Write clear, complete and logical proofs for mathematical hypotheses that are new to the student. Explain definitions of Brownian and martingales Explain the length in the queue and solve simple waiting time problems Construct a continuous-time Markov chain and identify its generator in settings of practical problems in a diverse range of applications. Explain the basic properties of the Poisson process and use these to solve problems. Construct a Poisson process and identify its parameter from practical problem settings in a diverse range of applications. Explain Gambler's ruin problem and calculate extinction probability Explain and be able to apply limit theorems of discrete-time Markov chains and use those to identify and interpret their stationary distribution Construct a discrete-time Markov chain and identify its transition probability matrix from practical problem settings. Explain and apply the theoretical concepts of probability theory and stochastic processes. Students who undertake STAT3921/4021 will be expected to have a deeper, more sophisticated understanding of the theory and to be able to work with more complicated applications than students who complete the regular STAT3021 unit. By completing this unit, you will develop a solid mathematical foundation of stochastic processes for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control, financial mathematics and statistical inference. Throughout the unit, various illustrative examples are provided in modelling and analysing problems of practical interest. This unit will also introduce basic concepts of Brownian motion and martingales. This unit will investigate simple queuing theory. This unit will derive key results of Poisson processes and simple continuous-time Markov chains. This unit will establish basic properties of discrete-time Markov chains including random walks and branching processes. We will see more complicated examples later on.A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. ![]() The random processes in the above examples were relatively simple in the sense that the randomness in the process originated from one or two random variables. Here, we measure $t$ in hours, but $t$ can take any real value between $9$ and $16$. $-$ Let $N(t)$ be the number of customers who have visited a bank from $t=9$ (when the bank opens at 9:00 am) until time $t$, on a given day, for $t\in$. Here are a few more examples of continuous-time random processes: In general, when we have a random process $X(t)$ where $t$ can take real values in an interval on the real line, then $X(t)$ is a continuous-time random process. The process $S(t)$ mentioned here is an example of a continuous-time random process. Therefore, a random process is a collection of random variables usually indexed by time (or sometimes by space).Ī random process is a collection of random variables usually indexed by time. ![]() When we consider the values of $S(t)$ for $t \in [0,\infty)$ collectively, we say $S(t)$ is a random process or a stochastic process. If you choose another time $t_2\in [0,\infty)$, you obtain another random variable $S(t_2)$ that could potentially have a different PDF. Based on your knowledge of finance and the historical data, you might be able to provide a PDF for $S(t_1)$. Vincent Granville, data scientist and author of Stochastic Processes and Simulations, introduces new point processes for data scientists. Note that at any fixed time $t_1 \in [0,\infty)$, $S(t_1)$ is a random variable. A collection of random variables that show the evolution of a given system over time is called a stochastic point process. Here, $S(t)$ is an example of a random process. ![]() Figure 10.1 shows a possible outcome of this random experiment from time $t=0$ to time $t=1$.įigure 10.1 - A possible realization of values of a stock observed as a function of time. Here, we assume $t=0$ refers to current time. In particular, let $S(t)$ be the stock price at time $t\in[0,\infty)$. For example, suppose that you are observing the stock price of a company over the next few months. In real-life applications, we are often interested in multiple observations of random values over a period of time.
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