Appendix

This appendix provides a concise review of fundamental concepts from stochastic analysis and calculus, which are essential for the rigorous development presented in this book. A.1.1. Slutsky's theoremTHEOREM A.1 (Slutsky's theorem).-Let(X n ) n≥1 and (Y n ) n≥1 be two sequences of random variables.If X n converges in distribution to a random variable X (denoted X n d -→ X), and Y n converges in probability to a constant c (denoted Y n p -→ c), then the following hold:A direct consequence is that if both X n and Y n converge in probability to constants c 1 and c 2 , respectively, then their sums, products and quotients also converge in probability to the corresponding constants, provided the denominator is non-zero.For instance, if X n p -→ c 1 and A.1.2. SemimartingalesDEFINITION A.1.-Let(Ω, F, (F t ) t≥0 , P ) be a filtered probability space.A stochastic process M = (M t ) t≥0 is called a martingale with respect to the filtration (F t ) if:The class of semimartingales is of paramount importance in stochastic calculus because it is the largest class of processes for which the Itô integral can be meaningfully defined. A.2. Appendix 2: essential mathematical toolsThis appendix collects several key mathematical results, ranging from matrix properties to analytical approximation techniques, which are applied throughout the book. A.2.1. On positive semi-definite matricesLEMMA A.1.-LetM be an n × n symmetric positive semi-definite matrix.If for some vector x ∈ R n , the quadratic form x M x = 0, then it must follow that M x = 0. 10.