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.

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3 Responses to Lecture 30. Stratonovitch stochastic differential equations

  1. alabair says:

    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.

    • 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.

  2. alabair says:

    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.

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