By Wolfgang Paul
This e-book offers with stochastic approaches that are vital in statistical physics and the rules of finance markets. It offers the foundation for estimation and calculation of greatly self sustaining approaches governing a extra advanced behaviour of platforms. The ebook is a needs to for monetary analysts and for physicists and mathematicians operating within the box of finance.
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Additional info for Stochastic processes: from physics to finance
Example text
The order ≤GDO3 -cx satisfies property (O1). We do not know whether ≤GDO3 -cx satisfies property (O2). 1 we conjecture that it does not. However, we have the following result. 2. The order ≤GDO3 -cx satisfies property (O3). Proof : Let (X, Y ) be a random vector. Furthermore, let φ be a oneto-one measurable function, and consider (φ(X), Y ). Let G(x, u) be defined as in Eq. 2), and define Gφ by Gφ (x, u) = G(φ−1 (x), u). Let U be a uniform random variable on [0, 1], which is independent of st X. Since, by Eq.
4. M. SHAKED, M. A. SORDO, AND A. SUAREZ-LLORENS 19 and hence, E[Y ⊥ U ⊥ ] = h(U ⊥ ). 4) Thus E[E[Y ⊥ U ⊥ ]] = E[h(U ⊥ )] = E[Y ⊥ ] = E[Y ] = E[E[Y U ]], where the first equality follows from Eq. 4) and the second equality follows from Eq. 3). Thus, the two variables in Eq. 2) have the same expected value. Now, let φ be a convex function. Then, E[φ(E[Y U ])] ≤ E[E[φ(Y ) U ]] = E[φ(Y )] = E[φ(Y ⊥ )] = E[φ(E[Y ⊥ U ⊥ ])], where the inequality follows from Jensen’s Inequality. In order to prove the second inequality in Eq.
SORDO, AND A. SUAREZ-LLORENS 9 (O1) The order ≤GDO is reflexive and transitive. st ˜ Y˜ ), and if (V, W ) and (V˜ , W ˜ ) satisfy V = (O2) If (X, Y ) ≤GDO (X, st st ˜ st st = Y , as well as E[W V ] = E[Y X] and V˜ = X and W = W st ˜ , then (V, W ) ≤GDO (V˜ , W ˜ ). ˜ V˜ = E Y˜ X E W ˜ Y˜ ) then (φ(X), l(Y )) ≤GDO (φ(X), ˜ l(Y˜ )), (O3) If (X, Y ) ≤GDO (X, where φ is a one-to-one measurable function, and l is a linear function. (O4) Let (X, Y ) be any random vector, then (X, Y ) ≥GDO (X ⊥ , Y ⊥ ).