I often come across similar statements:
- Let $X_t$ be a stochastic process that satisfies the stochastic differential equation: $$dX_t = \mu_t dt + \sigma_t dB_t$$
Where
- $B_t$ is a Weiner process
- $\mu_t$ , $\sigma_t$ are deterministic functions of time.
My Question: What does it actually mean to "satisfy" a stochastic differential equation?
It seems to me that these are all just definitions. We are defining $X_t$ such that the derivative of $X_t$ with respect to time is $dX_t = \mu_t dt + \sigma_t dB_t$. Therefore, it is satisfied by definition. We have chosen $X_t$ such that its derivative satisfies $\mu_t dt + \sigma_t dB_t$.
Thus, is the word "satisfy" really necessary here?
Could the above statement simply have been written as: Let $X_t$ be a stochastic process such that: $dX_t = \mu_t dt + \sigma_t dB_t$
Or does "satisfying a stochastic differential equation" have some other criteria that need to be met?
Thanks!
First to be clear, the process
$$X_t =x_{0}+ \int_{0}^{t}\mu_s ds + \int_{0}^{t}\sigma_s dB_s$$
is called an Itô process. So I agree "satisfying..." doesn't make much sense because $X_{t}$ is already given by the above integrals. It is akin to calling $f'(x)=0, f(0)=a$ an ODE, which has trivially the solution $f(x)=a$. It is not really thought of as differential equation where one tries to identify $X_{t}$ from a recursive-type SDE equation
$$dX_{t}=\mu(t,X_{t})dt+\sigma(t,X_{t})dB_{t}$$
eg. see Solution to General Linear SDE.