Stochastic Integration, Itô and Stratonovich Calculi

Abstract

In this chapter Itô and Stratonovich calculi are introduced and we prove the Itô lemma and describe Itô calculus for multiplicative noise. Finally Itô-Taylor expansion will be given for white noise-driven Langevin dynamics.

Problems

6.1

Nonanticipating function

Show the following relations hold for given nonanticipating functions f(t) and g(t),

$$\begin{aligned} (a)&\quad \left\langle \int _{t_0}^{t} ~f(t') ~ dW(t') \right\rangle = 0 \\\nonumber \\ (b)&\quad \left\langle \int _{t_0}^{t} ~f(t') ~dW(t') \int _{t_0}^{t} ~g(t'') ~ dW(t'') \right\rangle = \int _{t_0}^{t} dt' \left\langle f(t') ~g(t') \right\rangle . \end{aligned}$$

6.2

Itô and Stratonovich interpretations

For Langevin equation

$$\begin{aligned} \frac{d x}{dt}= a(x,t) + b(x,t) ~\varGamma (t) \end{aligned}$$

where $\varGamma (t)$ is Gaussian white noise, show that

(a) with Itô interpretation

$$\begin{aligned} \langle b(x,t) ~\varGamma (t) \rangle = 0. \end{aligned}$$

(b) with Stratonovich interpretation

$$\begin{aligned} \langle b(x,t) \varGamma (t) \rangle = \frac{1}{2} \langle b'(x,t) b(x,t) \rangle \end{aligned}$$

where $b'(x,t) = \frac{\partial }{\partial x} b(x,t)$.

6.3

Itô and Stratonovich interpretations

For the following Langevin equation,

$$\begin{aligned} \frac{d x(t)}{dt}= - x(t) ~\varGamma (t) , x(0)=x_0 \end{aligned}$$

derive $\langle x(t) \rangle$ in Itô and Stratonovich senses.

6.4

Itô’s lemma

Let W(t) be a Wiener process. Compute $(dx)^2$ using Itô’s lemma for:

$$ dx(t) = dt + x(t)~ dW(t). $$

6.5

Stochastic differential equation

Let W(t) be a Wiener process. Compute stochastic differential equation, i.e. dz(t), followed by:

(a) $z(t) = (x(t))^2$, where $dx(t) = \mu x(t) dt + \sigma x(t) dW(t)$,

(b) $z(t) = 3 + t + e^{W(t)}$,

(c) $z(t) = e ^{x(t)}$, where $dx(t) = \mu dt + \sigma dW(t)$,

with constant $\mu$ and $\sigma$.

6.6

Itô’s lemma

Using the Itô’s lemma, calculate the following double integral and show that

$$ \int _{t_0} ^ {t} \int _{t_0} ^ {s_1} dW(s_2) dW(s_1) = \frac{1}{2} [W(t) - W(t_0)]^2 - \frac{1}{2} (t-t_0) $$

6.7

Itô’s integral

Calculate the following multiple Itô integral and show that

$$ \int _{0} ^ {t} dW(s_1) \int _{0} ^ {s_1} dW(s_2) \cdots \int _{0} ^ {s_{n-1}} W(s_n) dW(s_n) = \frac{1}{(n+1)!} H_{n+1} (W(t)) $$

where $H_n(x)$ is the nth order Hermit polynomial with Rodrigues’ formula

$$ H_n(x) = e^{\frac{1}{2} x^2} \left( \frac{d}{dx}\right) ^n e^{-\frac{1}{2} x^2} \quad . $$

6.8

Itô-Taylor expansion

Show that solution of the Langevin equation

$$ dx(t)= a(x,t) ~dt + b(x,t) ~dW(t) $$

can be written as

$$\begin{aligned} x(t)= & {} x(t_0) + a[x(t_0)] \int _{t_0} ^ t ds_1 + b[x(t_0)] \int _{t_0} ^ t dW(s_1) \\\nonumber \\+ & {} b[x(t_0)] b'[x(t_0)] \int _{t_0} ^ t \int _{t_0} ^ {s_1} dW(s_2) dW(s_1) + R \end{aligned}$$

where R is remainder of order $\mathcal {O}(dt^{3/2})$ and higher if $dt=t -t_0$ is small. Keeping the first three and four terms in above expansion will give the Euler-Maruyama and Milstein schemes, respectively.