Introduction To Stochastic Calculus With Applications
(Fima C Klebaner)
Right now, stochastic calulus is applied in finance. Some financial models are set by asset processes which base on Brown Motion. This book is a mainintroduction for maths aspect. At the begging,this book gives the concept about integrals, such as Riemann integral and Stieltjes Integral. These are the foundation of the theory of shochastic calculus. The reader should know about the concepts of probability theory. The book introduces this part simply. Because the theory is built on the probability space , the reader should know about that. And Brown Motion is introduced. Such as the concept and the martingale property of Brown Motion.The reader will found that Brown Motion is used to describ the sample path of asset,such as stock, and so on. And the book tells the reader the Brown Motion calculus.In this part,the most important formula: Ito formula is introduced. That?s is one of the high lights of the stochastic theory. When you read here, you should pay you attention to this part ,because this formula is used broadly in finance models. The stochactic differential equations,which is usually called SDE for short, is usually used to describe the motion of all kind of finance tools ,such as stock and bond,and so on. Next,martingale as an important concept is brought in.This concept is used to get the concept of semimartingale which can build the integral.And then diffusion processes, martingales,calculus for semimartingales, are discussed. So,the continuous part about the sample path of asset?s motion.But some times, the asset?s fluction shows some discontinuous,the book calls that ?jump?.We can find that in real world.Stock price usually jumps. At last ,pure jump process,change of probability measure and the application of stochastic theory are discussed.We can know the theory is a powerful tool because stochatic theory are applied in finace, biology, engineering and physics broadly.
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