Editing Stochastic process (section)

== Application ==

=== Applications in Finance ===

==== Black-Scholes Model ====
One of the most famous applications of stochastic processes in finance is the '''[[Black-Scholes model]]''' for option pricing. Developed by [[Fischer Black]], [[Myron Scholes]], and [[Robert Solow]], this model uses '''[[Geometric Brownian motion]]''', a specific type of stochastic process, to describe the dynamics of asset prices.<ref>{{Cite journal |last1=Black |first1=Fischer |last2=Scholes |first2=Myron |date=1973 |title=The Pricing of Options and Corporate Liabilities |url=https://www.jstor.org/stable/1831029 |journal=Journal of Political Economy |volume=81 |issue=3 |pages=637–654 |doi=10.1086/260062 |jstor=1831029 |issn=0022-3808}}</ref><ref>{{Citation |last=Merton |first=Robert C. |title=Theory of rational option pricing |date=July 2005 |work=Theory of Valuation |pages=229–288 |url=http://www.worldscientific.com/doi/abs/10.1142/9789812701022_0008 |access-date=2024-09-30 |edition=2 |publisher=WORLD SCIENTIFIC |language=en |doi=10.1142/9789812701022_0008 |isbn=978-981-256-374-3|hdl=1721.1/49331 |hdl-access=free }}</ref> The model assumes that the price of a stock follows a continuous-time stochastic process and provides a closed-form solution for pricing European-style options. The Black-Scholes formula has had a profound impact on financial markets, forming the basis for much of modern options trading.

The key assumption of the Black-Scholes model is that the price of a financial asset, such as a stock, follows a '''[[log-normal distribution]]''', with its continuous returns following a normal distribution. Although the model has limitations, such as the assumption of constant volatility, it remains widely used due to its simplicity and practical relevance.

==== Stochastic Volatility Models ====
Another significant application of stochastic processes in finance is in '''[[Stochastic volatility|stochastic volatility models]]''', which aim to capture the time-varying nature of market volatility. The '''[[Heston model]]'''<ref>{{Cite journal |last=Heston |first=Steven L. |date=1993 |title=A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options |url=https://www.jstor.org/stable/2962057 |journal=The Review of Financial Studies |volume=6 |issue=2 |pages=327–343 |doi=10.1093/rfs/6.2.327 |jstor=2962057 |issn=0893-9454}}</ref> is a popular example, allowing for the volatility of asset prices to follow its own stochastic process. Unlike the Black-Scholes model, which assumes constant volatility, stochastic volatility models provide a more flexible framework for modeling market dynamics, particularly during periods of high uncertainty or market stress.

=== Applications in Biology ===

==== Population Dynamics ====
One of the primary applications of stochastic processes in biology is in '''[[population dynamics]]'''. In contrast to [[deterministic model]]s, which assume that populations change in predictable ways, stochastic models account for the inherent randomness in births, deaths, and migration. The '''[[birth-death process]]''',<ref name="Ross 2010">{{Cite book |last=Ross |first=Sheldon M. |title=Introduction to probability models |date=2010 |publisher=Elsevier |isbn=978-0-12-375686-2 |edition=10th |location=Amsterdam Heidelberg}}</ref> a simple stochastic model, describes how populations fluctuate over time due to random births and deaths. These models are particularly important when dealing with small populations, where random events can have large impacts, such as in the case of endangered species or small microbial populations.

Another example is the '''[[branching process]]''',<ref name="Ross 2010"/> which models the growth of a population where each individual reproduces independently. The branching process is often used to describe population extinction or explosion, particularly in epidemiology, where it can model the spread of infectious diseases within a population.

=== Applications in Computer Science ===

==== Randomized Algorithms ====
Stochastic processes play a critical role in computer science, particularly in the analysis and development of '''randomized algorithms'''. These algorithms utilize random inputs to simplify problem-solving or enhance performance in complex computational tasks. For instance, Markov chains are widely used in probabilistic algorithms for optimization and sampling tasks, such as those employed in search engines like Google's PageRank.<ref name="Randomized algorithms">{{Cite book |title=Randomized algorithms |date=1995 |publisher=Cambridge University Press |isbn=978-0-511-81407-5 |editor-last=Motwani |editor-first=Rajeev |location=Cambridge New York |editor-last2=Raghavan |editor-first2=Prabhakar}}</ref> These methods balance computational efficiency with accuracy, making them invaluable for handling large datasets. Randomized algorithms are also extensively applied in areas such as cryptography, large-scale simulations, and artificial intelligence, where uncertainty must be managed effectively.<ref name="Randomized algorithms"/>

==== Queuing Theory ====
Another significant application of stochastic processes in computer science is in '''queuing theory''', which models the random arrival and service of tasks in a system.<ref>{{Cite book |last=Shortle |first=John F. |title=Fundamentals of queueing theory |last2=Thompson |first2=James M. |last3=Gross |first3=Donald |last4=Harris |first4=Carl M. |date=2017 |publisher=John Wiley & Sons |isbn=978-1-118-94352-6 |edition=Fifth |series=Wiley series in probability and statistics |location=Hoboken, New Jersey}}</ref> This is particularly relevant in network traffic analysis and server management. For instance, queuing models help predict delays, manage resource allocation, and optimize throughput in web servers and communication networks. The flexibility of stochastic models allows researchers to simulate and improve the performance of high-traffic environments. For example, queueing theory is crucial for designing efficient data centers and cloud computing infrastructures.<ref>{{Cite book |title=Fundamentals of queueing theory |date=2018 |publisher=John Wiley & Sons |isbn=978-1-118-94356-4 |edition=5 |location=Hoboken}}</ref>