Question about find no arbitrage trading strategy

We got the stochastic process for stock price of n stocks at continues time.

We can find if there is a arbitrage trading strategy or dominant trading strategy.

I wonder if we cannot find such strategies, Can we conclude there is no mispricing in the stock prices.

In my opinion, although no such strategies exists in the stochastic process. The mispricing still possibly exists in stock prices. I don't know if I am right.

Philosphically, I agree with you.

Sometimes you will see people like Icahn, Kohlberg Kravis... buy a majority stake in a company and take it private, selling off parts of the company, restructuring others. One interpretation of this activity is exactly what you said: there is a mispricing in the stock (compared to assets, earnings, whatever), but no way to profit by setting up an arbitrage strategy involving stocks, bonds and other publicly traded securities of this and other firms, futures, ... . So these people have to do something else to earn a living [ ;-) ]. Arbitrage only takes care of situations where you can replicate one security by taking positions in other securities. In the real world (unlike the Arrow Debreu world) often you cannot do that, there aren't enough securities.

I'm trying to understand your question. To be concrete --- and trying to really putting your question in solid terms --- suppose for simplicity we are dealing with something like a standard geometric Brownian motion,

$$dS_t = \mu S_t dt + \Sigma S_t dW_t$$ where $S = (S_{1}, ..., S_{n})$ represent stock price of the $n$ risky assets at time, and $W = (W_1, ..., W_n)$ is an $n$-dimensional Brownian motion. Say $\mu$ is $n$-dimensional drift and $\Sigma$ is $n \times n$ variance-covariance matrix. Assume the usual conditions such that there exists a strong or weak solution to this SDE. For good measure, throw in a risk free asset with instantaneous risk free rate $r$.

So up to this point, I think most people would agree that this is a standard continuous time model for modeling $n$ risky assets (yes, you can put in other bells and whistles with jumps, stochastic volatility, more general stochastic integrator, etc, but let's just keep things simple).

But this is the next point where I start to lose you. You say,

We can find if there is a arbitrage trading strategy or dominant trading strategy.

I wonder if we cannot find such strategies, Can we conclude there is no mispricing in the stock prices.

In my opinion, although no such strategies exists in the stochastic process. The mispricing still possibly exists in stock prices. I don't know if I am right.

At this point, whether this model admits arbitrage or not is simply to invoke the First and Second Fundamental Theorem of Asset Pricing. But then are you deliberately rigging a model so that it admits arbitrage? But what is a good motivation to do so, especially in a continuous-time model? How would this be beneficial in pricing derivatives and/or understanding other asset pricing questions?