
2 answers
234 views
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 ...
- 159
1 vote
0 answers
225 views
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 $$m\ddot{x} = -m\...
- 732
1 vote
0 answers
74 views
A semimartingale interpolation problem
This question is a direct extension of this one. Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...
- 77
0 votes
1 answer
130 views
A martingale extension/interpolation problem
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a partition of $[0,T]$ with $t_0=0,t_n<t_{n+1},t_N=...
- 77
2 votes
0 answers
135 views
The stochastic approximation algorithm of Robbins-Monro
I am reading A Stochastic Approximation Method by Herbert Robbins and Sutton Monro and have a question concerning their algorithm. In below, I will basically follow their construction but will change ...
- 1,575
3 votes
0 answers
153 views
Applications of products of random matrices
I'm studying the paper "Matrix concentration for products" and I'm trying to find simple applications of the inequalities for the expected value of the spectral norm of products of random ...
- 183
2 votes
0 answers
92 views
Minimizing $\int\lambda({\rm d}y)\frac{\left|g(y)-\frac{p(y)}c\lambda g\right|^2}{r((i,x),y)}$ with respect to discrete parameter $i$
Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to(...
- 249
1 vote
0 answers
110 views
Reference for the positive probability of convergence to a stable point of a stochastic approximation algorithm
Consider a stochastic approximation process with $$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 ...
- 355