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Unformatted text preview: 1 MAT6082 Topics in Analysis II 1 st term, 2009-10 Teacher : Professor Ka-Sing Lau Schedule : Every Tuesday, 7:00pm to 9:30pm Venue : Room 222, Lady Shaw Building, CUHK Topics : Introduction to Stochastic Calculus In the past thirty years, there has been an increasing demand of stochastic calculus in mathematics as well as various disciplines such as mathematical finance, control theory, physics and biology. The course is a rigorous intro- duction to this topic. The material include conditional expectation, Markov property, martingales, stochastic processes, Brownian motions, Ito’s calculus, and stochastic differential equations. Prerequisites Students are expected to have good background in real analysis, probability theory and some basic knowledge of stochastic processes. References: 1. A Course in Probability Theory, K.L. Chung, (1974). 2. Measure and Probability, P. Billingsley, (1986). 3. Introduction to Stochastic Integration, H.H. Kuo, (2006). 4. Introduction to Stochastic Calculus with Application, F. Klebaner, (2001). 5. Brownian Motion and Stochastic Calculus, I. Karatzas and S.E. Shreve, (1998) 2 Everyone knows calculus deals with deterministic objects. On the other hand stochastic calculus deals with random phenomena. The theory was intro- duced by Kiyosi Ito in the 40’s, and therefore stochastic calculus is also called Ito calculus . Besides its interest in mathematics, it has been used extensively in statistical mechanics in physics, the filter and control theory in engineering. Nowadays it is very popular in the option price and hedging in finance. For example the well-known Black-Scholes model is dS ( t ) = rS ( t ) dt + σS ( t ) dB ( t ) where S ( t ) is the stock price, σ is the volatility, and r is the interest rate, and B ( t ) is the Brownian motion. The most important notion for us is the Brownian motion. As is known the botanist R. Brown (1828) discovered certain zigzag random movement of pollens suspended in liquid. A. Einstein (1915) argued that the movement is due to bombardment of particle by the molecules of the fluid. He set up some basic equations of Brownian motion and use them to study diffusion. It was N. Wiener (1923) who made a rigorous study of the Brownian motion using the then new theory of Lebesgue measure. Because of that a Brownian motion is also frequently called a Wiener process. Just like calculus is based on the fundamental theorem of calculus , the Ito calculus is based on the Ito Formula : Let f be a twice differentiable function on R , then f ( B ( t ))- f ( B (0)) = Z T f ( B ( t )) dB ( t ) + 1 2 Z T f 00 ( B ( t )) dt where B (0) = 0 to denote the motion starts at 0. There are formula for integration, for example, we have Z T B ( t ) dB ( t ) = 1 2 B ( t ) 2- 1 2 T ; Z T tdB ( t ) = TB ( T )- Z T B ( t ) dt....
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- Spring '08