Stochastic Modeling in Economics and Finance

Bulletin of the American Mathematical Society, 2002

2013

0 σ(s,Xs)dWs where W is a r-dimensional Brownian motion, b is a d × 1 drift vector and σ is a d × r dispersion matrix. Definition 5.1. A weak solution of (5.1) is a triple (X,W ), (Ω,F , P ), (Ft), where i) (Ω,F , P ) is a probability space, (Ft) is a filtration of sub-σ-fields of F satisfying usual conditions; ii) X = (Xt) is adapted to (Ft) and W is (Ft)-Brownian motion such that (5.1) is satisfied and P [ ∫ t 0 { |b(s,Xs)|+ σ ij(s,Xs) } ds < ∞] = 1 for every t > 0. The probability measure μ on B(R) defined by μ(Λ) = P (X0 ∈ Λ) is called the initial distribution of the solution. Definition 5.2. We say that uniqueness in the sense of probability law holds for (5.1) if, for any two weak solutions (X,W ), (Ω,F , P ), (Ft) and (X̂, Ŵ ), (Ω̂, F̂ , P̂ ), (F̂t) with the same initial distribution have the same law, i.e. for any t1, . . . , tn and n ≥ 1, P (Xt1 ∈ E1, . . . , Xtn ∈ En) = P̂ (X̂t1 ∈ E1, . . . , X̂tn ∈ En), where Ei ∈ B(R). Exercise 5.1. Let X be a weak solution of (5.1...

Choice Reviews Online, 2007

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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.