Evan Chow

Evan Chow

Quantitative researcher & data scientist at ADIA, the sovereign wealth fund of Abu Dhabi. Formerly @ Snap, Uber, Princeton, LSE. All views are mine.

Refresher on Fokker-Planck for mean field games

Published:

These are my quick refresher notes on Fokker-Planck (forward Kolmogorov) for the evolution of a probability density over time, and how this appears in mean field game theory.

Like my notes on Hamilton-Jacobi-Bellman, I focus on the main analytical aspects and intuitive understanding rather than the detailed proofs. I follow a similar format as well.

Disclaimers:

  • I prioritize intuition/readability linking to references where appropriate.
  • We will focus on the key equations, not go too deeply into proofs and may skip some parts for brevity.
  • There may be imprecisions: all feedback and corrections are welcome.
  • Long equations may not show up on mobile.

There are various ways to look at Fokker-Planck (Bogachev 2022; Sharma 2010). Here, we will derive the version of Fokker-Planck in the Achdou/Cardaliaguet introductory notes on mean field games (Achdou et al. 2020, 1.3.1), and also appearing in (Ryzhik 2018). This is a core ingredient of a standard MFG formulation, namely specifying how the distribution of many agents evolves over time, and so it is important to understand this result with some depth.

Specifically:

  1. We take a look at the (standard) drift-diffusion dynamics of the underlying states.
  2. We derive its forward Fokker-Planck equation that specifies how its probability distribution evolves over time.
  3. We tie this back to the MFG formulation in the Achdou/Cardaliguet notes.

For (1-2), the Fokker-Planck derivation primarily summarizes general notes on Fokker-Planck (Orlandini 2024; Wolschin 2024; Liang 2024; Brown 2024; Frouah 2024). The main contribution here is to formulate the derivation for the MFG formulation, and also gather treatments of boundary conditions & time/space integrals in a single place.

In a nutshell: the FP equation provides a way to model the evolution of a probability density over time

From Wikipedia:

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.

This is also called the forward Kolmogorov equation.

The main result we want to show

Setup and state dynamics

We have a \(d\)-dimensional vector of states \(X \in \mathbb{R}^d\), and a continuous but finite timeline \(t \in [0, T]\). Define a “drift” function \(\mu: \mathbb{R}^D \times [0, T] \to \mathbb{R}\) which is Hölder-continuous in space and continuous in time. The states \(X\) evolve over time according to the stochastic differential equation (SDE):

\[\begin{align} d X_t &= - \mu(X_t, t) dt + \sigma(X_t, t) dB_t & t \in [0, T] \hspace{0.5cm} \text{state evolution} \\ X_0 &= Z_0 & \text{initial state distribution} \end{align}\]

(Specifically \(-\mu\) will produce a \(-\text{div}\) in the final result, for MFG.)

Our goal: dynamics of the probability density over the states.

In the MFG world, we are interested in how the distribution of \(X\) evolves over time, as to model the crowd behavior of many (identical) agents. Since \(X\) evolves stochastically (via the Brownian), we need to focus on the probability density of \(X\). This is just a transformation of \(X\), which motivates using Ito’s Lemma to model its dynamics, and with more analysis, ultimately the Fokker-Planck forward equation.

Specifically we want to see how the above drift-diffusion system satisfies the Fokker-Planck forward equation below. This general formulation comes from (Ryzhik 2018, 4.2) and more rigorously in (Achdou et al. 2020, 1.3): it appears quite commonly in the MFG literature and so is important to understand.

\[\begin{align} \frac{\partial m}{\partial t} - \varepsilon \Delta m - \text{div} (mb) &= 0 & \text{in } \mathbb{R}^d \times (0,T) & \hspace{0.5cm} \text{density evolution} \\ m(x,0) &= m_0 (x) & \text{initial density} \end{align}\]

for some vector field \(b: \mathbb{R}^d \times [0, T] \to \mathbb{R}\). (Later in a mean field game, this will be based on the optimal controls.)

We introduce here the probability density of \(d-\)dimensional \(X\) at time \(t\) as \(m: \mathbb{R}^d \times [0,T] \to \mathbb{R}\).

  • This density is a scalar field: at any point \((X,t)\) it assigns a scalar value.
  • This density is a member of the Lebesgue space \(L^1\) (StackExchange). Informally, the absolute value of \(m\) is bounded everywhere.
  • We assume that the probability density vanishes at the boundaries so that \(m^c(\pm\infty)=0\) for all orders of differentiation \(c\). This additional regularity condition corresponds to the “natural boundary” condition referenced in (Orlandin 2024).

From the state dynamics to Fokker-Planck

State dynamics are as above.

Test function and Ito calculus

Introduce a smooth arbitrary test function \(\phi: \mathbb{R}^d \times (0, T) \to \mathbb{R}\), with boundary conditions \(\phi(X,T)=\phi(X,0)=0\). (We assume scalar-valued for simplicity.) We obtain the dynamics of \(\phi(x,t)\) via Ito’s lemma:

\[\begin{align} d \phi(X_t, t) &= \left[ \frac{\partial \phi(X_t, t)}{\partial t} - \mu \frac{\partial \phi(X_t, t)}{\partial X_t} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_t, t)}{\partial X_t^2} \right] dt + \left[ \sigma \frac{\partial \phi(X_t, t)}{\partial X_t} \right] dB_t \end{align}\]

This yields the Ito integral:

\[\begin{align} \phi(X_T, T) &= \phi(Z_0, 0) + \int_0^t \left[ \frac{\partial \phi(X_s, s)}{\partial s} - \mu \frac{\partial \phi(X_s, )}{\partial X_s} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_s, s)}{\partial X_s^2} \right] ds + \int_0^t \left[ \sigma \frac{\partial \phi(X_t, t)}{\partial X_t} \right] dB_s \end{align}\]

Obtaining the density via the expectation

Now, take the expectation wrt. \(X\) on both sides, and removing the Brownian integral which has expectation zero:

\[\begin{align} \mathbb{E}[\phi(X_T, T)] &= \mathbb{E}[\phi(Z_0, 0)] + \mathbb{E}\left( \int_0^T \left[ \frac{\partial \phi(X_s, s)}{\partial s} - \mu \frac{\partial \phi(X_s, s)}{\partial X_s} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_s, s)}{\partial X_s^2} \right] ds \right) \end{align}\]

Expand the expectation terms to reveal the probability density of \(X\):

\[\begin{align} \underbrace{\int_{\mathbb{R}^d} \phi(X, T) m(X, T) dX}_{\text{terminal integral}} &= \underbrace{\int_{\mathbb{R}^d} \phi(Z_0, 0) m(X, 0) dX}_{\text{initial integral}} + \underbrace{ \int_{\mathbb{R}^d} \int_0^T \left[ \frac{\partial \phi(X_s, s)}{\partial s} - \mu \frac{\partial \phi(X_s, s)}{\partial X_s} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_s, s)}{\partial X_s^2} \right] m(X_s, s) dX ds }_{\text{running integral}} \end{align}\]

Of course, at the boundaries \(\phi(X,T)=\phi(X,0)=0\), so both the terminal and initial integral are zero:

\[\begin{align} 0 &= \underbrace{ \int_{\mathbb{R}^d} \int_0^T \left[ \frac{\partial \phi(X_s, s)}{\partial s} - \mu \frac{\partial \phi(X_s, s)}{\partial X_s} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_s, s)}{\partial X_s^2} \right] m(X_s, s) dX ds }_{\text{running integral}} \end{align}\]

Hence we have an expression for the dynamics of the probability density \(m(X_t, t)\) for any time \(t \in (0,T)\).

Shifting dynamics of test function to the density

To evaluate the running integral, split up the sum into three smaller integrals, which we will handle separately. Our goal is to move the derivatives of \(\phi\) onto \(m\) so we can obtain its dynamics and therefore the Fokker-Planck equation. (For brevity we omit the parameters for \(m, \phi\), and also write \(X\) instead of \(X_s\) since \(s\) is given below.)

\[\begin{align} \int_{\mathbb{R}^d} \int_0^T &\left[ \frac{\partial \phi(X_s, s)}{\partial s} - \mu \frac{\partial \phi(X_s, s)}{\partial X_s} + \frac{1}{2} \sigma^2 \frac{\partial^2 \phi(X_s, s)}{\partial X_s^2} \right] m(X_s, s) dX ds = \\ & \underbrace{ \int_{\mathbb{R}^d} \int_0^T m \frac{\partial \phi}{\partial s} ds dX }_{\text{Integral }1} - \underbrace{ \int_{\mathbb{R}^d} \int_0^T m \mu \frac{\partial \phi}{\partial X} ds dX }_{\text{Integral }2} + \underbrace{ \frac{1}{2} \int_{\mathbb{R}^d} \int_0^T m \sigma^2 \frac{\partial^2 \phi}{\partial X^2} ds dX }_{\text{Integral }3} \end{align}\]

where \(m, \phi, \mu, \sigma\) are all functions of \((X_s, s)\). To take the integrals, we will use integration by parts and also our boundary conditions for \(\phi, m\).

\[\begin{align} \int_a^b u(x) v'(x) dx &= u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) dx &\text{integration by parts} \end{align}\]

Integral 1, with \(u=m, v'=\frac{\partial \phi}{\partial s}\):

\[\begin{align} \int_{\mathbb{R}^d} \int_0^T m \frac{\partial \phi}{\partial s} ds dX &= \int_{\mathbb{R}^d} \left[ m\phi |^T_0 - \int_0^T \frac{\partial m}{\partial s} \phi ds \right] dX = - \int_{\mathbb{R}^d} \int_0^T \frac{\partial m}{\partial s} \phi ds dX \end{align}\]

Integral 2, with \(u=m\mu, v'=\frac{\partial \phi}{\partial X}\) and recalling \(m(\pm \infty)=0\).

\[\begin{align} \int_{\mathbb{R}^d} \int_0^T m \mu \frac{\partial \phi}{\partial X} ds dX &= \int_0^T \int_{\mathbb{R}^d} m \mu \frac{\partial \phi}{\partial X} dX ds & \text{switch integration limits} \\ &= \int_0^T \left[ (m \mu) \phi |_{\mathbb{R}^d} - \int_{\mathbb{R}^d} \frac{\partial (m \mu)}{\partial X} \phi dX \right] ds & \text{integration by parts} \\ &= - \int_0^T \int_{\mathbb{R}^d} \frac{\partial (m \mu)}{\partial X} \phi dX ds & \text{apply boundaries } m^c(\pm\infty)=0 \\ &= - \int_{\mathbb{R}^d} \int_0^T \frac{\partial (m \mu)}{\partial X} \phi ds dX & \text{switch integration limits} \end{align}\]

Integral 3, where we will have to apply integration by parts twice so we get \(\frac{\partial^2 \phi}{\partial X^2} \to \frac{\partial \phi}{\partial X} \to \phi\) to yield its dynamics.

\[\begin{align} \frac{1}{2} \int_{\mathbb{R}^d} \int_0^T m \sigma^2 \frac{\partial^2 \phi}{\partial X^2} ds dX &= \frac{1}{2} \int_0^T \int_{\mathbb{R}^d} m \sigma^2 \frac{\partial^2 \phi}{\partial X^2} dX ds & \text{switch integration limits} \\ &= \frac{1}{2} \int_0^T \left[ m \sigma^2 \frac{\partial \phi}{\partial X} |_{\mathbb{R}^d} - \int_{\mathbb{R}^d} \frac{\partial (m \sigma^2)}{\partial X} \frac{\partial \phi}{\partial X} dX \right] ds & \text{by parts, } u=m\sigma^2 \text{ and } v'=\frac{\partial^2 \phi}{\partial X^2} \\ &= \frac{1}{2} \int_0^T \left[ - \int_{\mathbb{R}^d} \frac{\partial (m \sigma^2)}{\partial X} \frac{\partial \phi}{\partial X} dX \right] ds & \text{apply boundaries } m^c(\pm\infty)=0 \\ &= \frac{1}{2} \int_0^T \left[ -\frac{\partial (m\sigma^2)}{\partial X} \phi |_{\mathbb{R}^d} + \int_{\mathbb{R}^d} \frac{\partial^2 (m\sigma^2)}{\partial X^2} \phi dX \right] ds & \text{by parts, } u=\frac{\partial (m\sigma^2)}{\partial X} \text{ and } v'=\frac{\partial \phi}{\partial X} \\ &= \frac{1}{2} \int_0^T \left[ -\left( \frac{\partial m}{\partial X} \sigma^2 + m \frac{\partial \sigma^2}{\partial X} \right)\phi|_{\mathbb{R}^d} + \int_{\mathbb{R}^d} \frac{\partial^2 (m\sigma^2)}{\partial X^2} \phi dX \right] ds & \text{product rule} \\ &= \frac{1}{2} \int_0^T \int_{\mathbb{R}^d} \frac{\partial^2 (m\sigma^2)}{\partial X^2} \phi dX ds & \text{apply boundaries } m^c(\pm\infty)=0 \end{align}\]

Putting it all together, we obtain the expression:

\[\begin{align} 0 = \underbrace{- \int_{\mathbb{R}^d} \int_0^T \frac{\partial m}{\partial s} \phi ds dX}_{\text{Integral 1}} + \underbrace{\int_{\mathbb{R}^d} \int_0^T \frac{\partial (m \mu)}{\partial X} \phi ds dX}_{\text{Integral 2}} + \underbrace{\frac{1}{2} \int_0^T \int_{\mathbb{R}^d} \frac{\partial^2 (m\sigma^2)}{\partial X^2} \phi dX ds}_{\text{Integral 3}} \\ 0 = \int_{\mathbb{R}^d} \int_0^T \phi \left[ - \frac{\partial m}{\partial s} + \frac{\partial (m \mu)}{\partial X} + \frac{1}{2} \frac{\partial^2 (m\sigma^2)}{\partial X^2} \right] dX ds \end{align}\]

Eliminating the test function to obtain Fokker-Planck

Note we have specified this for any smooth arbitrary \(\phi\), with endpoints fixed at \(\phi(.,0)=\phi(.,T)=0\). This includes many possible function paths bounded far away from zero, which implies the rest of the integrand (bracketed) must be zero to always enforce the equality. Rearranging and adding back notation we obtain for any time \(t\):

\[\begin{align} \frac{\partial m}{\partial t} &= \frac{\partial (m \mu)}{\partial X} + \frac{1}{2} \frac{\partial^2 (m\sigma^2)}{\partial X^2} & \text{Fokker-Planck} \\ \frac{\partial m(X_t, t)}{\partial t} &= \frac{\partial (m(X_t, t) \mu(X_t, t))}{\partial X_t} + \frac{1}{2} \frac{\partial^2 (m(X_t, t)\sigma(X_t, t)^2)}{\partial X_t^2} & \text{parametrized} \end{align}\]

This is the Fokker-Planck (forward Kolmogorov) equation, specifying the dynamics of \(X\)’s probability density \(m(X, t)\) over \(t \in (0, T)\). (Note if we had used \(\mu\) in the states and not \(-\mu\), we would have \(-\frac{\partial (m \mu)}{\partial X}\) on the RHS, as in our references.)

Briefly, this says that the probability density of \(X\) evolves stochastically wrt. \(X\)’s drift and diffusion. It is important to take a moment to examine each term and importantly its dimensions.

  • Time derivative \(\frac{\partial m(X_t, t)}{\partial t}: \mathbb{R}^d \times [0, T] \to \mathbb{R}\): This is a scalar field, since at any given point \((X_t, t)\) it assigns a value indicating how it will change with respect to the “time-coordinate” only.
  • Drift term \(\frac{\partial (m(X_t, t) \mu(X_t, t))}{\partial X_t}: \mathbb{R}^d \times [0, T] \to \mathbb{R}\): This is also a scalar field since \(m, \mu\) are each scalar fields.
  • Diffusion term \(\frac{1}{2} \frac{\partial^2 (m(X_t, t)\sigma(X_t, t)^2)}{\partial X_t^2}\): This is also a scalar field since \(m, \sigma\) are each scalar fields.

Linking back to MFG, summary and simplification

Lastly, we need to tie this back to the mean field game notation in the Achdou/Cardaliguet notes (Achdou et al. 2020). Summarizing, we want to show equivalence between the Fokker-Planck density dynamics and the MFG density dynamics.

\[\begin{align} d X_t = - \mu(X_t, t) dt + \sigma(X_t, t) dB_t & &\text{state dynamics} \\ X_0 = Z_0 & & \text{initial state} \\ m, \mu, \sigma, b: \mathbb{R}^d \times [0, T] &\to \mathbb{R} & \text{scalar fields} \\ \underbrace{\frac{\partial m(X_t, t)}{\partial t}}_{\text{time derivative}} - \underbrace{\frac{1}{2} \frac{\partial^2 (m(X_t, t)\sigma(X_t, t)^2)}{\partial X_t^2}}_{\text{diffusion derivative}} - \underbrace{\frac{\partial \left(m(X_t, t) \mu(X_t, t)\right)}{\partial X_t}}_{\text{divergence derivative}} &= 0 & \text{Fokker-Planck density dynamics} \\ m(X_t, 0) &= Z_0 & \text{Fokker-Planck initial density} \\ \frac{\partial m}{\partial t} - \varepsilon \Delta m - \text{div} (mb) &= 0 &\text{MFG density dynamics} \\ m(x,0) &= m_0 (x) & \text{MFG initial density} \end{align}\]

To align this with the simple MFG formulation, we need to introduce a simple functional form for the diffusion coefficient.

Let \(\sigma(X_t, t)=\sqrt{2 \varepsilon}\) for some \(\varepsilon > 0\). (See Appendix). Moving this constant out of the diffusion derivative, this yields:

\[\begin{align} d X_t = - \mu(X_t, t) dt + \sigma(X_t, t) dB_t & &\text{state dynamics} \\ X_0 = Z_0 & & \text{initial state} \\ m, \mu, \sigma, b: \mathbb{R}^d \times [0, T] &\to \mathbb{R} & \text{scalar fields} \\ \underbrace{\frac{\partial m(X_t, t)}{\partial t}}_{\text{time derivative}} - \varepsilon \underbrace{\frac{\partial^2 (m(X_t, t))}{\partial X_t^2}}_{\text{diffusion derivative}} - \underbrace{\frac{\partial \left(m(X_t, t) \mu(X_t, t)\right)}{\partial X_t}}_{\text{divergence derivative}} &= 0 & \text{Fokker-Planck density dynamics} \\ m(X_t, 0) &= Z_0 & \text{Fokker-Planck initial density} \\ \frac{\partial m}{\partial t} - \varepsilon \Delta m - \text{div} (mb) &= 0 &\text{MFG density dynamics} \\ m(x,0) &= m_0 (x) & \text{MFG initial density} \end{align}\]

Linking back to MFG, showing equivalence

We expand the partial derivative terms on the Fokker-Planck side to show equivalence to the corresponding MFG terms. Recall states are a vector \(X_t = (X_t^1, X_t^2, \dots, X_t^d)^T \in \mathbb{R}^d\). For any given (\(X_t, t\)) we need to evaluate the derivative terms. (Each is also a scalar field and yields a scalar value.)

Before proceeding, recall that the Laplace operator is the sum of all the unmixed second partial derivatives, and the divergence operator is the sum of all the partial derivatives. Hence it is straightforward to see the equivalence below:

\[\begin{align} \frac{\partial m(X_t, t)}{\partial t} &= \frac{\partial m(X_t, t)}{\partial t} + \underbrace{\sum_{i=1}^d \frac{\partial m(X_t, t)}{\partial X_t^i} \frac{\partial X_t^i}{\partial t}}_{\text{chain rule}} = \frac{\partial m}{\partial t} & \text{time derivative} \\ \frac{\partial^2 (m(X_t, t))}{\partial X_t^2} &= \underbrace{\sum_{i=1}^d \frac{\partial^2 m(X_t, t)}{\partial (X_t^i)^2}}_{\text{Laplace operator }\Delta} \stackrel{\text{def.}}{=} \Delta m & \text{diffusion derivative} \\ \frac{\partial (m(X_t, t) \mu(X_t, t))}{\partial X_t} &= \underbrace{\sum_{i=1}^d \frac{\partial (m(X_t, t) \mu(X_t, t))}{\partial X_t^i}}_{\text{Divergence operator}} \stackrel{\text{def.}}{=} \text{div}(m\mu) & \text{divergence derivative} \end{align}\]

Therefore our Fokker Planck equation becomes:

\[\begin{align} \frac{\partial m(X_t, t)}{\partial t} - \varepsilon \frac{\partial^2 (m(X_t, t))}{\partial X_t^2} - \frac{\partial \left(m(X_t, t) \mu(X_t, t)\right)}{\partial X_t} &= 0 & \text{Fokker-Planck from above} \\ \frac{\partial m}{\partial t} - \varepsilon \Delta m - \text{div}(m \mu) &= 0 & \text{Fokker-Planck, with MFG notation} \end{align}\]

Setting the MFG notation such that \(b := \mu\) and the initial distribution \(m(x,0)=m_0(x)=Z_0\) finishes the equivalence. \(\square\)

Summary

In this article we:

  • discussed briefly how the probability density of states plays a role in mean field games
  • started with states that follow a drift-diffusion stochastic differential equation
  • obtained the dynamics of the state distribution (probability density), yielding the Fokker-Planck equation
  • showed equivalence to the common \(\varepsilon\)-formulation of Fokker-Planck in the mean field game literature

All feedback and suggestions are welcome.

Appendix: non-simplified expansion of Fokker-Planck

If we keep \(\sigma(X_t, t)\), then we obtain for the diffusion derivative term where it appears:

\[\begin{align} \frac{\partial^2 (m(X_t, t) \sigma(X_t, t)^2)}{\partial X_t^2} &= \sum_{i=1}^d \frac{\partial^2 (m(X_t, t) \sigma(X_t, t)^2)}{\partial (X_t^i)^2} & \text{diffusion derivative} \\ &= \sum_{i=1}^d \frac{\partial}{\partial X_t^i} \underbrace{\left[\sigma(X_t, t)^2 \frac{\partial m}{\partial X_t^i} + 2m(X_t, t) \sigma(X_t, t) \frac{\partial \sigma(X_t, t)}{\partial X_t^i}\right]}_{\text{first derivative via product rule}} \\ &= \sum_{i=1}^d \left[ \sigma(X_t, t)^2 \frac{\partial^2 m}{\partial (X_t^i)^2} + 4 \sigma(X_t, t) \frac{\partial \sigma(X_t, t)}{\partial X_t^i} \frac{\partial m}{\partial X_t^i} + 2m(X_t, t) \left( \frac{\partial \sigma(X_t, t)}{\partial X_t^i} \right)^2 + 2m(X_t, t) \sigma(X_t, t) \frac{\partial^2 \sigma(X_t, t)}{\partial (X_t^i)^2} \right] \end{align}\]

which makes things much more complex.

References

[1] Bogachev, Vladimir I., et al. Fokker–Planck–Kolmogorov Equations. Vol. 207. American Mathematical Society, 2022.

[2] Sharma, Shambhu N., and Hiren G. Patel. “The Fokker-Planck equation.” Stochastic Control. Rijeka: IntechOpen, 2010. 1-20.

[3] Achdou, Yves, et al. “An introduction to mean field game theory.” Mean Field Games: Cetraro, Italy 2019 (2020): 1-158.

[4] Ryzhik, Lenya. Notes on Mean Field Games. Stanford University, https://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf. Accessed 23 July 2024.

[5] Orlandini, Gianni. “Notes of the course in Statistical Mechanics.” University of Padova, https://userswww.pd.infn.it/~orlandin/fisica_sis_comp/fokker_planck.pdf. Accessed 23 July 2024.

[6] Wolschin, Georg. Statistical Mechanics and Stochastic Processes. University of Heidelberg, https://www.thphys.uni-heidelberg.de/~wolschin/statsem23_6.pdf. Accessed 23 July 2024.

[7] Liang, Xiaolong. “Deriving the Fokker-Planck Equation.” Xiaolong Liang’s Blog, 24 Oct. 2018, https://xl0418.github.io/2018/10/24/2018-10-24-derivingFPequ/. Accessed 23 July 2024.

[8] Brown, Kevin. Lecture Notes on the Fokker-Planck Equation. Woods Hole Oceanographic Institution, https://www.whoi.edu/cms/files/lecture07_21269.pdf. Accessed 23 July 2024.

[9] Frouah, Rachid. “Derivation of the Fokker-Planck Equation.” Finance Notes, https://www.frouah.com/finance%20notes/Derivation%20of%20the%20Fokker-Planck%20equation.pdf. Accessed 23 July 2024.