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I am trying to learn by myself stochastic calculus, and I have a question regarding semi-martingale stochastic process and geometric Brownian motion (GBM).

We know that a stochastic process $S_t$ is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): $\textrm{d}S_t = \mu_t S_t \textrm{d}t+\sigma S_t \textrm{d}W_t$ where $W_{t}$ is a Brownian motion

I was reading a paper where the authors assumed that under the null hypothesis, the real-valued logarithmic efficient price process denoted by $p_t^e$ is given by the Brownian semi-martingale: $\textrm{d}p_t^e = \mu_t \textrm{d}t+\sigma \textrm{d}W_t$

My question is, what is the difference between a GBM and a semi-martingale stochastic process? Are they the same stochastic process?

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I will provide a heuristic explanation.

A local martingale $(M_t)_t$ (mart) is a stochastic process whose expected increments are zero that is $E(\mathrm{d}M_t)=0$. Under suitable technical conditions, a local martingale is also a (true) martingale that is the current value is the best estimate for its future value: $E(M_t)=M_0$. The difference between the two cases is roughly that a local martingale might have paths under which it experiences explosive behavior.

A process $(A_t)_t$ with locally bounded variation (BV) is roughly speaking a process which is "relatively smooth" in the sense that it never experiences an arbitrarily large jump over a very small time interval. This process is also required to be right-continuous with left limits and adapted.

A semimartingale $(S_t)_t$ is a stochastic process allowing the decomposition: $$S\equiv M+A$$ i.e. it can be written as the sum of a martingale and a bounded variation process. Geometric Brownian Motion (GBM) is a semimartingale: $$\textrm{d}S_t=\underbrace{\mu S_t\textrm{d}t}_{\text{BV}}+\underbrace{\sigma S_t\textrm{d}W_t}_{\text{mart}}$$ Indeed on every infinitesimal interval $\textrm{d}t$ the GBM (1) increases/decreases by a proportional rate $\mu$, the amount being smaller the smaller the interval or the value $S_t$, nothing jumpy going on here, and (2) a random amount which depends on the Brownian Motion $W$ whose expected increment is zero: $E(\textrm{d}W_t)=0$.

Semimartingales are important in the theory of stochastic process because they are nice to integrate. Intuitively, they are made up of an "unbiased" stochastic term with no upward/downward trend, and a trend component which is restricted to bounded variation to avoid being explosive - and thus being integrable.

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