Questions tagged [continuous-time]
The continuous-time tag has no usage guidance.
1
vote
2answers
312 views
Solow model, time and steady state
Suppose we have a Solow model:
$$
Y(t)=C(t)+I(t)
$$
$$
I(t)=sY(t)
$$
$$
\dot K=I(t)-δK(t)
$$
With a given Cobb-Douglas:
$$
Y(t)=Z(t)K^aL^{1-a}
$$
$$
y(t)=Y(t)/L(t)
$$
$$
k(t)=K(t)/L(t)
$$
$$
y=...
8
votes
0answers
150 views
Proofs in the Appendix A of Sannikov (2007)
I have a few questions about the proofs in Appendix A of Sannikov (2007), Games with Imperfectly Observable Actions in Continuous Time.
In lemma 4, when he shows the Lipschitz continuity of $H_a(w,\...
2
votes
2answers
56 views
Matching problem in continuous time
Let's split each period into $n$ intervals. There's a continuum $u$ of unemployed and $v$ of vacancies. During each interval, there is a total of $X/n$ job offers. That means that each unemployed gets ...
1
vote
1answer
301 views
Update of value function in continuous time - HJB
When solving (numerically, by value function iteration) a dynamic programming problem in discrete time, such as
$$V_1(a) = \max_{c} \ u(c) + \dfrac{1}{1+\rho}V_0(a')$$
we maximize with respect to ...
5
votes
1answer
111 views
From Discrete to Continuous time: Total Differential
I'm trying to derive an HJB from a discrete time setting. At some point, I am left with
$$ \lim_{\Delta\to 0} \frac{v(c_{t+\Delta}, u_{t+\Delta}, t+\Delta) - v(c_{t}, u_{t}, t)}{\Delta}$$
and am not ...
13
votes
1answer
397 views
Stochastic growth in continuous time
Literature: See Chang (1988) for theoretical part and Achdou et
al. (2015) for numerical part respectively.
Model
Consider the following stochastic optimal growth problem in per capita notation.
\...
5
votes
2answers
397 views
Intuition of the Kolmogorov Equations
So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:
"The backward equation involves time $t$ and ...
3
votes
1answer
40 views
Help understanding expression for continuous discounting
In footnote 3 of their paper "Patent Breadth, Patent Life, and the Pace of Technological Progress", O'Donoghue Scotchmer and Thisse take a discount rate of $r$ and write
If the flow of profit $\...
1
vote
0answers
42 views
Forward-Looking HJB: Rewrite as PDV
We live in continuous time. Let there be some discount rate $D(t)$, which consists of a discount rate, and some death probability. $V(t)$ contains the flow value of, say, being alive. If you are alive,...
7
votes
1answer
232 views
Compute evolution of a distribution over time
We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the ...
1
vote
1answer
54 views
Computing the continuous time survival rate
We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the ...
4
votes
1answer
184 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 ...
5
votes
1answer
190 views
Optimal consumption in Merton-like portfolio choice model with constant wage
My Questions
Consider the following problem. It is almost identical to
the classic Merton portfolio choice problem. Here I'm solving it using
the so-called Martingale method. I have provided my ...
3
votes
0answers
127 views
Kolmogorov-Forward-Equation with Entry and Exit
I'll try to ask this question in the most simplistic environment possible, so lets think about a household that can consume and save. After some cumbersome work, we have solved for the optimal savings ...
10
votes
6answers
896 views
References to learn continuous-time dynamic programming
Does anyone know of good references to learn continuous-time dynamic programming? The references don't have to be books. They could be links to online resources as well. Links to clear, concise ...
15
votes
2answers
600 views
Complete Markets in Continuous Time
In the standard discrete time economies with a finite number of states, $n$, a complete markets economy is simply an economy with $n$ independent assets (Think Ljunqvist and Sargent Chapter 8). This ...
