constituents

• \(V\) a vector space
• \((u,v)\), an ordered pair of vectors in \(V\) (its not commutative!)

requirements

We define \(\langle u, v \rangle \in \mathbb{F}\) as the inner product of \((u,v)\) in that order!. It carries the following properties:

1. positivity: \(\langle v, v\rangle \geq 0, \forall v \in V\)
2. definiteness: \(\langle v, v\rangle = 0\) IFF \(v = 0\)
3. additivity in the first slot: \(\langle u+v, w\rangle = \langle u, w \rangle + \langle v, w \rangle\)
4. homogeneity in the first slot: \(\langle \lambda u, v \rangle = \lambda \langle u, v \rangle\)
5. conjugate symmetry: \(\langle u,v \rangle = \overline{\langle v,u \rangle}\)

additional information

Inner Product Space

An Inner Product Space is a vector space with a well-defined inner product. For instance, \(\mathbb{F}^{n}\) has the canonical inner product named Euclidean Inner Product (see below, a.k.a. dot product for reals). The existence of such a well-defined inner product makes \(\mathbb{F}^{n}\) an Inner Product Space.

an integer (\(\mathbb{Z}\)) is the natural numbers, zero, and negative numbers: …,-4,-3,-2,-1,0,1,2,2,3

representing integers

• what are the limitations of computational arithmetic
• how to perform efficient arithmetic
• how to encode data more compactly and efficiently

See also computer number system

The integrating factor \(\rho(x)\) is a value that helps undo the product rule. For which:

\begin{equation} log(\rho(x)) = \int P(x)dx \end{equation}

for some function \(P(x)\).

Separating the \(\rho(x)\) out, we have therefore:

\begin{equation} e^{\int P dx} = \rho(x) \end{equation}

Why is this helpful and undoes the product rule? This is because of a very interesting property of how \(\rho(x)\) behaves.

Goal

We are going to solve the inter-temporal choice problem, for ten time stamps, and perform some numerical optimization of the results

Main Methods

We do this by solving backwards. We will create a variable \(k\) to measure asset, and \(k_{t}\) the remaining asset at time \(t\).

Let us first declare the function for power utility. \(k\) is our asset holding, \(\gamma\) our relative margin of risk, and \(U\) the power utility.