Mathematical Finance

One is concerned with defining the following financial concepts in terms of continuous-timeStochasticProcesses:

Assets: what people/entities own and it’s cash value
Security: an asset that can be traded. Can be based on debt (a deposit to a bank that is owed to the depositor but can be invested by the bank), on equity (stocks) or on both (hybrid)
Bond: a bond is a security that is a form of loan/IOU with interest
Risk: uncertainty about financial returns. Can arise due to various aspects of the market leading to various forms of risk
Markets: the system in which trading occurs
Portfolios: collection of investments
Gains: when the market value of an asset exceeds purchase price. Can be realized (sold for cash) or unrealized.
Wealth: Collection of assets that can be used for trading/transactions

A financial market is modeled as a collection of stochastic processes on an underlying filtered probability space that model stocks (risky assets) and bond (risk-free):

Initial stock prices i.e. share of bond/market. That is continuous, -adapted, and has finite variation:

Misplaced & S_0(t) &> 0 \\ S_0(0) &= 1 \end{aligned}$$ 2. A finite-variation process $A(t)$ (singular part of $S$) $$A\left(t\right):=\int_0^{t}\frac{1}{S_0\left(s\right)}\space dS_{0^{}}^{\perp}\left(s\right)$$ 3. A risk-free bond (absolutely continuous part of $S$): $$\begin{aligned} r\left(t\right) &\in L^1([0, T]) \\ r\left(t\right) &:=\frac{1}{S_0\left(t\right)}\space \frac{d}{dt} S_{0^{}}^{AC}\left(t\right) \end{aligned} $$ 4. The average and dividend rates of return: $$\begin{aligned} b: [0,T] \times \mathbb{R}^N \to \mathbb{R} \in L^2([0, T]) \\ \delta: [0,T] \times \mathbb{R}^N \to \mathbb{R} \in L^2([0, T]) \end{aligned}$$ 5. The volatility: $$\sigma: [0,T] \times \mathbb{R}^{N \times D} \to \mathbb{R} \in L^2([0, T])$$ such that $$ \sum_{n=1}^N \sum_{d=1}^D \int_0^T \sigma_{n,d}^2(s) \space ds < \infty$$

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Mathematical Finance

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