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

[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?

 

[pasmith]

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?

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 $\theta(t) > \pi$.

 

[Only a Mirage]

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 $\theta(t) > \pi$.

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 $\theta(t) > \pi$?