I am struggling to understand this question, and require some assistance. The question reads:
Consider the Random Walk with a drift. Fix a grid size $\Delta x$ and a time step $\Delta t$. Let $\delta>0$ be some small number. Then, consider the random walk where at each multiple of $\Delta t$, we move to the right by $\Delta x$ with probability $\frac{1}{2}+\delta$ and we move to the left by $\Delta x$ with probability $\frac{1}{2}-\delta$. Suppose at $t=0$ we are at the origin $x=0$, adn we enforce that for a fixed $\sigma>0$, and $\beta>0$, we have: $$\sigma^2 = c\frac{(\Delta x)^2}{2\Delta t}, \>\>\>\> \delta=\beta\Delta x$$ Consider the limit $\Delta x\rightarrow0$. Derive the PDE associated with being at point $x$ at time $t$.
Now, I found the Equation ($p$ denoting probability): $$p(x,t+\Delta t) = (\frac{1}{2}+\delta) p (x+\Delta x, t)+(\frac{1}{2}-\delta) p(x-\Delta x, t)$$ We saw that if the probabilities were one half each, from this step we would subtract $p(x,t)$ from both sides to get: $$p(x,t+\Delta t) - p(x,t) = \frac{1}{2} (p(x+\Delta x, t)+p(x-\Delta x, t)-2p(x,t))$$ And then divided both sides by $\Delta t$ and multiplied the right hand side by $1=\frac{(\Delta x)^2}{(\Delta x)^2}$ to achieve our desired PDE. But, with the $\delta$ in the probability, we can't seem to do this method anymore. In fact, I'm struggling to see how we can shape it into this form. If anyone had some tips, it'd be greatly appreciated.
As I commented, the balance equation should be
$$p(x,t+\Delta t) = \left(\frac{1}{2}-\delta\right) p (x+\Delta x, t)+\left(\frac{1}{2}-\delta \right) p(x+\Delta x, t)$$
Subtracting $p(x,t)$ gives, after some algebra:
$$p(x,t+\Delta t) - p(x,t) = \frac{1}{2} (p(x+\Delta x, t)+p(x-\Delta x, t)-2p(x,t)) + \delta(p(x-\Delta x, t)-p(x+\Delta x, t))$$
Add/subtract $p(x,t)$ to/from the $\delta$ term on the RHS, then multiply the LHS by $\frac{1}{\Delta t}$ and the RHS by $\frac{2\sigma^2/c}{(\Delta x)^2}$. Also substitute $\delta = \beta\Delta x$ to get:
$$\frac{p(x,t+\Delta t) - p(x,t)}{\Delta t} = \frac{\sigma^2}{c} \frac{p(x+\Delta x, t)+p(x-\Delta x, t)-2p(x,t)}{(\Delta x)^2} + \frac{2\beta\sigma^2}{c}\frac{p(x-\Delta x, t)-p(x,t)+p(x,t)-p(x+\Delta x, t)}{\Delta x}$$
Take limits as $\Delta x \to 0, \Delta t \to 0$ and you get the convection-diffusion equation in 1 dimension:
$$ \frac{\partial p}{\partial t} = \frac{\sigma^2}{c} \frac{\partial^2 p}{\partial x^2} -\frac{4\beta\sigma^2}{c}\frac{\partial p}{\partial x} $$