Stochastic Calculus for Finance, Volume I and II

Commentationes Mathematicae Universitatis Carolinae, 2017

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This text has evolved from mathematics courses in the Master of Science in Computational Finance (MSCF) program at Carnegie Mellon University. The content of this hook has been used successfully with students whose mathematics background consists of calculus and calculus-based probability The text given precise statements of results, plausibility arguments, and even some proofs, but more importantly, intuitive explanations developed and refined through classroom experience with this material are provided. Exercises conclude every chapter. Some of these extend the theory and others are drawn from practical problems in quantitative finance.

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2020

This paper serves as a rigorous introduction to probability theory and stochastic calculus. It will first introduce basic facts about probability over uncountable spaces, and subsequently transition to a discussion of brownian motion and stochastic calculus.

2006

Exercise 1.1: Show that for each n, the random variables K(1), . . . , K(n) are independent.

This, somewhat unusual collection of problems in Measure-Theoretic Probability and Stochastic Analysis, should have been more appropriately called: "The Problems I Like". The problems borrowed from the existing literature have solutions different than in the original sources, and for those full references and occasional historical remarks are provided. There are also original problems:

UNITEXT for Physics

For simplicity again, in this chapter we will only consider processes with just one component. We already remarked in the Sect. 6.3 that a white noise is a singular process whose main properties can be traced back to the non differentiability of some processes. As a first example we have shown indeed that the Poisson impulse process (6.63) and its associated compensated version (6.65) are white noises entailed by the formal derivation respectively of a simple Poisson process N (t) and of its compensated variant N (t). In the same vein we have shown then in the Example 6.22 that also the formal derivative of the Wiener process W (t)-not differentiable according to the Proposition 6.18-meets the conditions (6.69) to be a white noise, and in the Appendix H we also hinted that the role of the fluctuating force B(t) in the Langevin equation (6.78) for the Brownian motion is actually played by such a white noiseẆ (t). We can now give a mathematically more cogent justification for this identification in the framework of the Markovian diffusions. The Langevin equation (6.78) is a particular case of the more general equatioṅ X (t) = a(X (t), t) + b(X (t), t) Z (t) (8.1) where a(x, t) and b(x, t) are given functions and Z (t) is a process with E [Z (t)] = 0 and uncorrelated with X (t). From a formal integration of (8.1) we find X (t) = X (t 0) + t t 0 a(X (s), s) ds + t t 0 b(X (s), s)Z (s) ds so that, being X (t) assembled as a combination of Z (s) values with t 0 < s < t, to secure the non correlation of X (t) and Z (t) we should intuitively require also the non correlation of Z (s) and Z (t) for every pair s = t. Since moreover Z (t) is presumed

2002

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