On the Distribution of Wealth, Income, and Functions—Part 1

The Generator of a Stochastic Process

How do functions of a noisy state evolve over time? A tour of the generator of an Itô process, guided by a simple macroeconomic model of wealth and income.

Important“Will I be reading AI output here?”

No, you won’t. The entire text was written by myself. This does not mean that I did not use AI tools, I did: mostly to understand the topic better, to find and discuss helpful examples, and to create (interactive) visualizations. I can assure you, each and every single em-dash was added, intentionally, by me (I looove em—dashes). Enjoy the read :)

This text aims to shed some light onto important, fundamental properties of stochastic differential equations (SDEs) that govern the noisy dynamics of Itô processes. It has grown out of a lengthy note in which I explained to myself a modeling framework from macroeconomics; specifically the Aiyagari–Bewley–Huggett model for distributions of wealth and income in the continuous-time formulation of Yves Achdou et al. (Achdou et al. 2014, 2022). In the process, I realized that I have gained a much more useful understanding of properties of stochastic (Itô) processes, objects I have been working with for quite a while. So the economics model has not only been interesting in and of itself, but also turned out to be a valuable vehicle for conveying certain aspects of the rich theory behind Itô processes, which I would like to share with you here.

As the post got more and more lengthy, I decided to split it into two parts. This first part will, aside from briefly motivating and establishing the basics, introduce the generator of a stochastic process. In the next part, this generator will reappear in the two other results I want to demonstrate: the Feynman–Kac formula and the Fokker–Planck (or Kolmogorov forward) equation. All of these concepts are going to be introduced …

1. … specifically for Markovian Itô processes, so processes without memory, without jumps, processes in which all noise comes from a driving Wiener process (or Brownian motion), and which can be mapped through smooth functions to return Itô processes again.
2. … by following along a practical example from macroeconomics that most readers, also non-economists (like myself), should be able to relate to without any prior knowledge.

TipBy the way…

For more details on SDEs, I always recommend the (publicly and freely available) book on “Applied Stochastic Differential Equations” by S. Särkkä (Särkkä and Solin 2019). As for notation, I tried to converge to the least confusing mix between Särkkä’s book and the macroeconomics literature by Achdou et al. (Achdou et al. 2014, 2022). The vast majority of the content in this post is only my personal perspective on, and understanding of, the work by (Särkkä and Solin 2019; Achdou et al. 2014, 2022).

Dynamics and uncertainty

Differential equations provide a universal language that is shared across many disciplines of scientific modeling and computing. This framework allows for understanding and predicting complex phenomena via rigorous analysis and simulation. Many laws describe deterministic variations of the state of a system with one free variable (usually time) or with multiple (usually (time and) space). Such deterministic relationships are then formulated as ordinary (ODEs) and partial differential equations (PDEs). However, there are cases—and we will get to know a specific one—in which it makes sense to deliberately model parts of a system as “random”. From my experience, stochasticity is usually incorporated into a model if …

• … parts of the system are understood but explicitly modeling them would introduce a degree of complexity that is consciously avoided.
• … parts of the system are explained by different plausible models and choosing one of them is consciously avoided.
• … parts of the system are not (properly, completely) understood and random noise is the best we got to model this component at all.

Introducing a stochastic component into the model allows for explicitly encoding uncertainty about the model or individual components. “Random noise”, after all, does not mean that you know nothing. It just implies that multiple—even infinitely many—solutions of the same system are plausible. (And, yes, perhaps one also knows nothing). Communicating uncertainty is crucial in many applications—in particular critical ones, such as weather- and climate modeling, finance, disease spread, and many more.

The formal language that allows us to introduce stochasticity into a dynamical system are, you guessed it, stochastic differential equations (SDEs). Unlike ODEs and PDEs, solutions of SDEs are not functions, but probability distributions over functions. These solutions are known as stochastic processes. Drawing samples from a stochastic process yields trajectories that are plausible realizations of the dynamics system under the chosen noise model. But more on that in a second. Before we dive into the technical details, let me briefly introduce the practical example that allows us to connect technical terms and symbols to interpretable concepts and quantities.

A simple economy

We have a closer look at SDEs in the context of a specific dynamical system: an economy in which multiple (actually: infinitely many) agents or individuals earn income, accumulate wealth, and consume goods. Here’s a rough overview.

In this massively simplified model of an economy there exists one single good, say apples. People in our economy, whom I at times call “individuals” or “agents”, receive a certain amount of this good as income. We decide against a realistic, complicated model for income dynamics. Instead, the income each agent receives follows the trajectory of a stochastic process and the agents have no way to influence what they earn. They cannot work harder to earn more or tend their apple tree in hope of a better harvest. This is what economists would call an exogenous force; it enters the model “from outside” without the possibility for anyone to control it.

Agents can do exactly two things with each apple they receive: consume it immediately or give it to another individual in the economy. Consumption, of course, brings joy—or utility—for the individual. This utility increases further and further—though at a diminishing rate—the more an individual consumes. But the economy would be quite boring if everybody just received a random amount of apples, which they immediately eat and gain a varying degree of joy from it. So there is a second aspect to the individuals’ dynamics, other than income.

Whenever an individual chooses not to consume everything they earn—and we will see in Part 2 why and how they decide that—their only other option is to hand a portion of their apples to other individuals in the economy. In other words, individuals can lend apples to other individuals, who in turn borrow them. In this exchange, the borrower will hand the lender the corresponding amount of IOUs; claims on apples of theirs that they receive at some point in the future. By lending out goods and compiling claims on goods produced in the future, you accumulate wealth. If an unlucky individual’s income situation has been dire for some time, they had to borrow and give out claims on their future goods, which pushed their wealth into the negative. An apple-loan, if you will. If they are really unlucky they reach a point at which they are not allowed anymore to borrow more. They hit a limit, which we will call the borrowing constraint and have to live from what they receive as income. A time limit at which you have to have paid back your debt does not exist. But there is an interest rate that borrowers have to pay to the wealthy people who lent them their apples.

A simple economy in the language of SDEs

This gives us a high-level overview regarding the laws that govern our little economy. It’s very simple: there is no money, no complex interactions between agents, no storage of goods, no insurance—many complicated mechanisms we take for granted in our day-to-day lives, yet would quickly lead to an infeasibly complex model, at least for the purpose of this post. In order to properly investigate the model, let us now translate these vague concepts into a formal language.

The agents

Say we are interested in the distribution of wealth and income in the economy. This requires, first of all, that we consider different individuals in different economic situations and that behave differently, depending on their own individual situation. This is what economists call heterogenous agents (versus a single, representative agent, which is a different approach that is less suitable for studying “relational” quantities like inequality). So, each agent gets their own dynamics.

An agent in our economy is fully determined by their

• wealth \(a\) and
• income \(z\).

Over time, the agents move as \((a_t, z_t)\)-particles through the wealth-income state space. Next, we define these dynamics.

Wealth dynamics

The dynamics of wealth follow a simple rule: An agent receives income \(z_t\) and consumes some amount \(c_t\) of goods. What an agent chooses not to consume, they give away to other agents in exchange for a promise to receive an equal amount of the borrower’s goods at some point in the future. Lending out goods increases wealth—the amount of claims on other people’s future goods. Individuals receive (or pay) interest at a rate \(r > 0\) on the wealth they hold (or owe). Together, this yields the following differential equation: \[\dot{a}_t = z_t + r a_t - c_t, \tag{1}\] where we used the short-hand notation \(\dot{a}_t = {\mathrm{d}a_t} / {\mathrm{d}t}\). Three terms: income plus interest payments minus consumption. Makes sense, that’s pretty much what happens to me every month. Eq. (1) is called a budget identity or budget constraint, and it governs the dynamics of the wealth dimension. As mentioned before, we add the borrowing constraint \(\underline a\), a limit under which an agent’s wealth cannot fall. At any time \(t\), \[ a_t \geq \underline{a}. \tag{2}\]

Important: In order to study properties of a specific stochastic process, everything on the right-hand side of eq. (1) has to be specified. Therefore, we take the interest rate \(r\) and the consumption policy \(c\), which determines the amount an agent consumes at time \(t\) as \(c_t = c(a_t, z_t)\), as given. How \(r\) is determined and how agents choose \(c\) is the topic of Part 2. In this part, we will next focus on the income dynamics. Each individual’s income path \(z_t\) is random and unknown in advance. But each path is generated from a shared law, which we do know.

Income dynamics

As mentioned earlier, in this model, the agents have no control over their income—the income process is a given exogenous force. To add insult to injury, an individual’s income is subject to regular shocks, making the received amount of apples increase or decrease randomly at any given time instant. This is what makes each agent’s situation unique. Even if everybody started out at the exact same situation, over time, their respective income paths will spread them apart. Concretely, the dynamics of income are governed by a stochastic process that solves the following SDE: \[\mathrm{d}z_t = \mu(z_t) \mathrm{d}t + \sigma(z_t) \mathrm{d}W_t. \tag{3}\] This object splits the income dynamics into two parts:

1. a deterministic part, which specifies the direction in which the income moves from each point \(z\). This is governed by the drift, which we model here as a function \(\mu(z)\) that we assume to be given.
2. a stochastic part, which introduces a noise (or diffusion) component into the income process. Effectively, at each point the income dynamics receive a shock in a random direction and of a random magnitude. The magnitude is scaled by the dispersion coefficient \(\sigma(z_t)\), which we assume to be given. The object \(W_t\) is called a Wiener process (or Brownian motion), which is itself a stochastic process. It follows very simple stochastic dynamics: over each time step of length \(h\), \(W_t\) takes a step into a random direction that is drawn from a zero-centered Gaussian random variable. The variance is given by the step size \(h\). Formally, \[W_{t+h} - W_t \sim \mathcal{N}\left(0, h\right) \qquad \Leftrightarrow \qquad W_{t+h} = W_t + \sqrt{h}\epsilon, \quad \text{with} \quad \epsilon \sim \mathcal{N}(0, 1). \tag{4}\] Crucially, the Wiener increment itself has zero mean and its variance corresponds to the step size of the increment.

This interactive visualization aims to demonstrate both the deterministic and stochastic part in isolation.

import {stepFigure} from "./figures/steps.js"
viewof stepSize = Inputs.range([0.005, 1], {value: 0.1, step: 0.005, label: "step size h"})
stepFigure({d3, h: stepSize, width})

On the left, one single step is shown: a starting value is displaced by the drift, but where it ends up, exactly, is sampled from a Gaussian around the end point. Press the “Take 1 step” button to see this process animated. The right shows a whole trajectory of many steps. You can move the slider in order to change the step size \(h\) and determine how many steps are taken.

Think of the individuals drifting and wiggling along the income dimension, without being able to exercise any control over this process. An agent that currently earns \(z_t\) will drift according to the direction specified by \(\mu(z_t)\) and receive a random shock, positive or negative, according to their individual draw from the increment \(\mathrm{d} W_t\). This shock is amplified by the dispersion \(\sigma(z_t)\)—the larger \(\sigma\), the stronger the income shock.

Wealth is coupled to income, directly, and inherits the stochasticity from the Wiener diffusion. Therefore, the \((a_t, z_t)\)-process for the agents’ dynamics can be written as one joint, vector-valued SDE:

NoteAn individual’s wealth-income dynamics

\[\mathrm{d} \begin{pmatrix} a_t \\ z_t \end{pmatrix} = \begin{pmatrix} z_t + r a_t - c_t \\ \mu(z_t) \end{pmatrix} \mathrm{d}t + \begin{pmatrix} 0 \\ \sigma(z_t) \end{pmatrix} \mathrm{d}W_t. \tag{5}\]

The first term, the drift function, now a vector with two components, captures the deterministic part of the dynamics of both dimensions that make up an individual in our model: the budget identity (eq. (1)) governs wealth dynamics, the drift \(\mu(z_t)\) of the income process governs income dynamics. The Wiener diffusion \(W_t\) enters only via the income process, hence the zero in the dispersion function’s wealth component.

With eq. (5) we now have the object that we want to study. The law that governs the dynamics of an individual through the wealth- and income space. And the best thing is: we have an interpretable setup in which we can get to know this model class a bit more. Not just symbols \(a_t\), \(z_t\), and \(c_t\), but wealth, income, and consumption. Here is what it looks like when a bunch of agents follow these dynamics for a while:

import {agentsFigure} from "./figures/agents.js"
agentsFigure({width, invalidation})

On the right, we see two time series, one for wealth (top) and a really rough one for income (bottom). A few agents (including the one from the right panels, highlighted in purple) are visualized in the joint wealth-income space on the left.

Now, we understand how agents move. Sometimes, however, it might be more interesting to track the changes of some quantity related to the agents. The next section introduces how functions of the state move together with the state.

The generator of an Itô process: how do functions of the state evolve?

We are interested in the rate at which any (scalar) function \(\psi(a, z)\) changes, in expectation, with time. This rate is obtained by applying an operator \(\mathcal{A}\), called the (infinitesimal) generator of the process. Instead of pasting in the textbook definition, I will try and provide the intuitive idea.

Let \(\psi(a, z)\) be some (scalar) function of the state. An example: \(\psi(a, z) = z + r a\) could be defined as the total income (“labor” + capital income) that an agent gains. We will think a similar (and slightly more interesting) example through in more detail, later. Since the state evolves dynamically over time and the function \(\psi\) reacts to changes in its arguments, the state dynamics—both deterministic and stochastic—should somehow transfer to the expected drift of \(\psi\). But how?

The deterministic part

The (deterministic) drift of the wealth-income process transfers to \(\psi\) as a product of state velocity \(\times\) sensitivity of \(\psi\), for each state dimension. For example, in the wealth dimension, the state changes at rate \(z + r a - c\). That’s the velocity part. And how \(\psi\) varies with changes in wealth, the sensitivity part, is simply determined by the wealth-component of the gradient of \(\psi\) with respect to wealth, i.e. \(\partial_a \psi\).¹ The same holds for the income dimension. This is nothing but the chain rule of differentiation. Writing \(x_t = (a_t, z_t)\) for simplicity, we know that \[ \frac{\mathrm{d}\psi}{\mathrm{d}t} = \frac{\mathrm{d}\psi}{\mathrm{d}x} \cdot \frac{\mathrm{d}x}{\mathrm{d} t}, \tag{6}\] sensitivity \(\times\) velocity, or: how \(\psi\) varies with the state \(\times\) how the state varies with time. Plugging in the respective derivatives, evaluated at the current state \(x_t = (a, z)\), we get \[ \frac{\mathrm{d}\psi}{\mathrm{d}x} = \begin{pmatrix}\partial_a\psi \\ \partial_z\psi\end{pmatrix}, \qquad \frac{\mathrm{d}x}{\mathrm{d}t} = \begin{pmatrix}z + r a - c \\ \mu(z)\end{pmatrix}, \tag{7}\] and taking the inner product (right-hand side of eq. (6)) gives \[ \frac{\mathrm{d}\psi}{\mathrm{d}t} = (z + r a - c)\partial_a\psi + \mu(z)\partial_z\psi. \tag{8}\] Both of these terms will appear in the generator. But before we get to the actual definition, something is missing.

The stochastic part

So a part of the generator—how the drift of the state transfers to the drift of \(\psi\)—just follow from basic calculus. Which leaves the stochastic part of the state: the income shocks from the Wiener process \(W_t\), scaled by the dispersion coefficient \(\sigma(z)\)—see eqs. (3) and (5). As we saw in eq. (4), these shocks are centered at zero, i.e., they have zero mean—income is as likely to be pushed up by \(+ \Delta z\) as down by \(-\Delta z\). So…on average, the shocks cancel out. Does that mean that they do not affect the expected rate of change of \(\psi\)?

Well, not necessarily. If \(\psi\) is curved in the dimension driven by noise, then the shocks do not cancel. Why that’s the case is best explained visually:

import {generatorFigure} from "./figures/generator.js"
viewof step = Inputs.form({
  theta: Inputs.range([0, 1.5], {value: 0.55, step: 0.01, label: "drift strength"}),
  sigma: Inputs.range([0, 1.2], {value: 0.7, step: 0.01, label: "dispersion σ"}),
})
generatorFigure({Plot, ...step, width})

The figure shows what happens to a bunch of agents, each starting from the same income. Each agent is mapped through a function \(\psi\) and advanced one time step. Crucially, in this example, the function \(\psi\) is curved upward—convex—in \(z\). Left: Without the noise, the drift moves each agent by the same amount, in the same direction (upward, as \(\psi\) is increasing). Middle: Without the drift, adding noise from the Wiener process spreads the agents out across \(\psi\). On average, the value of \(\psi\) increases, as the convex curvature in \(\psi\) increases upward shocks more than it decreases downward shocks. Right: A full step of the process—drift + noise—increases the function value of \(\psi\) and the state (income) itself.

So, in expectation, for convex functions \(\psi\), shocks will lead to larger values of \(\psi\). Analogously, if \(\psi\) is curved downward—concave—in \(z\), then the income volatility will decrease \(\psi\) on average. To account for this effect, the generator of our process contains a third term, \(+\tfrac12 \sigma^2(z)\cdot \partial_{zz}\psi\), referred to as the Itô correction. The math tells us that …

• …wherever the function \(\psi\) is (strictly) convex in \(z\), \(\partial_{zz}\psi(a, z) > 0\). Therefore, the term \(+\tfrac12 \sigma^2(z) \partial_{zz}\psi(a, z) > 0\) is strictly positive, which implies that income volatility increases the rate at which \(\psi\) changes.
• …wherever the function \(\psi\) is (strictly) concave in \(z\), \(\partial_{zz}\psi(a, z) < 0\). Therefore, the term \(+\tfrac12 \sigma^2(z) \partial_{zz}\psi(a, z) < 0\) is strictly negative, which implies that income volatility decreases the rate at which \(\psi\) changes.

To summarize: even though the noise that drives the income process is symmetric around zero, not influencing \(\psi\) to first order, the curvature \(\partial_{zz}\psi\) introduces an asymmetric effect with respect to the spread of the income process.

The generator in full

Let us collect what I described above in a formal statement: For any (scalar) function \(\psi\) of the state of the process from eq. (5), the rate at which \(\psi\) changes, in expectation, over an infinitesimal time increment is given by an operator \(\mathcal{A}\) called the generator of the process. The generator of the wealth-income process is defined as follows:

NoteGenerator of the process that solves eq. (5)

\[\mathcal{A}\psi = (z + r a - c) \partial_a \psi + \mu(z) \partial_z \psi + \tfrac12 \sigma^2(z) \partial_{zz}\psi. \tag{9}\]

TipBy the way…

The generator generalizes standard calculus, in particular the chain rule of differentiation, to stochastic processes. How closely the concept is related to derivatives can be seen from the more general definition.

The (infinitesimal) generator \(\mathcal{A}\) of a stochastic process \((x_t)_{t\geq 0}\) is defined as an operator that, applied to any (scalar) function \(\psi\) of the state, gives the rate at which \(\psi\) changes, in expectation, over an infinitesimal time increment: \[\mathcal{A}\psi(x) = \lim_{h \to 0}\frac{\mathbb{E}\left[\psi(x_{t+h}) \mid x_t = x\right] - \psi(x)}{h}.\]

Let’s make this more clear by looking at an example.

Example: Relative change of income

The function we are going to look at is the logarithm of total income \(y \coloneqq z + ra\), i.e., \[ \psi(a, z) = \log (z + ra) = \log y. \] What is so interesting about this quantity? The answer is: its rate of change.

For a second, let \(y_t\) be a deterministic and continuously differentiable path of an agent’s total income over time. Applying the chain rule of differentiation gives \[ \frac{\mathrm{d}}{\mathrm{d} t} \log y_t = \frac{\mathrm{d}\log y_t}{\mathrm{d}y_t} \cdot \frac{\mathrm{d}y_t}{\mathrm{d}t} = \frac{\dot{y}_t}{y_t}. \tag{10}\] The quantity \(\dot{y}_t / y_t\) is the instantaneous growth rate of the quantity \(y_t\). Think: the rate of change, relative to the current level.

In our economy, however, income \(z_t\) is stochastic, which makes \(y_t\) stochastic, as well. And this creates two challenges when it comes to the equality in eq. (10). We will adress both with the generator \(\mathcal{A}\) (see eq. (9)) of the wealth-income process:

1. The time derivative \(\dot{y}_t\) does not exist, in the classical sense, due to the stochastic process \(y_t\) being nowhere differentiable. Its stochastic-process analogue would be² \[\mathcal{A} y = r (y - c) + \mu(z) \qquad \Rightarrow \qquad \frac{\mathcal{A} y}{y} = \frac{r (y - c)}{y} + \frac{\mu(z)}{y}. \tag{11}\]

2. The second challenge is that the equality comes from applying the chain rule to a strictly concave function, the logarithm. We learned above that the generator does generalize the chain rule to stochastic processes. But it needs a correction term whenever the noise and the curvature of the function are non-zero. Therefore, \[ \mathcal{A}\log y \neq \frac{\mathcal{A}y}{y}, \tag{12}\] and the gap is quantified by the Itô correction term. This is where it gets really interesting, so let’s have a closer look!

With \(\psi(a, z) = \log (z + r a)\), the derivatives appearing inside the generator are \[ \partial_a \psi(a, z) = \frac{r}{z + ra}, \qquad \partial_z\psi(a, z) = \frac{1}{z + ra}, \qquad \partial_{zz}\psi(a, z) = -\frac{1}{(z + ra)^2}. \tag{13}\] Plugging these into the generator of the wealth-income process gives \[ \begin{aligned} \mathcal{A}\psi(a, z) &= \frac{r (z + ra - c)}{z + ra} + \frac{\mu(z)}{z + ra} - \frac{\sigma^2(z)}{2(z + ra)^2} \\[2ex] &= \underbrace{\frac{r (y - c)}{y} + \frac{\mu(z)}{y}}_{= \mathcal{A}y / y} - \frac{\sigma^2(z)}{2y^2}. \end{aligned} \tag{14}\]

This tells us that \[ \mathcal{A} \log y = \frac{\mathcal{A}y}{y} -\frac{1}{2} \left(\frac{\sigma(z)}{y}\right)^2. \tag{15}\]

This is the instantaneous growth-rate equality from eq. (10) but corrected for the stochasticity in the income process. The reason why volatility in income introduces a gap between the two rates—a gap that does not exist for deterministic income paths—is due to an effect called volatility drag. Intuitively, noisy fluctuations in \(y\) make average relative changes (\(\mathcal{A}y / y\)) look higher than what they actually add up to over time (\(\mathcal{A}\log y\)).

An example: take an agent with a total income of \(y= 2\) apples. If this agent loses \(1\) apple and gets it back immediately afterwards, this amounts to relative changes of \(-50\%\) and \(+100\%\). The average growth is \(\tfrac12 (-50\% + 100\%) = 25\%\) per step. But the agent did not gain anything, so something is off. Taking the logarithm of the growth factors (\(-50\% \to \times 0.5\) and \(+100\% \to \times 2\)) fixes this: \(\log 0.5 + \log 2 = -0.69 + 0.69 = 0\).

The Itô-correction term \(\tfrac12 \left(\sigma(z)/y\right)^2\) quantifies the size of the gap between the average relative change and the expected logarithmic change. Besides the income volatility \(\sigma(z)\), the size of the gap also depends on the level of total income, which is individual to each agent. Therefore, different agents suffer to different degrees from volatility drag.

There is another interesting aspect to this: The same absolute income shock can mean very different things for differently situated individuals. For example, a loss of \(1\) apple would mean a loss of \(50\%\) for someone with a total income of \(2\) apples but only a \(1\%\) loss for someone receiving \(100\) apples. These unequal conditions are mirrored in the Itô-correction, which involves the term \(\sigma(z) / y\). This term can be interpreted as a relative volatility in income: how large the shocks are, relative to an agent’s current total income. For equal-sized shocks, individuals with high values of \(y\) are strictly less impacted by income volatility—and the resulting volatility drag—than low-income individuals. The higher \(y\), the lower \(\sigma(z) / y\). This effect is weaker in models, in which the magnitude of the shocks, \(\sigma(z)\), scales proportionately with income \(z\). But regardless of \(\sigma(z)\), capital income \(ra\), which enters total income alongside \(z\), depends only on an agent’s wealth. Hence, accumulating higher levels of wealth increases \(y\) and thus reduces the relative impact of income shocks.

Conclusion & what’s next

This post introduced a simple macroeconomics model for how heterogenous agents drift and diffuse through a landscape of wealth and income. These dynamics are governed by an SDE, in which income follows a diffusion process and wealth accumulates according to a simple budget identity: what an agent saves at each time instant equals their total income (“labour + capital”) minus their consumption. Assuming a fixed and known interest rate \(r\) and consumption policy \(c\) allowed us to begin studying properties of the resulting process.

The generator gives us the expected rate of change of any (scalar) function of the stochastic state. It fixes two things that standard calculus does not account for: (1) that the function argument (the state) is nowhere differentiable, and (2) that the stochasticity in the state has a nontrivial effect on how a curved function changes over time in expectation. The meaning of the correction became more clear by thinking this through in the context of a practical example.

What’s left?

From the economics perspective, fixing the interest rate and the consumption is precisely removing the interesting bits from the model. So, it would be a shame to not have a look at how to determine these quantities in a sliiiightly more realistic setting. Luckily, determining both of these quantities connects beautifully to two more general insights about stochastic processes: the Feynman–Kac formula and the Fokker–Planck equation. The next part of this post will introduce both, which will also teach us how the agents choose their individual consumption, repeatedly, over time, and how the interest rate is determined by the aggregate state of the economy. So, if you’re interested, stay tuned :)

References

Achdou, Yves, Francisco J Buera, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. 2014. PDE Models in Macroeconomics.

Achdou, Yves, Jiequn Han, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. 2022. “Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.” The Review of Economic Studies 89 (1): 45–86. https://doi.org/10.1093/restud/rdab002.

Särkkä, Simo, and Arno Solin. 2019. Applied Stochastic Differential Equations. 1st ed. Cambridge University Press. https://doi.org/10.1017/9781108186735.

Footnotes

1. I use the short-hand notation \(\partial_x = \frac{\partial}{\partial x}\) throughout the post.

2. This follows simply from taking the relevant derivatives of \(y(a, z) = z + r a\): \(\partial_a y = r, ~ \partial_z y = 1, ~ \partial_{zz} y = 0\) and plugging them into the definition of the generator (see eq. (9)).