What is the definition of a martingale in financial theory?

In financial theory, a martingale is defined as a stochastic process where the conditional expectation of the next value, given all past values, is equal to the most recent value. This effectively means that, on average, you cannot expect to gain or lose value from one time period to the next based solely on past information; the future expected value is equal to the present value.

This property is crucial in financial modeling, particularly in pricing derivative securities and in the assessment of fair games. It implies that the best prediction of a future price is simply the current price, reinforcing the concept of a "fair game" in gambling or betting scenarios.

Understanding this definition helps in grasping more complex concepts related to randomness, probability, and pricing in financial environments. The other choices do not accurately capture the essence of what a martingale represents in the context of stochastic processes and their applications in finance.