Questions tagged [continuous-time]

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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 ...
Oliver Queen's user avatar
1 vote
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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 ...
Oliver Queen's user avatar
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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 ...
kokoro's user avatar
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4 votes
2 answers
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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-\...
Albert Zevelev's user avatar
5 votes
1 answer
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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} \...
Albert Zevelev's user avatar
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 ...
Max's user avatar
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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_{...
Albert Zevelev's user avatar
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}...
Mr. Fafa's user avatar
3 votes
1 answer
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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 ...
mathsquestions1's user avatar
2 votes
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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 ...
studentp's user avatar
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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 ...
Mark's user avatar
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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=...
inquirius's user avatar
12 votes
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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,\...
Theoretical Economist's user avatar
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 ...
FooBar's user avatar
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4 votes
1 answer
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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 ...
Sophie's user avatar
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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 ...
FooBar's user avatar
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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. \...
clueless's user avatar
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5 votes
2 answers
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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 ...
pdevar's user avatar
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3 votes
1 answer
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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 $\...
Ubiquitous's user avatar
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1 vote
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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,...
FooBar's user avatar
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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 ...
FooBar's user avatar
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1 vote
1 answer
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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 ...
FooBar's user avatar
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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 ...
Metta World Peace's user avatar
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 ...
jmbejara's user avatar
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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 ...
FooBar's user avatar
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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 ...
jmbejara's user avatar
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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 ...
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