Questions tagged [continuous-time]
The continuous-time tag has no usage guidance.
24
questions
2
votes
2answers
356 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-\...
4
votes
1answer
81 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} \...
4
votes
1answer
46 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 ...
3
votes
0answers
125 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_{...
4
votes
2answers
184 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}...
3
votes
1answer
85 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 ...
2
votes
0answers
60 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 ...
1
vote
0answers
19 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 ...
1
vote
2answers
371 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:
$$
Y(t)=Z(t)K^aL^{1-a}
$$
$$
y(t)=Y(t)/L(t)
$$
$$
k(t)=K(t)/L(t)
$$
$$
y=...
10
votes
0answers
258 views
Proofs in the Appendix A of Sannikov (2007)
I have a few questions about the proofs in Appendix A of Sannikov (2007), Games with Imperfectly Observable Actions in Continuous Time.
In lemma 4, when he shows the Lipschitz continuity of $H_a(w,\...
3
votes
2answers
68 views
Matching problem in continuous time
Let's split each period into $n$ intervals. There's a continuum $u$ of unemployed and $v$ of vacancies. During each interval, there is a total of $X/n$ job offers. That means that each unemployed gets ...
2
votes
1answer
559 views
Update of value function in continuous time - HJB
When solving (numerically, by value function iteration) a dynamic programming problem in discrete time, such as
$$V_1(a) = \max_{c} \ u(c) + \dfrac{1}{1+\rho}V_0(a')$$
we maximize with respect to ...
6
votes
1answer
150 views
From Discrete to Continuous time: Total Differential
I'm trying to derive an HJB from a discrete time setting. At some point, I am left with
$$ \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
483 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
895 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
48 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
46 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
324 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
61 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 ...
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
1answer
377 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
159 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 ...
12
votes
6answers
1k 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 ...
16
votes
3answers
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 ...
