Questions tagged [continuous-time]
The continuous-time tag has no usage guidance.
27
questions
1
vote
0
answers
42
views
Are there any ``sophisticated'' mathematical modelling where they solve for the utility function?
Are there any references in literature of any ``sophisticated'' mathematical modelling where they solve for the utility function under specific conditions using differential equations theory?
In such ...
- 184
1
vote
0
answers
34
views
What is the intuition behind the different information structures from the static to the continuous time ones?
It is a difficult and challenging problem at the same time, to model the information structure in theoretical models of economics and finance. The information structure in most of the literature is ...
- 184
0
votes
2
answers
63
views
The definition of Poisson process
I find that in many economics model, many events are often called "Poisson process" even though they will only occur once, like the death of an agent or the exit of a firm or transition from ...
- 21
4
votes
2
answers
457
views
Multiple solutions to an HJB, how to pin down the optimal "viscosity" solution?
Consider the deterministic consumption-savings problem:
$
V(a_t)
=
\underset{c}{\max}
\int_{\tau =t}^{\tau = \infty} e^{-\rho (\tau - t) } u(c_{\tau}) d\tau
$ w/ $u(c)=\frac{c^{1-\gamma}-1}{1-\...
- 340
5
votes
1
answer
141
views
What is the relationship between the HJB and "Hamiltonian"? Why is the Hamiltonian H(p) inside the HJB?
Deterministic Optimal Control Problem
$
V(a_t)
=
\underset{c}{\max}
\int_{\tau =t}^{\tau = \infty} e^{-\rho (\tau - t) } u(c_{\tau}) d\tau
$
s.t.
$
\frac{da}{d\tau} = \left( r a_{\tau} - c_{\tau} \...
- 340
4
votes
1
answer
60
views
Relationship of continuous and discrete time models
I am trying to understand two comments from a colleague. I have no clue about continuous time models. He said something like "just replace $\delta$ with $e^{-rdt}$ the Poisson process becomes a ...
- 49
4
votes
0
answers
173
views
What are the boundary value conditions for generic HJBs in economics?
Consider a routine continuous time optimization problem:
$
V(t,a_{t}) :=
\max \int_{\tau=t}^{\tau = T}
e^{-\rho (\tau -t)} u(c_{\tau})d\tau
$ $\text{ s.t. }$
$\dot{a}_{t} = y + ra_{t} - c_{t}$,
$a_{...
- 340
3
votes
2
answers
220
views
Solving a HJB with a probability to transit to a new state
I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time.
The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}...
- 41
3
votes
1
answer
101
views
He, Krishnamurthy (2013)
How do you derive equation (10) on page 740 from He, Krishnamurthy (2013 AER)?
They say that "Given the log objective function in equation (8), the risky asset household chooses $\alpha_t^h$ to solve ...
- 313
2
votes
0
answers
71
views
Finding optimal path in continuous time
I have the following optimal control problem
$$\max_{c_t,l_t} \int_0^{\infty} [ln(c_t)+\theta ln(1-l_t)]e^{-pt}dt$$
st. $$\dot{k_t}=k_t^{1/2}l_t^{1/2}-c_t-\beta l_t$$ $$k_0>0$$
I do big part of ...
- 120
1
vote
0
answers
24
views
Continuous trading economies
Continuous trading economies. I see this expression in many papers but I do not know what it means in the precise technical sense. And I'm not an economist.
Maybe it could refer to an economic ...
- 111
1
vote
2
answers
425
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:
$$
Y(t)=Z(t)K^aL^{1-a}
$$
$$
y(t)=Y(t)/L(t)
$$
$$
k(t)=K(t)/L(t)
$$
$$
y=...
- 51
12
votes
0
answers
327
views
Proofs in the Appendix A of Sannikov (2007)
I have a few questions about the proofs in Appendix A of Sannikov (2007), Games with Imperfectly Observable Actions in Continuous Time.
In lemma 4, when he shows the Lipschitz continuity of $H_a(w,\...
- 2,156
3
votes
2
answers
75
views
Matching problem in continuous time
Let's split each period into $n$ intervals. There's a continuum $u$ of unemployed and $v$ of vacancies. During each interval, there is a total of $X/n$ job offers. That means that each unemployed gets ...
- 10.7k
4
votes
1
answer
869
views
Update of value function in continuous time - HJB
When solving (numerically, by value function iteration) a dynamic programming problem in discrete time, such as
$$V_1(a) = \max_{c} \ u(c) + \dfrac{1}{1+\rho}V_0(a')$$
we maximize with respect to ...
- 43
7
votes
1
answer
204
views
From Discrete to Continuous time: Total Differential
I'm trying to derive an HJB from a discrete time setting. At some point, I am left with
$$ \lim_{\Delta\to 0} \frac{v(c_{t+\Delta}, u_{t+\Delta}, t+\Delta) - v(c_{t}, u_{t}, t)}{\Delta}$$
and am not ...
- 10.7k
15
votes
1
answer
578
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.
\...
- 1,559
5
votes
2
answers
1k
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 ...
- 163
3
votes
1
answer
67
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 $\...
- 16.9k
1
vote
0
answers
52
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,...
- 10.7k
8
votes
1
answer
398
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 ...
- 10.7k
1
vote
1
answer
69
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 ...
- 10.7k
7
votes
2
answers
421
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 ...
- 1,416
8
votes
1
answer
453
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 ...
- 9,345
4
votes
0
answers
185
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 ...
- 10.7k
12
votes
6
answers
2k
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 ...
- 9,345
17
votes
3
answers
1k
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 ...
- 1,613