Calculus: Differential, Integral, and Motion

Calculus Preview

• Calculus focuses on change, providing tools to model it in areas including physics, engineering, and economics.

Two Main Branches of Calculus

• Differential Calculus centers on the instantaneous rate of change, introducing derivatives for tangents, optimization, and rates of change.
• Integral Calculus deals with the accumulation of quantities, using integrals to find areas, volumes, and average values.
• The Fundamental Theorem of Calculus connects differential and integral calculus.

Motion Example

• Quantities for motion of an object along a line include position, velocity and acceleration
• $v(t) = s'(t)$ describes velocity as the derivative of position.
• $a(t) = v'(t) = s''(t)$ describes acceleration as the derivative of velocity, or the second derivative of position.
• $s(T) - s(0) = \int_{0}^{T} v(t) dt$ presents displacement as the integral of velocity.

Historical Overview

• Ancient Greece: Archimedes had preliminary ideas on limits and areas.
• 17th Century: Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus.
  • Newton: Focus on physics, laws of motion, and gravitation.
  • Leibniz: Focus on mathematical formalism and notation.
• 18th Century: Mathematicians like Euler and the Bernoulli family developed and applied calculus.
• 19th Century: Rigorous formalization of calculus concepts by Cauchy, Riemann, and Weierstrass.
• 20th Century: Calculus became a fundamental tool in science, engineering, economics, and computer science.

Conceptual Understanding

• Understanding concepts is essential in Calculus
• Core concepts include limits, continuity, derivatives, and integrals.
• Conceptual understanding allows application of calculus to new and unfamiliar problems.

Keys to Success in Calculus

• Attend lectures and participate, read the textbook, do homework, ask questions, work with peers, and seek resources like office hours.
• Build a foundation in algebra and trigonometry.
• Practice consistently.

Definition of a Vector Space

• A vector space over a field $\mathbb{K}$ (either $\mathbb{R}$ or $\mathbb{C}$) is a nonempty set $E$ with addition and scalar multiplication operations.
• Addition: $+ : E \times E \rightarrow E$ associates $(u, v) \in E \times E$ with $u + v \in E$.
• Scalar Multiplication: $ \cdot : \mathbb{K} \times E \rightarrow E$ associates $(\lambda, u) \in \mathbb{K} \times E$ with $\lambda \cdot u \in E$.

Properties of Vector Space Operations

• Associativity of addition: $\forall u, v, w \in E, (u + v) + w = u + (v + w)$
• Commutativity of addition: $\forall u, v \in E, u + v = v + u$
• Existence of an additive identity: $\exists 0_E \in E, \forall u \in E, u + 0_E = 0_E + u = u$
• Existence of additive inverse: $\forall u \in E, \exists (-u) \in E, u + (-u) = (-u) + u = 0_E$
• Compatibility with scalar multiplication: $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, (\lambda \mu) \cdot u = \lambda \cdot (\mu \cdot u)$
• Distributivity of scalar multiplication over vector addition: $\forall \lambda \in \mathbb{K}, \forall u, v \in E, \lambda \cdot (u + v) = \lambda \cdot u + \lambda \cdot v$
• Distributivity of scalar multiplication over scalar addition: $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, (\lambda + \mu) \cdot u = \lambda \cdot u + \mu \cdot u$
• Identity element for scalar multiplication: $\forall u \in E, 1_{\mathbb{K}} \cdot u = u$

Example of a Vector Space

• $C^0(\mathbb{R}, \mathbb{R})$ represents the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, forming a vector space over $\mathbb{R}$.
  • Addition: $(f + g)(x) = f(x) + g(x)$ for all $f, g \in C^0(\mathbb{R}, \mathbb{R})$
  • Scalar Multiplication: $(\lambda \cdot f)(x) = \lambda f(x)$ for $\lambda \in \mathbb{R}$ and $f \in C^0(\mathbb{R}, \mathbb{R})$

Definition of a Vector Subspace

• Given a vector space $E$ over field $\mathbb{K}$, a subset $F$ of $E$ is a vector subspace of $E$ if:
  • $F$ is nonempty.
  • $F$ is stable under addition: $\forall u, v \in F, u + v \in F$.
  • $F$ is stable under scalar multiplication: $\forall \lambda \in \mathbb{K}, \forall u \in F, \lambda \cdot u \in F$.

Theorem for Vector Subspaces

• $F$ is a vector subspace of $E$ if and only if:
• $0_E \in F$
• $\forall u, v \in F, \forall \lambda \in \mathbb{K}, \lambda u + v \in F$

Alternative Definition of Vector Subspace

• A subset F of E is a SV if:
  • $F \subset E$
  • $F \neq \emptyset$
  • $\forall (x, y) \in F^2, \forall \lambda \in \mathbb{K}, \lambda x + y \in F$

Vector Subspace Example

• Let $E = \mathbb{R}^2$, then $F = {(x, y) \in \mathbb{R}^2 \mid x + y = 0 }$ is a vector subspace of $\mathbb{R}^2$.

Algorithmic Trading Definition

• Algorithmic trading is executing buy or sell orders using automated pre-programmed trading instructions that considers price, timing, and volume.
• Algorithmic trading is also known as automated trading, black-box trading, or algo-trading
• Algorithmic trading is widely used by institutional investors and trading firms.

Advantages of Algorithmic Trading

• Reduced transaction costs with algorithms finding the best prices.
• Improved order execution with algorithms monitoring market conditions.
• Increased speed and efficiency by automating the trading process.
• Reduced emotional bias by removing human emotion from the trading process.

Disadvantages of Algorithmic Trading

• Technical issues from complex algorithmic trading systems.
• Market manipulation through techniques like spoofing and layering.
• System failures causing technical glitches.
• Over-optimization reduces live trading performance.

Trend Following Strategies

• Trend following strategies buy or sell assets as trends continue.
• Moving averages are used in trend following.
• Moving Average Convergence Divergence (MACD) indicates the relationship between two moving averages of a security's price.
  • $MACD = 12\text{-day EMA} - 26\text{-day EMA}$
• Moving Average (MA) smooths out price data by creating an average price.

Mean Reversion Strategies

• Mean reversion strategies buy or sell as they revert prices to the mean.
• Bollinger Bands measure a security's price volatility, uses moving average, an upper and lower band.

Arbitrage Strategies

• Arbitrage strategies exploit price differences for the same asset in different markets.

Sentiment Analysis Strategies

• Sentiment analysis strategies use NLP to gauge investor sentiment.

High-Frequency Trading Strategies

• High-Frequency Trading (HFT) strategies uses high speeds, high turnover rates, and high order-to-trade ratios.

Transfer Function

• $G(s) = \frac{1}{s(s+1)(s+5)}$ is the open loop transfer function.

Calculating Gain for Phase Margin of 45°

• Needed is the frequency where the angle G(jw) is -135^\circ.
• Phase margin of 45^0 can be attained by making the magnitude equal to 1.
• $\arctan(\omega) + \arctan(\frac{\omega}{5}) = 45^\circ$

Formula

• $\arctan(x) + \arctan(y) = \arctan(\frac{x+y}{1-xy})$

Solving for Omega

• $\frac{\frac{6\omega}{5}}{1 - \frac{\omega^2}{5}} = 1$
• $\omega^2 + 6\omega - 5 = 0$
• $\omega = -3 + \sqrt{14} \approx 0.7417$

Magnitude of G(jw)

• $|G(j\omega)| = \frac{1}{\omega \sqrt{\omega^2 + 1} \sqrt{\omega^2 + 25}}$
• $|G(j0.7417)| = \frac{1}{0.7417 \sqrt{0.7417^2 + 1} \sqrt{0.7417^2 + 25}} \approx \frac{1}{0.7417 \cdot 1.245 \cdot 5.05} \approx 0.213$

Finding K

• $K |G(j\omega)| = 1$
• $K = \frac{1}{|G(j\omega)|} \approx \frac{1}{0.213} \approx 4.69$

Transfer Function and System Type

• This is a type 1 system (one integrator in the open loop).
• $G(s) = \frac{K}{s(s+1)(s+5)}$

Calculating Steady-State Error

• Ramp input of $R(s) = \frac{1}{s^2}$
• $e_{ss} = \lim_{s \to 0} \frac{sR(s)}{1 + G(s)} = \lim_{s \to 0} \frac{5}{K}$
• $e_{ss} = \frac{5}{4.69} \approx 1.066$

Time Delay Transfer Functions

• $G(s) = \frac{1}{s(s+1)(s+5)}e^{-0.5s}$ with a time delay

Calculating Changes in Angle

• $\angle e^{-j\omega T} = -\omega T$ radians $= -\omega T \cdot \frac{180}{\pi}$ degrees
• Using the omega 0.7417
• Change in the angle due to delay approximates 21.25^\circ

The New Phase Margin

• Nuevo margen de fase $= 45^\circ - 21.25^\circ \approx 23.75^\circ$

What Time Delay Does

• Reduced Stability
• More Oscillator Response Transisiton
• Increased Response Time

Partial Differential Equations (PDEs)

• These are equations involving functions of several variables and their partial derivatives.
• PDEs are useful for modeling many phenomena in science and engineering like fluid flow and heat transfer

Examples of PDEs Applications

• Heat Equation: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ describes the diffusion of heat.
• Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ describes wave propagation.
• Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ describes steady-state phenomena.

Complexity of PDEs

• There is no general method for solving all PDEs.
• PDE solutions can be much more complicated than ODE solutions.
• The theory of PDEs is still under development.

Order Classification

• The equation containing $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ is a second order.

Linearity

• A PDE is linear if it is linear in the unknown function and derivatives.
• $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ is linear, while $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x}$ is nonlinear.

Homogeneity

• A linear PDE is homogeneous if the equation contains no terms that do not involve the unknown function or its derivatives.
• $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ is homogeneous.
• $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + f(x, t)$ is nonhomogeneous.

Solution Techniques

• Separation of Variables and Fourier Series are viable solution techniques

Further Solution Techniques

• A tool for solving linear PDEs with nonhomogeneous terms or boundary conditions is known as Green's functions.
• Finite difference method, Finite element method, and Finite volume method, are three numerical methods.

Applications

• Heat transfer, fluid flow, and electromagnetism all rely on PDEs

Helpful Software

• FEniCS: an open-source software
• COMSOL
• MATLAB PDE Toolbox

What is Game Theory?

• Game theory is the study of multi-agent decision problems.

Concepts in Game Theory

• Each agent has actions, utility functions over action profiles, and acts to maximize their utility
• Games can be cooperative vs. non-cooperative, zero-sum vs. non-zero sum, complete vs. incomplete information, static vs. dynamic, and discrete vs. continuous

Traditional vs. Algorithmic Game Theory

• Traditional GT assumes players are computationally unbounded.
• Algorithmic GT Bridges Computer Science and Game Theory.
• Algorithmic GT examines games with computationally bounded players

Topics in Algorithmic Game Theory

• Mechanism Design
• Price of Anarchy
• Solution Concepts
• Complexity of Nash Equilibrium
• Game Representation

Selfish Routing Example

• Network of n players seeks to minimize travel time

Selfish Rating and Braess's Paradox

• Adding a network link increases average latency at equilibrium.
• Define $G = (V, E)$ as a graph then for each $e \in E$, let $l_e(x)$ be the latency function, where $x$ is the amount of flow on edge $e$.

Social Cost

• $SC(f) = \sum_{e \in E} f_e \cdot l_e(f_e)$
• Where Let $s, t \in V$ be the source and destination, respectively, and let $f$ be the total amount of flow from $s$ to $t$.

Calculating Price of Anarchy

• $PoA = \frac{\text{Social cost of worst-case Nash equilibrium}}{\text{Social cost of optimal solution}}$

What is Mechanism Design?

• Mechanism design is reverse game theory
• Mechanism design is the how to design game rules to achieve a desired outcome.

Revelation principle

• Revelation Principle states: Any social choice function that can be implemented can be implemented truthfully
• There is nothing lose by restricting attention to truthful mechanisms by letting players report their true values.

Nash equilibrium

• A set of strategies (one for each player) such that no player can improve their utility by unilaterally changing their strategy.

• A frequency response of a system is the response of the system to a sinusoidal input signal.

Stable Linear Time Invariant Systems

• Stable LTI systems with impulse response have output of y(t) = $\int_{-\infty}^{\infty} h(\tau)x(t-\tau)d\tau$
• Let $x(t) = e^{jwt}$, where $w$ is the frequency of the input signal. Then the output is
• $y(t)= e^{jwt} H(jw)$
• Where $H(jw) = \int_{-\infty}^{\infty} h(t)e^{-jwt}dt$ is the Fourier Transform of $h(t)$.
• Changes in magnitude and phase is determined by $H(jw)$.

Definition Magnitude and Phase

• The frequency response $H(jw)$ is a complex function of frequency $w$.
• $H(jw) = |H(jw)|e^{j\angle H(jw)}$

• Fourier Transform represents signal frequency content

The Funtions of Fourier

• $X(jw) = \int_{-\infty}^{\infty} x(t)e^{-jwt}dt$
• $x(jw) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(jw)e^{jwt}dw$

Fourier Transform Properties

• Operations include linearity, time scaling, time shifting, convolution, and multiplication

Useful Fourier Table

• $\delta(t)$ has a Fourier Transform of 1.
  • 1 has a Fourier Transform of $2\pi\delta(w)$.
  • $e^{jwt}$ has a Fourier Transform of $2\pi\delta(w - w_0)$.

Blackbody Definition

• A blackbody is an object that absorbs all electromagnetic radiation.
• A blackbody when heated emits radiation.
• The spectrum of this radiation depends upon temperature
  • Temperature not material
  • Classical physics is unable to explain blackbody radiation

Planck's Breakthrough

• Planck solved UVCatastrophe proposing energy is quantized
• Energy can only be emitted/absorbed with discrete packets called quanta
• Quantum equation $E = hf$,
• E is Energy
• h is Planck's Constant - f is radiation frequency - Planck's Constant Relates photon energy to frequency.

Photoelectric Effect

• Light shines on a metal emitting electrons

• The kinetic energy depends on light frequency.

• Independent of light intensity

• Einstein proposed light made up of particles (photons).

• When a photon strikes a metal, it can transfer its energy to an electron.

  • If the energy of the photon is greater than the work function of the metal, the electron will be emitted from the metal.

• Kinetic energy of the emitted electron is given by: KE = hf - ∅.

Ordinary Differential Equations (ODEs)

• An equation with a function and its derivatives.
• Function only dependent on one variable
• Referred, ODE Examples: $\frac{dy}{dx} = 5x^2$, $\frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0$, and $\frac{d^3y}{dt^3} + t\frac{dy}{dt} = e^t$
• Order - The order of a ODE is dependent on highest derivative, -Linearity - An ODE is linear if function is linear.

Key Terms for Ordinary Differential Equations (ODEs)

• Non-linearity of function exists with no products or functions
  • Example: $a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = f(x)$
• $a_n(x), a_{n-1}(x), \dots, a_1(x), a_0(x)$ y $f(x)$ are functions and must be x.

Examples for Ordinary Differential Equations (ODEs)

• $\frac{dy}{dx} + 5y = e^x$ and, $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = \sin(x)$
• These are linear terms
• $\frac{dy}{dx} = y^2$ and $\frac{d^2y}{dx^2} + \sin(y) = 0$ are Non-Linear

General Solutions and Particular Solutions

• "General" is when the equation returns with constants
  • "Particular" is given specified constants Initial Value Problems (PVI) Initial problems and its solutions

Algorithmic Trading

• Automated black box system trades according to a certain algorithm strategy
• Increased speed, less bias, accuracy, and cheap!
• But! Requires dev costs, technical expertise, less safety, requires regulatory approval
• Over-Optimization requires backtesting

Orders by Algorithm

• VWAP
• Executes to match with weighted volume average price
• TWAP
• Balances equally in amount and time
• Shortfall
• Aims to minimize actual price and decision price

High-Frequency Trading and Flash Crashes

• HFT is quick and short term
• Quickness could become instability and unfair

Dark Pools and Alternative Trading Systems (ATS)

• Dark Pools Trade securities with zero transparency
• Allows trading of large amounts without volatility - But limits visibility overall.