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Questions tagged [continuous-time]

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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=...
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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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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 ...
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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 ...
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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
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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
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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
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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 $\...
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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
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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
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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
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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
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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 ...
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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 ...
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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 ...