Questions tagged [continuous-time]

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complete market in a two country context, continuous time

In a standard two country macro context (think Obstfeld-Rogoff), there is the unit root problem which prevents local analysis around the steady state. One way to get around this is to assume complete ...
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Mechanism Design and game theory with stochastic caclulus

Does anybody know if there is a literature that combines mechanism design or market games with dynamic programming or stochastic calculus in general? I have heard about stochastic differential game ...
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3 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-\...
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4 votes
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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} \...
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4 votes
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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 ...
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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_{...
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3 votes
2 answers
189 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}...
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3 votes
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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 ...
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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 ...
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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 ...
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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=...
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11 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,\...
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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 ...
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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 ...
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6 votes
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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 ...
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15 votes
1 answer
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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. \...
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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 ...
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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 $\...
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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,...
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7 votes
1 answer
346 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 ...
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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 ...
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7 votes
2 answers
315 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 ...
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8 votes
1 answer
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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 ...
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3 votes
0 answers
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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 ...
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12 votes
6 answers
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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 ...
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16 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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