Stochastic differential equations

Today I worked with a colleague of mine understanding stochastic differential equations.

Notes I took during the seminar.

And this is the translation:

Stochastic processs

Let $(\Omega, B, P)$ be a measure space, with Borel $\sigma$-algebra B, and measure P such that P is a probability measure.

Definition A stochastic process is a parametrized collection of random variables:

$\{X_t \mid t \geq 0\},$

such that each $X_t$ is a random variable in the measure space.

Brownian movement or Wiener process

Let $\{ w_t \mid t \geq 0 \}$ be a stochastic process, such that $w_t$ is continous in the weak sense with respect to $t$. $w_t$ is a Wiener process if:

$ 0 \leq t1 \leq t2 $ implies $w_{t_2} - w_{t_1} \sim N(0, t_2 - t_1)$
For any $t_1 < t_2 < t_3$, $w_{t_3} - w_{t_2}$ is independent of $w_{t_2} - w_{t_1}$ .
The probability $w_0 = 0$ satisfies $P(w_0 = 0) = 1$.

Note: In general, $w_t$ is non differentiable in any point.

Ito integral

Let $f(t, x_t) = f(t)$, with $x_t$ an stochastic process, such that

$\int_a^b E(f(t)) dt < \infty.$

We will say that $f(t)$ is a random function. Let

$\{a = t_1 < \cdots < t_{n + 1} = b\},$

be a partition of $[a, b]$, with equally spaced points, and let $\Delta t = (b - a)/n$ and $\Delta w_i = w_{t_{i + 1}} - w_{t_i}$ . Then, Ito’s integral is

$\int_a^b f(t) dw_t = \lim\sum_{i = 1}^n f(t_i) \Delta w_i.$

Notes:

If $s_n$ represents the nth-partial sum, we say that $\lim s_n = I$ if

$\lim E((s_n - I)^2) = 0,$

(convergence in probability).

Note that there are two stochastic processes involved in the definition: $x_t$, which is implicit in $f(t)$ and $w_t$, which after the seminar we concluded, represents noise or decoherence, depending the problem.

Ito’s stochastic differential equation

Definition: $x_t$ is a solution of the stochastic differential equation,

$dx_t = \alpha(x_t,t) dt + \beta(x_t, t)dw_t,$

if for any $t > 0$, $x_t$ satisfies

$x_t = x_0 + \int_0^t \alpha(x_t, t)dt + \int_0^t \beta(x_t, t)dw_t.$

Under some conditions, it can be proved that the solution to this equation is unique (proof?).

Teorem (chain rule) If $x_t$ is the solution of a stochastic differential equation, and $F(x, t)$ is a real function such that the partial derivatives

$\partial_t F, \qquad \partial_x F, \qquad \partial_{xx} F$

are continous functions, then

$dF(x_t, t) = f(x_t, t) dt + g(x_t, t) dw_t,$

where

$f(x, t) = \partial_tF + \alpha(x_t, t) \partial_x F + \frac{1}{2} \beta^2(x_t, t) \partial_{xx}F$

and

$g(x_t, t) = \beta(x_t, t) \partial_x F.$

Stochastic maltus model

Theorem The solution to the stochastic differential equation

$dx_t = r x_t dt + c x_t dw_t$

is $x_t = x_0\exp((r - c^2/2)t + c \cdot w_t)$.

Proof Let $F = \ln(x)$, according to the chain rule:

$d\ln(x_t) = \left(rx_t\cdot \frac{1}{x_t} + \frac{1}{2} c^2 x_t^2 \left(- \frac{1}{x_t^2}\right)\right) dt + c x_t \cdot \frac{1}{x_t} dw_t,$

Simplifying the equation, we get

$d\ln(x_t) = \left(r - \frac{c^2}{2}\right) dt + c \cdot dw_t.$

We didn’t work the details, but we could show that the fundamental theorem of calculus is valid for stochastic integrals, at least in the case of integrating a constant function. Therefore, applying it to the previous equation, we get:

$\ln\left(\frac{x_t}{x_0}\right) = \left(r - \frac{c^2}{2}\right) t + c\cdot w_t,$

the theorem follows.

Numerical methods

I. Euler-Murayama method

$dx_i = \Delta x_i \qquad dw_i = \Delta w_i,$ $x_{i + 1} = x_i + \alpha(x_i, t_i) \Delta t + \beta(x_i, t_i) \Delta w_i.$

Note that in order to implement this method, we should select the $\Delta w_i$ randomly with distribution $N(0, \Delta t)$.

II. Milstein method

Modify the last equation into

$x_{i +1} = x_i + \alpha(x_i, t_i) \Delta t + \beta(x_i, t_i) \Delta w_i + \frac{1}{2} \beta(x_i, t_i) \frac{\partial \beta}{\partial x}(x_i, t_i) \left((\Delta w_i)^2 - \Delta t\right).$

This last method resembles predictor-corrector methods in ordinary differential equations.