(Markov property for geometric Brownian motion and its...

(Markov property for geometric Brownian motion and its maximum to date). Recall the geometric Brownian motion $S(t)$ of (7.4.1) and its maximum-to-date process $Y(t)$ of (7.4.3). According to Definition 2.3.6, in order to show that the pair of processes $(S(t), Y(t))$ is Markov, we must show that whenever $0 \leq t \leq T$ and $f(x, y)$ is a function, there exists another function $g(x, y)$ such that $$ \mathbb{E}[f(S(T), Y(T)) \mid \mathcal{F}(t)]=g(S(t), Y(t)) . $$ Use the Independence Lemma, Lemma 2.3.4, to show that such a function $g(x, y)$ exists. 7.8 Exercises 335

(Markov property for geometric Brownian motion and its maximum to date). Recall the geometric Brownian motion $S(t)$ of (7.4.1) and its maximum-to-date process $Y(t)$ of (7.4.3). According to Definition 2.3.6, in order to show that the pair of processes $(S(t), Y(t))$ is Markov, we must show that whenever $0 \leq t \leq T$ and $f(x, y)$ is a function, there exists another function $g(x, y)$ such that
$$
\mathbb{E}[f(S(T), Y(T)) \mid \mathcal{F}(t)]=g(S(t), Y(t)) .
$$

Use the Independence Lemma, Lemma 2.3.4, to show that such a function $g(x, y)$ exists.
7.8 Exercises
335

Key Concepts

Geometric Brownian Motion

A geometric Brownian motion (GBM) is a continuous-time stochastic process used to model phenomena such as stock prices. It is characterized by exponential growth driven by a combination of a deterministic drift component and a random diffusion component, typically modeled with Brownian motion. The multiplicative nature of GBM makes its logarithm normally distributed, which simplifies many calculations in finance and physics.

Maximum-to-Date Process

The maximum-to-date process records the highest value reached by a stochastic process up to any given time. This process is particularly important in applications where the record values influence decision-making or risk assessment, such as in the pricing of lookback options in finance. Its behavior relative to the underlying process helps in determining extremal events.

Markov Property

The Markov property is a fundamental characteristic of many stochastic processes asserting that the future evolution of the process depends only on its present state and not on the path that led to it. This 'memorylessness' simplifies the analysis and prediction of such processes because it reduces the complexity of conditional probabilities and expectations to those based solely on current information.

Conditional Expectation and Filtration

Conditional expectation is the expected value of a random variable given a certain sigma-algebra, which represents information available at a specific time. A filtration is an increasing sequence of these sigma-algebras that models the accumulation of information over time. Together, these concepts are crucial in describing how the processes adapt and evolve as more information becomes available.

Independence Lemma

The Independence Lemma, which relies on the independent increments property of Brownian motion, allows one to decompose a stochastic process into parts that evolve independently of the past. This lemma is instrumental in showing that, for processes like GBM, the future evolution conditional on the present state is independent of the historical path, thereby offering a route to proving the Markov property.

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