Let $\mu$ be a probability measure on $\mathbb R$ with Lebesgue density, i.e. $\mu(dx)=\mu(x)dx$. Let $\phi_{+}: \mathbb R_+\to \mathbb R_{+}$ and $\phi_{-}: \mathbb R_+\to \mathbb R_{-}$ be respectively increasing and decreasing s.t. $\phi_{\pm}(0)=0$. Assume that $\phi_{\pm}$ satisfy the following differential system:

$$\phi_{\pm}'(x)~~=~~\frac{1~-~F_{\mu}\big(\phi_+(x)\big)~+~F_{\mu}\big(\phi_-(x)\big)}{2\phi_{\pm}(x)\mu\big(\phi_{\pm}(x)\big)} \mbox{ for all } x>0,$$

where $F_{\mu}$ denotes the cumulative distribution function of $\mu$. My question is how to specify the density $\mu$? If there is some numerical shcema, I'm equally glad to know about it.

I find a solution if $\mu$ is symmetric, i.e. $\mu(x)=\mu(-x)$ for all $x\in\mathbb R$. Then it is easy to guess that $\phi_{\pm}=\pm\phi$ with

$$\phi^{-1}(x)~~=~~\int_0^x\frac{y\mu(y)}{1~-~F_{\mu}(y)}dy \mbox{ for all } x\in\mathbb R_+.$$

If someone knows how to treat the general case, please let me know. Thanks a lot!

Let $\mu$ be a probability measure on $\mathbb R$ with Lebesgue density, i.e. $\mu(dx)=\mu(x)dx$. We aime to find increasing and decreasing functions $\phi_{+}: \mathbb R_+\to \mathbb R_{+}$ and $\phi_{-}: \mathbb R_+\to \mathbb R_{-}$ s.t. $\phi_{\pm}(0)=0$ and

$$\phi_{\pm}'(x)~~=~~\frac{1~-~F_{\mu}\big(\phi_+(x)\big)~+~F_{\mu}\big(\phi_-(x)\big)}{2\phi_{\pm}(x)\mu\big(\phi_{\pm}(x)\big)} \mbox{ for all } x>0,$$

where $F_{\mu}$ denotes the cumulative distribution function of $\mu$. My question is how to specify $(\phi_+,\phi_-)$ in terms of $\mu$? If there is some numerical shcema, I'm equally glad to know about it.

I find a solution if $\mu$ is symmetric, i.e. $\mu(x)=\mu(-x)$ for all $x\in\mathbb R$. Then it is easy to guess that $\phi_{\pm}=\pm\phi$ with

$$\phi^{-1}(x)~~=~~\int_0^x\frac{y\mu(y)}{1~-~F_{\mu}(y)}dy \mbox{ for all } x\in\mathbb R_+.$$

If someone knows how to treat the general case, please let me know. Thanks a lot!