Questions tagged [stochastic-processes]
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22
questions
3
votes
0answers
24 views
Value function iteration with stochastic productivity's standard deviation
Hello I would like to know how would you discretize the AR(1) process of technology in a standart RBC model when there is stochastic productivity's standard deviation. Namely I have:
Technology $Z_t$ ...
4
votes
0answers
37 views
Value function iteration with habit
I would like to know how I could write a value function when there are habits in preferences. I have the following equations:
$$
u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
4
votes
1answer
81 views
Stochastic AK model derivation
Consider the following version of the stochastic Ak model written as a Bellman equation:
$$v(A,k)=max\ log(c)+\beta E[v(A',k')|A]$$
$$s.t\ k'+c\leq Ak$$ and non-negatitvities.
$A$ is a stationary ...
1
vote
0answers
23 views
Intuitive/Practical meaning of non-stationarity of GDP Data
As i just read in a time series book that a particular GDP data under consideration is non-stationary verified through various tests. From non-stationarity definition this means that the process has ...
0
votes
1answer
136 views
Are overlapping generation (OLG) models extensions of a DSGE model?
Are overlapping generation models (OLG) extensions of a dynamic stochastic general equilibrium (DSGE) model? Or aren't these DSGE per se?
1
vote
1answer
40 views
Expectational stability: adaptive learning of RE equilibria in dynamic systems
There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand.
Step 1.
What does this formula below mean, ...
0
votes
1answer
36 views
Limit of random walk auto correlation function
Given the random walk process $y_{t}=y_{t-1}+e_{t}$, the auto correlation function is given by $corr(y_{t}, y_{t-h})=(\frac{t-h}{t})^{1/2}=(1-\frac{h}{t})^{1/2}$, which tends to 0 as t tends to ...
0
votes
2answers
190 views
Autocorrelation function of a random walk process
What is the intuition behind the result that the autocorrelation function of a random walk process $y_{t}=y_{t-1}+e_{t}$ tends to 1 as $t\rightarrow 0$? Thank you.
2
votes
1answer
71 views
Purpose of Semidefinite Integral
I want to know the meaning of Semidefinite Integral.
I am used to read definite and indefinite integral but I want to know the meaning of such equation :
$\pi(e)\left(1-F\left[-\frac{a}{\pi(e)}\...
-1
votes
1answer
112 views
Stochastic process difference equation: stationary distribution
How can I find the stationary distribution (as t goes to infinity) of stochastic difference equations in the form:
$x_{t+1} = a*x_t + b*N(0,1)$
where N(0,1) is a standard normal pdf
I have ...
1
vote
0answers
99 views
Stochastic Dynamic Programming: Deriving the Steady-State for a Lottery
I am working through the basic examples of the stochastic RBC models in the book by McCandless (2008): The ABCs of RBCs, pp. 71 - 75
A Standard Stochastic Dynamic Programming Problem
Here is a ...
3
votes
1answer
72 views
Profit maximization under uncertainity
I have a seller say S and I have a buyer say B. Buyer’s willing to pay is equal to x which is private information. But Seller believe that it falls in the range [0,x1]. Seller’s belief distribution is ...
1
vote
0answers
33 views
Generalization of Tauchen 1986 approach to a case of time-varying volatility
My question is about generalization of Tauchen'86 approach to a case of time-varying volatility.
Say, I have a process $$z_{t+1}=\rho z_t+\sigma_t \varepsilon_{t+1}$$ where $\varepsilon\sim \mathcal{...
1
vote
0answers
88 views
How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?
Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
2
votes
1answer
69 views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
7
votes
2answers
299 views
Application of Poisson process in economic modelling
To understand the emergence of constitution, Myerson (2008) models a scernario that a political leader gathers supports from captains in order to defeat challengers whose arrival is modelled by a ...
8
votes
2answers
1k views
Transition Matrix: Discrete -> Continuous Time
I have the code corresponding to Tauchen (1986) (Python equivalent of this), which generates a discrete approximation of a discrete time AR(1) process.
For example, if you set up grid size as 3, it ...
4
votes
1answer
1k views
Proving there exists no arbitrage opportunities given 3 states and 2 assets
Assume there are 3 states of the world: w1, w2, and w3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with Return R1 in state w1, R2 in state w2, and R3 ...
5
votes
2answers
131 views
Find probability that payoff function is in $[10,20]$
In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:
$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$
where $S_t$ satisfies
$$dS_t=15dW_t$$
$$S_0=95$$
...
5
votes
2answers
147 views
Decomposition of an additive functional into a Martingale part and other
This question relates to a theorem about the decomposition of additive functionals---a technique especially useful in macroeconomics and finance. This question has two objective. First, I don't have a ...
6
votes
1answer
103 views
Showing that a transformation is measure preserving
Note: This question is related to this question about the construction of stochastic processes. Specifically, it relates to the transformation $\mathbb S: \Omega \rightarrow \Omega$ that is mentioned. ...
12
votes
3answers
925 views
Understanding the construction of stochastic processes
I've seen stochastic processes modeled/constructed in the following way.
Consider the
probability space $(\Omega, \mathcal F, Pr)$ and let $\mathbb S$ be the (measurable)
transformation $\...
