-1
$\begingroup$

I am reading mathematical finance on my own. I am wondering how are differentiation wrt different variables defined.

In stochastic calculus, we can see equations like, $dX_{t} = \mu dt + \sigma dW_{t}$. But I am wondering how to differentiate some equations to get this. My questions arouse from my intuition from ODEs. In ODEs, an equation $y = f(x)$ only can be differentiated as, for example, $\frac{dy}{dx}=t$, then we can multiply $dx$ to both sides to get $dy = tdx$.

So what should $X_t$ look like, and especially the operator look like, in order for us to multiply it on both sides. In other words, given an arbitrary stochastic process, are we able to differentiate it?

$\endgroup$
3
  • 2
    $\begingroup$ "Differentiate" is used in a loose sense here. $\endgroup$ Commented Oct 13, 2021 at 22:40
  • $\begingroup$ Theses kind of processes typically are not differentiable at all. The SDE essentially is just a short-hand notation for a corresponding integral equation, which gives rigorous meaning to those objects. $\endgroup$
    – Tobsn
    Commented Oct 14, 2021 at 6:31
  • $\begingroup$ @Tobsn Thank you for your comment. It sounds like what I thought. So basically, stochastic calculus only consists of integrations, while differentiation is only a shorthand rather than a proper operation? $\endgroup$
    – David
    Commented Oct 14, 2021 at 20:47

1 Answer 1

1
$\begingroup$

Consider the following stochastic process from time $[0, t]$ then given $dX_s = \mu ds + \sigma dW_s$ where $W_s$ is the standard Wiener process, we have

$\int_0^t dX_s = X_t - X_0 = \int_0^t \mu ds + \int_0^t \sigma dW_s$.

If the drift and diffusion coefficients are constants, we can make a further simplification to obtain $X_t = X_0 + \mu (t+0) + \sigma (W_t - W_0) = X_0 + \mu t + \sigma W_t$ because $W_0$ is defined to be $0$ almost surely.

There are differentiation rules for stochastic calculus, like the Ito's lemma. However, please note that we cannot differentiate stochastic processes (in the undergraduate calculus sense) that contain a Wiener process since the Wiener process is continuous everywhere but nowhere differentiable and consequential the differential of the stochastic process does not exist.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .