Transform a SDE Into a PDE and Vise Versa

Ito's calculus and SDEs
On this page, we learn how to transform a partial differential equation into a stochastic differential equation and vice versa. Dynkin formula and Feynman-Kac formula are defined in this section, but the proofs of these formulae are omitted. Let's denote a random variable $\mathrm{X}_t$ starting at $x$ (i.e. $\mathrm{X}_0=x$)) by $\mathrm{X}_t^x$, and the expectation with respect to $\mathrm{X}_t^x$ by $\mathrm{E}^x$.

Definition.
Let ${\mathrm{X}_t}$ be a time homogeneous Ito process in $\mathfrak{R}^n$ such that $(\mathrm{X}_{t+\delta}-\mathrm{X}_{t})$ depends on $\delta$, but not on $t$. The generator of $\mathrm{X}_t$ is then defined by $$\mathcal{A} f(x)=\lim_{t\rightarrow 0}\frac{E^x[f(\mathrm{X}_t)]-f(x)}{t}; \text{ } x \in \mathfrak{R}^n.$$ $D_\mathcal{A}$ denotes the set of functions $f:\mathfrak{R}^n \rightarrow \mathfrak{R}$ for which the limit exists $\forall x \in \mathfrak{R}^n$.

Theorem. Dynkin formula
Let $X_t$ be a time homogeneous Ito process defined as $$\mathrm{d} \mathrm{X}_t = b(\mathrm{X}_t)\mathrm{d}t + v(\mathrm{X}_t)\mathrm{d}B_t,$$ and let $\tau$ be a stopping time for the process, $\mathrm{E}^x[\tau]<\infty$. If $f\in C^2(\mathfrak{R}^n)$ is a bounded function, and $b(\mathrm{X}_t)$ and $v(\mathrm{X}_t)$ satisfy bounded conditions , then $f\in D_\mathcal{A}$ and $$\mathcal{A} f(x) =\sum_{i=1}^n b_i(x) \frac{\partial f}{\partial x_i}+\frac{1}{2} \sum_{i,j}(v v^{T})_{i,j}(x) \frac{\partial ^2f}{\partial x_i\partial x_j},$$ where $v^T$ is the transpose of $v$.

Exercise
Given the following $2-$dimensional stochastic process $$\mathrm{d} \mathrm{X}_t= \left[ \begin{array}{c} \mathrm{d} \mathrm{X}_t^{[1]}\\ \mathrm{d} \mathrm{X}_t^{[2]} \end{array} \right] =\left[ \begin{array}{c} b_1\\ b_2 \end{array} \right]\mathrm{d}t + \left[ \begin{array}{cc} v_1 &v_2\\ v_3 &v_4 \end{array} \right] \left[ \begin{array}{c}\mathrm{d}B_t^{[1]}\\\mathrm{d}B_t^{[2]} \end{array} \right],$$ where $B_t^{[1]}$ and $B_t^{[2]}$ are independent Brownian motions.

Find the generator $\mathcal{A}$ of the above stochastic process.

Solution.
We have $$b= \left[ \begin{array}{cc} b_1 \\ b_2 \end{array} \right] ~~~\text{and}~~~ v= \left[ \begin{array}{cc} v_1 & v_2 \\ v_3 & v_4 \end{array} \right]$$ where $b_i$'s and $v_i$'s are functions of $x \in \mathfrak{R}$. This gives the following partial differential equation $$\mathcal{A} f(x)=b_1 \frac{\partial f}{\partial x_1}+b_2 \frac{\partial f}{\partial x_2}+\frac{1}{2} ( v_1^2+v_2^2 ) \frac{\partial ^2f}{\partial x_1\partial x_1} + (v_1 v_3 +v_2 v_4) \frac{\partial ^2f}{\partial x_1\partial x_2} + \frac{1}{2} (v_3^2+v_4^2 )\frac{\partial ^2f}{\partial x_2\partial x_2}. \nonumber$$

Theorem. Feynman-Kac formula
Let $f\in C^2(\mathfrak{R}^n)$ be a bounded function and let $\tau$ be a stopping time for the process. Assume that $q\in C(\mathfrak{R}^n)$ is a lower bounded function and that $$y(\tau,x)=E^x \left[\mathrm{exp}\left(-\int_0^{\tau} q(\mathrm{X}_z)\mathrm{d}z\right) f(\mathrm{X}_{\tau}) \right].~~~(1)$$ Then $$\begin{array}{c c} q~ y+ \frac{\partial y}{\partial \tau}= \mathcal{A} y: & \tau>0,x\in \mathfrak{R}^n;~~~(2)\\ y(0,x)=f(x)~:& x \in \mathfrak{R}^n.~~~(3) \end{array}$$ If $y(\tau,x)$ is bounded on $\mathfrak{R} \times \mathfrak{R}^n$ and $y$ satisfies the conditions in (2) and (3), then the solution for $y$ is given by the formula in (1).

Let's take an example when the function $y(\tau,x)$ is defined over $\tau \in[0,T]$. We transform the variable $\tau$ by $t=T-\tau$ and replace $x$ by the security underlying price $S$. Assume that q is a constant risk-free interest rate with $q(S_t)=r$. If a bounded function $y(t,S_t)$ is defined over $[0,T] \times [0,\infty)$, and satisfy $$\begin{array}{c c} r~ y-\frac{\partial y}{\partial t}= \mathcal{A} y: & t\in[0,T],~S_t \geq 0; \\ y(T,S_T)=f(S_T)~:& S_T \geq 0, \end{array}$$ where $y(T,S_T)$ is the termination value of an option at maturity time $T$. The solution of $y(t,S_t)$ is then
$$y(t,S_t)=E^{S_t} \left[\mathrm{exp}\left(-\int_0^{T-t} r\mathrm{d}z\right) f(S_{T-t}) \right]=e^{- r (T-t)} E^{S_t} \left[ f(S_{T-t}) \right].$$

---

References

1. Oksendal B., Stochastic Differential Equations: An Introduction With Applications, 5th ed. Springer, 2000.