Stochastic processes and boundary value problems#Characteristic operator

The link between semi-elliptic operators and stochastic processes, followed by their use to solve boundary value problems, is repeatedly and independently rediscovered in the early-mid-20th century.

The connection that Kakutani makes between stochastic differential equations and the Itō process is effectively the same as Kolmogorov's forward equation, made in 1931, which is only later recognized as the Fokker–Planck equation, first presented in 1914-1917. The solution of a boundary value problem by means of expectation values over stochastic processes is now more commonly known not under Kakutani's name, but as the Feynman–Kac formula, developed in 1947.

These results are founded on the use of the Itō integral, required to integrate a stochastic process. But this is also independently rediscovered as the Stratonovich integral; the two forms can be translated into one-another by an offset.

Let $D$ be a domain (an open and connected set) in $\backslash mathbb\{R\}^\{n\}$. Let $\backslash Delta$ be the Laplace operator, let $g$ be a bounded function on the boundary $\backslash partial\; D$, and consider the problem:

:

It can be shown that if a solution $u$ exists, then $u(x)$ is the expected value of $g(x)$ at the (random) first exit point from $D$ for a canonical Brownian motion starting at $x$. See theorem 3 in Kakutani 1944, p. 710.

Let $D$ be a domain in $\backslash mathbb\{R\}^\{n\}$ and let $L$ be a semi-elliptic differential operator on $C^\{2\}(\backslash mathbb\{R\}^\{n\};\backslash mathbb\{R\})$ of the form:

:$L\; =\; \backslash sum\_\{i\; =\; 1\}^\{n\}\; b\_\{i\}\; (x)\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_\{i\}\}\; +\; \backslash sum\_\{i,\; j\; =\; 1\}^\{n\}\; a\_\{ij\}\; (x)\; \backslash frac\{\backslash partial^\{2\}\}\{\backslash partial\; x\_\{i\}\; \backslash ,\; \backslash partial\; x\_\{j\}\}$

where the coefficients $b\_\{i\}$ and $a\_\{ij\}$ are continuous functions and all the eigenvalues of the matrix $\backslash alpha(x)\; =\; a\_\{ij\}(x)$ are non-negative. Let $f\backslash in\; C(D;\backslash mathbb\{R\})$ and $g\backslash in\; C(\backslash partial\; D;\backslash mathbb\{R\})$. Consider the Poisson problem:

:

The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion $X$ whose infinitesimal generator $A$ coincides with $L$ on compactly-supported $C^\{2\}$ functions $f:\backslash mathbb\{R\}^\{n\}\backslash rightarrow\; \backslash mathbb\{R\}$. For example, $X$ can be taken to be the solution to the stochastic differential equation:

:$\backslash mathrm\{d\}\; X\_\{t\}\; =\; b(X\_\{t\})\; \backslash ,\; \backslash mathrm\{d\}\; t\; +\; \backslash sigma\; (X\_\{t\})\; \backslash ,\; \backslash mathrm\{d\}\; B\_\{t\}$

where $B$ is n-dimensional Brownian motion, $b$ has components $b\_\{i\}$ as above, and the matrix field $\backslash sigma$ is chosen so that:

:$\backslash frac1\{2\}\; \backslash sigma\; (x)\; \backslash sigma(x)^\{\backslash top\}\; =\; a(x),\; \backslash quad\; \backslash forall\; x\; \backslash in\backslash mathbb\{R\}^\{n\}$

For a point $x\backslash in\backslash mathbb\{R\}^\{n\}$, let $\backslash mathbb\{P\}^\{x\}$ denote the law of $X$ given initial datum $X\_\{0\}\; =\; x$, and let $\backslash mathbb\{E\}^\{x\}$denote expectation with respect to $\backslash mathbb\{P\}^\{x\}$. Let $\backslash tau\_\{D\}$ denote the first exit time of $X$ from $D$.

In this notation, the candidate solution for (P1) is:

:

provided that $g$ is a bounded function and that:

:

It turns out that one further condition is required:

:

For all $x$, the process $X$ starting at $x$ almost surely leaves $D$ in finite time. Under this assumption, the candidate solution above reduces to:

:$u(x)\; =\; \backslash mathbb\{E\}^\{x\}\; \backslash left[\; g\; \backslash big(\; X\_\{\backslash tau\_\{D\}\}\; \backslash big)\; \backslash right]\; +\; \backslash mathbb\{E\}^\{x\}\; \backslash left[\; \backslash int\_\{0\}^\{\backslash tau\_\{D\}\}\; f(X\_\{t\})\; \backslash ,\; \backslash mathrm\{d\}\; t\; \backslash right]$

and solves (P1) in the sense that if $\backslash mathcal\{A\}$ denotes the characteristic operator for $X$ (which agrees with $A$ on $C^\{2\}$ functions), then:

:

Moreover, if $v\; \backslash in\; C^\{2\}(D;\backslash mathbb\{R\})$ satisfies (P2) and there exists a constant $C$ such that, for all $x\backslash in\; D$:

:$|\; v(x)\; |\; \backslash leq\; C\; \backslash left(\; 1\; +\; \backslash mathbb\{E\}^\{x\}\; \backslash left[\; \backslash int\_\{0\}^\{\backslash tau\_\{D\}\}\; \backslash big|\; g(X\_\{s\})\; \backslash big|\; \backslash ,\; \backslash mathrm\{d\}\; s\; \backslash right]\; \backslash right)$

then $v=u$.

{{reflist}}

• {{cite journal

| doi = 10.3792/pia/1195572706

| last = Kakutani

| first = Shizuo

|authorlink= Shizuo Kakutani

| title = Two-dimensional Brownian motion and harmonic functions

| journal = Proc. Imp. Acad. Tokyo

| volume = 20

| issue = 10

| year = 1944

| pages = 706–714

|doi-access = free

}}

• {{cite journal

| doi = 10.3792/pia/1195572742

| last = Kakutani

| first = Shizuo

|authorlink= Shizuo Kakutani

| title = On Brownian motions in n-space

| journal = Proc. Imp. Acad. Tokyo

| volume = 20

| issue = 9

| year = 1944

| pages = 648–652

|doi-access = free

}}

Category:Boundary value problems

Category:Partial differential equations

Category:Stochastic differential equations