Deriving the Fokker-Planck equation
Motivation
The Fokker-Planck equation is a partial differential equation (PDE) which describes the evolution of the probability density function of an Ito diffusion. Since it is a PDE, it admits solutions in certain special cases and is amenable to numerical methods for PDEs in the general case.
Derivation
Consider the SDE
\[\mathrm{d}X_{t}=a(t,X_{t})\mathrm{d}t+b(t,X_{t})\mathrm{d}W_{t}\]with bounded coefficients: (i.e., $\sup_{t,x}|a(t,x)|+\sup_{t,x}|b(t,x)|<\infty$). This requirement is used below to apply Fubini’s theorem but can be relaxed.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be smooth with compact support. By Ito’s lemma,
\[\mathrm{d}f=f_{x}\mathrm{d}X_{t}+\frac{1}{2}f_{xx}\mathrm{d}X_{t}^{2}=\left(af_{x}+\frac{1}{2}b^{2}f_{xx}\right)\mathrm{d}t+bf_{x}\mathrm{d}W_{t}.\]Taking expectations of both sides,
\[\mathbb{E}\left[\mathrm{d}f\right]=\mathbb{E}\left[\left(af_{x}+\frac{1}{2}b^{2}f_{xx}\right)\mathrm{d}t\right].\]The above is formalism for the expression
\[\mathbb{E}f(X_{T})-\mathbb{E}f(X_{t})=\mathbb{E}\left[\int_{t}^{T}a(s,X_{s})f_{x}(X_{s})+\frac{1}{2}b(s,X_{s})^{2}f_{xx}(X_{s})\mathrm{d}s\right].\]By Fubini’s theorem, we can interchange the expectation and integral on the right hand side. Moreover, by the Mean Value Theorem, we can find $\xi$ between $t$ and $T$ such that
\[\frac{\mathbb{E}f(X_{T})-\mathbb{E}f(X_{t})}{T-t}=\mathbb{E}\left[a(\xi,X_{\xi})f_{x}(X_{\xi})+\frac{1}{2}b(\xi,X_{\xi})^{2}f_{xx}(X_{\xi})\right].\]Taking limits as $T\downarrow t$ and applying the Dominated Convergence Theorem,
\[\frac{\partial}{\partial t}\left[\mathbb{E}f(X_{t})\right]=\mathbb{E}\left[a(t,X_{t})f_{x}(X_{t})+\frac{1}{2}b(t,X_{t})^{2}f_{xx}(X_{t})\right].\]Let $p(t,\cdot)$ be the density of $X_{t}$. Then, the above is equivalent to
\[\frac{\partial}{\partial t}\int p(t,x)f(x)\mathrm{d}x=\int p(t,x)\left(a(t,x)f_{x}(x)+\frac{1}{2}b(t,x)^{2}f_{xx}(x)\right)\mathrm{d}x.\]Applying integration by parts to the right hand side,
\[\frac{\partial}{\partial t}\int f(x)p(t,x)\mathrm{d}x=\int f(x)\left(-\frac{\partial}{\partial x}\left[p(t,x)a(t,x)\right]+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}\left[p(t,x)b(t,x)^{2}\right]\right)\mathrm{d}x.\]Since this holds for all functions $f$, it follows that
\[\frac{\partial p}{\partial t}(t,x)=-\frac{\partial}{\partial x}\left[p(t,x)a(t,x)\right]+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}\left[p(t,x)b(t,x)^{2}\right].\]This is the Fokker-Planck equation in one dimension. The derivation for multiple dimensions is similar.