# What is the solution to this ODE (and SDE)?

1. Apr 18, 2014

### Only a Mirage

I'm trying to analyze the following Ito stochastic differential equation:

$$dX_t = \|X_t\|dW_t$$

where $X_t, dX_t, W_t, dW_t \in \mathbb{R}^n$. Here, $dW_t$ is the standard Wiener process and $\|\bullet\|$ is the $L^2$ norm. I'm not sure if this has an analytical solution, but I am hoping to at least find an analytical expression for the expected value $E[X_t]$.

In order to gain intuition for this problem, I'm considering the following ordinary differential equation:

$$\dot{z}(t) =\|z(t)\|b(t)$$

where $z(t), b(t) \in \mathbb{R}^n$ and everything is completely deterministic. Does anyone know the analytical solution to this second equation, and under what conditions it exists?

2. Apr 18, 2014

### pasmith

It is most easily solved in spherical polar coordinates where $\|z\| = r$ so that $z = r\mathbf{e}_r$ and $\mathbf{b} = b_r\mathbf{e}_r + \mathbf{b}_n$ where $\mathbf{b}_n \cdot \mathbf{e}_r = 0$. We then have $$\dot r \mathbf{e}_r + r \dot{\mathbf{e}_r} = r(b_r\mathbf{e}_r + \mathbf{b}_n).$$ Since $\mathbf{e}_r$ and $\dot{\mathbf{e}_r}$ are orthogonal we have $$\dot r = rb_r(t), \\ \dot{\mathbf{e}_r} = \mathbf{b}_n.$$ The radial and angular components thus decouple and the radial component has solution $$r(t) = r(0) \exp\left( \int_0^t b_r(s)\,ds\right).$$ The angular component represents the motion of a point on the unit $(n-1)$-sphere. If $\mathbf{b}$ is continuous then a solution should exist, but finding a co-ordinate representation of it valid for all time may be impossible, as one can see from the angular components in $\mathbb{R}^3$, $$\dot \theta = b_\theta(t), \\ \dot \phi = \frac{b_\phi(t)}{\sin \theta(t)}$$ with solution $$\theta(t) = \theta(0) + \int_0^t b_\theta(s)\,ds, \\ \phi(t) = \phi(0) + \int_0^t \frac{b_\phi(s)}{\sin \theta(s)}\,ds,$$ which becomes invalid if ever $\theta(t) < 0$ or $\theta(t) > \pi$.

3. Apr 18, 2014

### Only a Mirage

Thanks a lot for the answer. Can you explain why $$\theta(t) = \theta(0) + \int_0^t b_\theta(s)\,ds, \\ \phi(t) = \phi(0) + \int_0^t \frac{b_\phi(s)}{\sin \theta(s)}\,ds,$$ becomes invalid if
ever $\theta(t) < 0$ or $\theta(t) > \pi$?