Lecture 30. Stratonovitch stochastic differential equations

As usual, let $\left( \Omega , (\mathcal{F}_t)_{t \geq 0} , \mathbb{P} \right)$ be a filtered probability space which satisfies the usual conditions. It is often useful to use the language of Stratonovitch ‘s integration to study stochastic differential equations because the Itō’s formula takes a much nicer form. If $(N_t)_{0 \leq t \leq T}$ , $T > 0$ , is an $\mathcal{F}$ -adapted real valued local martingale and if $(\Theta_t)_{0 \leq t \leq T}$ is an $\mathcal{F}$ -adapted continuous semimartingale satisfying $\mathbb{P} \left( \int_0^T \Theta_t^2 d \langle N \rangle_t < +\infty \right)=1$ , then by definition the Stratonovitch integral of $(\Theta_t)_{0 \leq t \leq T}$ with respect to $(N_t)_{t \ge 0}$ is defined as
$\int_0^T \Theta_t \circ d N_t =\int_0^T \Theta_t d N_t+\frac{1}{2} \langle \Theta, N \rangle_T,$
where:

$\int_0^T \Theta_t d N_t$ is the Itō integral of $(\Theta_t)_{0 \leq t \leq T}$ against $(N_t)_{0 \leq t \leq T}$ ;
$\langle \Theta, N \rangle_T$ is the quadratic covariation at time $T$ between $(\Theta_t)_{0 \leq t \leq T}$ and $(N_t)_{0 \leq t \leq T}$ .

By using Stratonovitch integral instead of Itō’s, the Itō formula reduces to the classical change of variable formula.

Theorem. Let $(X_t)_{t \geq 0}=\left( X^1_t , \cdots , X^n_t \right)_{t \geq 0}$ be a $n$ – dimensional continuous semimartingale. Let now $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^2$ function. We have
$f(X_t) =f(X_0)+\sum_{i=1}^n \int_0^t \frac{\partial f}{\partial x_i} (X_s) \circ dX^i_s, \quad t \ge 0.$

Let $\mathcal{O} \subset \mathbb{R}^n$ be a non empty open set. A smooth vector field $V$ on $\mathcal{O}$ is simply a smooth map
$\begin{array}{llll} V: & \mathcal{O} & \rightarrow & \mathbb{R}^{n} \\ & x & \rightarrow & (v_{1}(x),...,v_{n}(x)). \end{array}$
The vector field $V$ defines a differential operator acting on smooth functions $f: \mathcal{O} \rightarrow \mathbb{R}$ as follows:
$Vf(x)=\sum_{i=1}^n v_i (x) \frac{\partial f}{\partial x_i}.$
We note that $V$ is a derivation, that is a map on $\mathcal{C}^{\infty} (\mathcal{O} , \mathbb{R} )$ , linear over $\mathbb{R}$ , satisfying for $f,g \in \mathcal{C}^{\infty} (\mathcal{O} , \mathbb{R} )$ , $V(fg)=(Vf)g +f (Vg).$
An interesting result is that, conversely, any derivation on $\mathcal{C}^{\infty} (\mathcal{O} , \mathbb{R} )$ is a vector field.

Let now $(B_t)_{t \geq 0}=(B^1_t,...,B^d_t)_{t \geq 0}$ be a $d$ -dimensional Brownian motion and consider $d+1$ $C^1$ vector fields $V_i : \mathbb{R}^n \rightarrow \mathbb{R}^n$ , $n \geq 1$ , $i=0,...,d$ . By using the language of vector fields and Stratonovitch integrals, the fundamental theorem for the existence and the uniqueness of solutions for stochastic differential equations is the following:

Theorem. Assume that $V_0,V_1,\cdots,V_d$ are bounded vector fields with bounded derivatives up to order 2. Let $x_0 \in \mathbb{R}^n$ . On $\left( \Omega , (\mathcal{F}_t)_{t \geq 0} , \mathbb{P} \right)$ , there exists a unique continuous and adapted process $(X_t^{x_0})_{t \geq 0}$ such that for $t \geq 0$ ,
$X_t^{x_0}=x_0 + \sum_{i=0}^d \int_0^t V_i (X_s^{x_0}) \circ dB^i_s,$
with the convention that $B^0_t=t$ .

Thanks to Itō’s formula the corresponding Itō’s formulation is
$X_t^{x_0} =x_0 + \frac{1}{2} \sum_{i=1}^d \int_0^t \nabla_{V_i} V_i (X_s^{x_0}) ds +\sum_{i=0}^d \int_0^t V_i (X_s^{x_0}) dB^i_s,$
where for $1 \leq i \leq d$ , $\nabla_{V_i} V_i$ is the vector field given by
$\nabla_{V_i} V_i (x)=\sum_{j=1}^n \left( \sum_{k=1}^n v_i^k (x) \frac{\partial v^j_i}{\partial x_k}(x)\right)\frac{\partial}{\partial x_j}, \text{ }x \in \mathbb{R}^n.$
If $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a $C^2$ function, from Itō’s formula, we have for $t \geq 0$ ,
$f(X_t^{x_0})=f(x_0) + \sum_{i=0}^d \int_0^t (V_i f) (X_s^{x_0}) \circ dB^i_s,$
and the process
$\left( f(X_t^{x_0})-\int_0^t (Lf)(X_s^{x_0})ds \right)_{t \geq 0}$
is a local martingale where $L$ is the second order differential operator
$L = V_0+\frac{1}{2} \sum_{i=1}^d V_i^2.$

3 Responses to Lecture 30. Stratonovitch stochastic differential equations

alabair says:

October 24, 2012 at 5:56 am

my sincere thanks for this blog.
I’d like to know why it prefers the integral of Stratonovich rather than Itô on solving stochastic differential equations.

- Fabrice Baudoin says:
  
  October 24, 2012 at 1:49 pm
  
  Thanks for the interest. We sometimes prefer to use Stratonovitch integrals for SDEs because the Ito’s formula takes a very simple form. Also, it is the correct way to define SDEs on manifolds.
  
alabair says:

October 24, 2012 at 11:27 pm

This is what my observations while flying over your book “An Introduction to the Geometry of Stochastic Flows.”
This is a very encouraging and promising Blog.
thank you again.