In an economics paper which has model written in continuous time, I came across a budget constraint showing stochastic evolution of wealth:
$$\mathrm{d}w_{t} = (r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t})\mathrm{d}t + k_{t}\sigma\mathrm{d}W_{t}.$$
Here $w_{t}$ is total wealth, $r_{t}$ is return on wealth, $k_{t}$ is capital, $\alpha_{t}$ is excess return on capital, $c_{t}$ is consumption, $m_{t}$ is real money (nominal money divided by price level), $i_{t}$ is nominal interest rate rate, $\sigma$ is an idiosyncratic shock and $W_{t}$ is a Brownian Motion. When I first saw such equations, I tried to think if they could be written in usual $\frac{\mathrm{d}x_{t}}{\mathrm{d}t} = a_{t} + b_{t} +h_{t}$ form which can then be recast as $\mathrm{d}x_{t} = (a_{t} + b_{t} + h_{t})\mathrm{d}t$ but apparently in the above case, it does not seem possible because of an additive term on the right hand side. What I exactly mean is, I reckon I cannot write the above equation as
$$\frac{\mathrm{d}w_{t}}{\mathrm{d}t} = r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t}$$
since I will be missing the $k_{t}\sigma\mathrm{d}W_{t}$ term on the right hand side. And I think I cannot also write is as
$$\frac{\mathrm{d}w_{t}}{\mathrm{d}t} = r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t} + k_{t}\sigma\mathrm{d}W_{t}$$
since in the original first equation, there is no $\mathrm{d}t$ term multiplying $k_{t}\sigma\mathrm{d}W_{t}$ on the right hand side.
Is there any way these equations can be written as usual differential equations like I just mentioned? What I am trying to do is to understand if there is some math here that I am missing. Am new to stochastic analysis. Will appreciate any help.
After doing some digging around, I think I have found the answer. I am posting it here so that it might help somebody who may have the same question.
As written in An Introduction to Computational Stochastic PDEs (page 314) and Oksendal's Stochastic Differential Equations: An Introduction with Applications, 6e (page 65), a White Noise process in continuous time can be written as time derivative of a Brownian Motion, i.e.
$$\zeta(t) = \frac{\mathrm{d}W_{t}}{\mathrm{d}t}$$
So the differential form of the SDE which represents the budget constraint in the question is
$$\frac{\mathrm{d}w_{t}}{\mathrm{d}t} = r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t} + k_{t}\sigma\frac{\mathrm{d}W_{t}}{\mathrm{d}t}$$
where the last additive term contains a White Noise shock (which is time derivative of Brownian Motion) to capital. Multiplying both sides by $\mathrm{d}t$, one gets the following differential form:
$$\mathrm{d}w_{t} = (r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t})\mathrm{d}t + k_{t}\sigma\mathrm{d}W_{t}$$