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Notations

Mathematical Notations

The following notations will be used throughout the course:

  • Natural Numbers: \(\mathbb{N} = \{1, 2, \ldots\}\), \(\mathbb{N}_0 = \{0, 1, 2, \ldots\}\).
  • Integers: \(\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\)
  • Rational Numbers: \(\mathbb{Q} = \{ p/q\colon p \in \mathbb{Z}, q \in \mathbb{N}\}\)
  • Real Numbers: \(\mathbb{R}\)
  • Vectors in \(\mathbb{R}^d\) are denoted in bold font, \(\boldsymbol{x} = (x^1, \dots, x^d)\), and are assumed to be column vectors.
  • Vectors with positive components \(\mathbb{R}^d_+ = \{\boldsymbol{x} \in \mathbb{R}^d : x^k \geq 0, k=1,\ldots,d\}\) and vectors with strictly positive components \(\mathbb{R}^d_{++} = \{\boldsymbol{x} \in \mathbb{R}^d : x^k > 0, k=1,\ldots,d\}\).
  • Scalar Product: \(\boldsymbol{x} \cdot \boldsymbol{y} := \sum x_k y_k\) denotes the scalar product of \(\boldsymbol{x}\) and \(\boldsymbol{y}\) in \(\mathbb{R}^d\).
  • \(\beta \boldsymbol{x} := (\beta x_1, \ldots, \beta x_d)\) represents the multiplication of \(\boldsymbol{x}\) in \(\mathbb{R}^d\) by a scalar \(\beta \in \mathbb{R}\).
  • \(\boldsymbol{x} + \boldsymbol{y} := (x_1 + y_1, \ldots, x_d + y_d)\) represents vector addition in \(\mathbb{R}^d\).
  • Component wise operations: \(\boldsymbol{x}\boldsymbol{y}= (x_1 y_1, \ldots, x_d y_d)\), \(\boldsymbol{x}/ \boldsymbol{y} = (x_1/ y_1, \ldots, x_d/y_d)\), \(f(\boldsymbol{x}) = (f(x_1), \ldots, f(x_d))\) for any function \(f\colon \mathbb{R}\to \mathbb{R}\).

  • For scalars \(x, y \in \mathbb{R}\), the following notations are used:

    \[ x \vee y = \max\{x, y\}, \quad x \wedge y = \min\{x, y\}, \quad x^+ = \max\{x, 0\}, \quad x^- = \max\{-x, 0\}. \]

    Notably, \(x = x^+ - x^-\) and \(|x| = x^+ + x^-\).

Color/Environment conventions

Definition

For a ... we define

Remark

Note that

Example

As an example we consider

Theorem

Let \((\Omega, \mathcal{F}, P)\) be a probability space...

Proposition

Assuming no-arbitrage for the financial market, the followign assertions holds...

Corollary

As a corrolary to the previous proposition, it holds

Lemma

In the case where \(P^\ast\) is equivalent to \(P\), it holds...

Proof

In a first step we show that \((i)\) implies \((ii)\)...

Exercise

Solve in a a binomial financial market...