Ornstein-Uhlenbeck process

An important Ito process in fiance is Ornstein-Uhlenbeck process (or Mean-reverting process). In fiance, It’s called as Hull-White model to describe a stochastic interest rate.

It is given by OU process of the following form.

$\displaystyle d r_t = (a_t - b_t r_t) dt + \sigma_t dW_t, \ \ \ \ \ (1)$

where ${W}$ is a 1-dim BM, ${b_{t}}$ and ${\sigma_{t}}$ are positive processes. (1) has an explicit form of

$\displaystyle r_t = r_0 e^{-\int_0^t b(s) ds} + \int_0^t e^{-\int_s^t b(u) du} a(s) ds + \int_0^t \sigma(s) e^{-\int_s^t b(u) du} dW_s. \ \ \ \ \ (2)$

–QED–

Example 1

Consider, for constants ${q}$ and ${\theta}$

$\displaystyle d r_t = q(\theta - r_{t}) dt + \sigma_t dW_t.$

One can write it explicitly

$\displaystyle r_{t} = r_{0} e^{-qt} + \theta (1 - e^{-qt}) + \int_{0}^{t} e^{-q(t-s)} dW_{s}.$

Note that the last term is a random variable of normal distribution ${\mathcal N (0, \frac{1}{2q} (1 - e^{-2qt}))}$ . Therefore,

${r_{t} \Rightarrow \mathcal N(\theta, \frac{1}{2q})}$ as ${t\rightarrow \infty}$ .

–QED–

Example 2

In this below, we are going to consider an example of system of SDE with mean field effect. Let ${N}$ be a given natural number, and ${N}$ -dimensional process ${X = (X^{i})_{i= 1, 2, \ldots N}}$ satisfies

$\displaystyle d X^{i}_{t} = (\bar X_{t} - X^{i}_{t}) dt + dW^{i}_{t}, \ X_{0}^{i} = x^{i}, \ i = 1, 2, \ldots, N$

where ${\bar X_{t} = \frac 1 N \sum_{i=1}^{N} X^{i}_{t}. }$ Existence and uniqueness of the solution is well known with Lipschitz coefficients. We are going to study the asymptotic distribution of ${X^{i}}$ as ${N}$ goes to infinity. By summing up all ${N}$ equations, we observe that

$\displaystyle d \bar X_{t} = \frac 1 N \sum_{i=1}^{N} d W^{i}_{t} = \frac 1 {\sqrt N} d B_{t}$

for some standard Brownian motion ${B}$ . So we can write ${\bar X_{t} = \bar x + \frac 1 {\sqrt N} B_{t} := B^{N, \bar x}_{t}}$ . We rewrite the equation as

$\displaystyle d X^{i}_{t} = (B^{N, \bar x}_{t} - X^{i}_{t}) dt + dW^{i}_{t}, \ X_{0}^{i} = x^{i}, \ i = 1, 2, \ldots, N$

and apply the formula , then we have

$\displaystyle X_{t}^{N, i} = x^{i} e^{-t} + \int_{0}^{t} e^{-t + s} B^{N, \bar x}_{s}ds + \int_{0}^{t} e^{-t+s} d W_{s}^{i}.$

Note that ${B_{s}^{N, \bar x} \rightarrow \bar x}$ both in almost surely and in ${L^{2}}$ . Heuristically, one can replace ${B^{N, \bar x}}$ by constant process ${\bar x}$ , and have asymptotic process ${Y^{i} := \lim_{N\rightarrow \infty}X^{N, i}}$ satisfying

$\displaystyle Y^{i}_{t} = x^{i} e^{-t} + \int_{0}^{t} e^{-t + s} \bar x ds + \int_{0}^{t} e^{-t+s} d W_{s}^{i} = \bar x + e^{-t} (Y_{0}^{i} - \bar x) + \int_{0}^{t} e^{-t+s} d W_{s}^{i}.$

This is indeed a solution of McKean-Vlasov equation of the following type, i.e. ${(Z, W) = (Y^{i}, W^{i})}$ satisfies

$\displaystyle d Z_{t} = (\mathbb E[Z_{t}] - Z_{t}) dt + dW_{t}, \ \hbox{ s.t. } \mathbb E[Z_{0}] = \bar x.$

Omega

Deterministic Randomness

Ornstein-Uhlenbeck process

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