Questions tagged [stochastic-processes]
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34
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Granger-Sims causality and subtle differences
For a bivariate process $(\textbf{X},\textbf{Y})=( (X_t, Y_t)^\top, t\in\mathbb{Z})$, we say that the process $\textbf{X}$ Sims-causes the process $\textbf{Y}$ (notation $\textbf{X}\overset{Sims}{\to}...
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transition probabilities from a AR(1) stochastic process
I have a stochastic volatility model for commodity price which follows an AR(1) process:
ln(pt ) − m = ρ (ln(pt−1) − m) + exp(σt)ut ut ∼ IID(0, 1)
σt − μ = ρσ(σt−1 − μ) + ηεt εt ∼ IID(0, 1)
...
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Stochastic control of jumps of random size
Consider the problem of maximizing expected lifetime utility
$$
V(a_t) \equiv \max_c\mathrm{E}_t \int_t^\infty e^{\rho (s - t)}u(c_t)\mathrm{d}t
$$
subject to a state process $\mathrm{d}a_t$ which is ...
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Stochastic optimal control problem (calculus)
I am reading a paper with a stochastic optimal control problem. At one point the author faces the following Hamilton-Jacobi-Bellman (HJB) equation:
$$\rho V(A)=\max_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)...
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Applications of a certain type of stochastic processes in macroeconomic, macroeconometric or finance
A compound Poisson random vector $Y$ is well defined in this site in wikipidia.
Nothing prevents me from compound strictly stationary stochastic processes instead of compound random vectors. The ...
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Dynamic Information Provision model setup - It generalizes Dirk Bergemann and Stephen Morris
The following model setup is from the paper Dynamic Information Provision: Rewarding the Past and Guiding the Future by Ian Ball. It generalizes both the ideas of strategic information transmission of ...
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Market price of interest rate risk under the CIR model
My goal is to find the market price of risk associated with the interest rate under the CIR model whose stochastic differential equation under the physical measure is given:
\begin{eqnarray}\label{...
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Bayes’ rule in "The sources of capital misallocation"
I am reading a paper titled "The sources of capital misallocation". In the model, firms are facing incomplete information about their future productivities. In particular, the productivity ...
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A conceptual question about the limitation of the MA processes
We know that linear time-series techniques are frequently used in macroeconometrics. The Wold Representation Theorem states that any covariance-stationary process may be expressed as an MA process ...
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Derivation of autocovariances Lewis (2021) RES
I am studying this paper, and I don't understand the derivation of the covariances at the bottom of page 3090.
Basically I have two shocks: $\varepsilon_{1t}$ has constant volatility $E[\varepsilon_{...
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Is playing against state of nature considered Stochastic game or Bayesian game?
Say if there is a team of gamblers betting on stock exchange. We can model the outcome of stock exchange as state of nature, because it is not deterministic. So the objective of team of gamblers is to ...
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Bond Price expression
I've researching some mathematical finance and I've stumbled upon something I can't seems to find sources on. I'm probably overlooking something, but I hope someone can enlighten me and give me some ...
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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$ ...
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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}\...
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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 ...
- 1,048
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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 ...
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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?
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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, ...
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80
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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 ...
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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.
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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)}\...
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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 ...
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130
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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 ...
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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 ...
user15967
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49
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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{...
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How to use Girsanov theorem to prove $\hat{W_t}$ is $\hat{\mathbb P}$-Brownian motion?
Assumptions:
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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$dS_t=15dW_t$$
$$S_0=95$$
...
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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 ...
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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. ...
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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 $\...
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