I am working on a mean reverting stochastic differential equation in this form $dC(t)=(c_{in}-c(t))\frac{r_{in}}{V}dt+\frac{r_{in}}{V}\alpha dB_t$ where $B_t $ is a standard wiener process (Brownian motion),as a simple form $$\frac{dC}{dt}=\theta(\mu-c)+\sigma \frac{dB_t}{dt}$$ I am trying to solve it by radial basis functions $$\phi_i=\sqrt{||c(t)-c(t_i)||+k^2}$$ where k is a shape parameter, and $\frac{dB_t}{dt}$ is Gaussian noise
so I did like below
$$c(t)\sim\sum_{i=1}^{n} w_i\phi_i\\c(t)\sim\sum_{i=1}^{n} w_i\sqrt{(c(t)-c(t_i))^2+k^2} \\c'(t)\sim \sum_{i=1}^{n}w_i\frac{(c(t)-c(t_i))}{\sqrt{(c(t)-c(t_i))^2+k^2}}$$ so I putthem into equation $$\frac{dC}{dt}=(c_{in}-c)\frac{r_{in}}{V}+\frac{r_{in}}{V}\alpha \frac{dB_t}{dt}\\ \sum_{i=1}^{n}w_i\frac{(c(t)-c(t_i))}{\sqrt{(c(t)-c(t_i))^2+k^2}}=(c_{in}-\sum_{i=1}^{n} w_i\sqrt{(c(t)-c(t_i))^2+k^2})\frac{r_{in}}{V}+\frac{r_{in}}{V}\alpha \frac{dB_t}{dt}\\ \sum_{i=1}^{n}w_i(\frac{(c(t)-c(t_i))}{\sqrt{(c(t)-c(t_i))^2+k^2}}+\sqrt{(c(t)-c(t_i))^2+k^2})\frac{r_{in}}{V})=\frac{r_{in}}{V}(c_{in}+\alpha \frac{dB_t}{dt})$$ then turn it to $$ a_{i,j}=\frac{(c(t_j)-c(t_i))}{\sqrt{(c(t_j)-c(t_i))^2+k^2}}+\sqrt{(c(t_j)-c(t_i))^2+k^2})\frac{r_{in}}{V} \\ \begin{pmatrix} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{m1} & \cdots & a_{mn} \end{pmatrix} \begin{pmatrix} w_1 \\ w_2\\ \vdots\\ w_n \end{pmatrix} = \begin{pmatrix} \frac{r_{in}}{V}(c_{in}+\alpha \dfrac{dB_{t_1}}{dt})\\ \frac{r_{in}}{V}(c_{in}+\alpha \dfrac{dB_{t_2}}{dt})\\ \vdots \\ \frac{r_{in}}{V}(c_{in}+\alpha \dfrac{dB_{t_n}}{dt}) \end{pmatrix} $$Now my first question is : $$\textit{Is my work true ?}$$second question: Is this true that, I put $\frac {dB_{t_n}}{dt}\sim $ rand(0,1)?