Integration Rules and Formulas

Reglas de Integración (Integration Rules)

Formulario de Integrales (Integral Formulas)

Integrales Indefinidas Básicas (Basic Indefinite Integrals)

• The integral of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.
• The integral of $\frac{1}{x}$ is $\ln |x| + C$.
• The integral of $e^x$ is $e^x + C$.
• The integral of $a^x$ is $\frac{a^x}{\ln a} + C$.
• The integral of $\sin x$ is $-\cos x + C$.
• The integral of $\cos x$ is $\sin x + C$.
• The integral of $\sec^2 x$ is $\tan x + C$.
• The integral of $\csc^2 x$ is $-\cot x + C$.
• The integral of $\sec x \tan x$ is $\sec x + C$.
• The integral of $\csc x \cot x$ is $-\csc x + C$.
• The integral of $\frac{1}{x^2 + 1}$ is $\arctan x + C$.
• The integral of $\frac{1}{\sqrt{1 - x^2}}$ is $\arcsin x + C$.
• The integral of $\sinh x$ is $\cosh x + C$.
• The integral of $\cosh x$ is $\sinh x + C$.

Reglas Básicas de Integración (Basic Integration Rules)

• The integral of $cf(x)$ is $c \int f(x) dx$.
• The integral of $[f(x) + g(x)]$ is $\int f(x) dx + \int g(x) dx$.
• The integral of $[f(x) - g(x)]$ is $\int f(x) dx - \int g(x) dx$.

Integración por Sustitución (Integration by Substitution)

• If $u = g(x)$, then $du = g'(x) dx$.
• $\int f(g(x))g'(x) dx = \int f(u) du$

Integración por Partes (Integration by Parts)

• $\int u dv = uv - \int v du$

Integrales Trigonométricas (Trigonometric Integrals)

• The integral of $\tan x$ is $\ln |\sec x| + C$ or $-\ln |\cos x| + C$.
• The integral of $\cot x$ is $\ln |\sin x| + C$.
• The integral of $\sec x$ is $\ln |\sec x + \tan x| + C$.
• The integral of $\csc x$ is $-\ln |\csc x + \cot x| + C$.

Reducción de Potencia (Power Reduction)

• $\sin^2 x = \frac{1 - \cos(2x)}{2}$.
• $\cos^2 x = \frac{1 + \cos(2x)}{2}$.

Fórmulas Diversas (Miscellaneous Formulas)

• The integral of $\frac{1}{a^2 + x^2}$ is $\frac{1}{a} \arctan \frac{x}{a} + C$.
• The integral of $\frac{1}{\sqrt{a^2 - x^2}}$ is $\arcsin \frac{x}{a} + C$.
• The integral of $\frac{1}{x\sqrt{x^2 - a^2}}$ is $\frac{1}{a} \operatorname{arcsec} \left| \frac{x}{a} \right| + C$.

Fórmulas Hiperbólicas Inversas (Inverse Hyperbolic Formulas)

• The integral of $\frac{1}{\sqrt{x^2 + a^2}}$ is $\sinh^{-1} \frac{x}{a} +C$.
• The integral of $\frac{1}{\sqrt{x^2 - a^2}}$ is $\cosh^{-1} \frac{x}{a} +C$.
• The integral of $\frac{1}{a^2 - x^2}$ is $\frac{1}{a} \tanh^{-1} \frac{x}{a} +C$, where $|x| < a$.
• The integral of $\frac{1}{x^2 - a^2}$ is $-\frac{1}{a} \coth^{-1} \frac{x}{a} +C$, where $|x| > a$.
• The integral of $\frac{1}{a^2 - x^2}$ is $\frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right| + C$.
• The integral of $\frac{1}{x^2 - a^2}$ is $\frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right| + C$.
• These formulas assume functions are defined and $C$ is added.

Algorithmic Game Theory (AGT)

What is Game Theory?

• Game theory studies mathematical models of strategic interactions among rational agents.
• It applies to social science, logic, systems science, and computer science.

Cooperative vs. Non-Cooperative

• Cooperative game theory allows binding commitments.
• Non-cooperative game theory does not allow binding commitments.

What is a Game?

• A game consists of players, rules for their moves and information at each move, and payoff specifications.

Example: Prisoner's Dilemma

• Two suspects are arrested and questioned separately.
• If both confess, both get 10 years.
• If neither confess, both get 1 year.
• If one confesses, they get 0 years, and the other gets 25 years.
• The strategies and the number of years in prison are shown in a table:

Nash Equilibrium

• A Nash equilibrium is a set of strategies where no player is incentivized to unilaterally change their strategy.

Example: Prisoner's Dilemma

• The Nash equilibrium is (Confess, Confess).

Algorithmic Game Theory (AGT)

• AGT is an interdisciplinary field combining game theory and computer science.

Challenges

• Computational complexity makes finding Nash equilibria difficult.
• Incomplete information occurs when players lack knowledge of other players' payoffs.
• Dynamics involve repeated games where players learn over time.

Topics in AGT

• Mechanism design creates games with desirable properties.
• Price of anarchy measures efficiency loss from selfish behavior.
• Learning in games considers how players adapt over time.

Mechanism Design

• Mechanism design involves designing games to achieve desired outcomes.
• It is used even when players possess private information.

Example: Auctions

• English auctions involve increasing bids until one bidder remains.
• Sealed-bid auctions involve bidders submitting bids in secret envelopes.
• Vickrey auctions involve the highest bidder winning but paying the second-highest bid.

Revelation Principle

• The revelation principle states any mechanism can be replaced by a direct mechanism.
• In this mechanism, players truthfully reveal their private information.

Price of Anarchy

• The price of anarchy (PoA) measures the ratio between the worst-case Nash equilibrium and the social optimum.

Example: Traffic Routing

• A network has two routes from A to B.
• Route 1 has a fixed cost of 1, and Route 2 has a cost of $x$, where $x$ is the amount of traffic.
• If one player uses each route, the total cost is $1+1=2$.
• If both players use the second route, the total cost is $0.5+0.5=1$.
• Nash Equilibrium: Both users take the second route.
• Social Optimum: One user takes Route 1, and the other takes Route 2.
• Therefore, the Price of anarchy = $2/1 = 2$.

Learning in Games

Regret Minimization

• Regret is the difference between the payoff a player received and the payoff they could have received with a different strategy.

No-Regret Learning

• No-regret learning algorithms ensure players have no regret long-term.

Chapter 4: Applications of Derivatives

4.1: Related Rates

What are Related Rates?

• Related Rates problems find the rate at which a quantity changes.
• This is done by relating it to other quantities with known rates of change.
• Think: A ladder leaning against a wall has its base slide away. How fast is the top sliding down?

How to Solve Related Rates Problems

• Draw a diagram: Label changing quantities if possible.
• List known quantities and rates: Note given and required rates of change.
• Write an equation: Connect quantities using geometry, trig, or other known relationships.
• Differentiate with respect to time: Differentiate both sides with respect to $t$.
• Substitute and solve: Input known values and solve for the unknown rate of change.

Example 1

• Air is pumped into a spherical balloon at $100 cm^3/s$.
• How fast is the radius increasing when the diameter is 50 cm?

Solution:

• Step 1: Imagine a sphere (balloon) being inflated and create a diagram.
• Step 2: $\frac{dV}{dt} = 100 \text{ cm}^3/\text{s}$ (rate of change of volume).
• We must find $\frac{dr}{dt}$ when $d = 50 \text{ cm}$, which means $r = 25 \text{ cm}$.
• Step 3: $V = \frac{4}{3} \pi r^3$ represents the volume of a sphere.
• Step 4: $\frac{dV}{dt} = \frac{d}{dt} (\frac{4}{3} \pi r^3)$ becomes $\frac{dV}{dt} = \frac{4}{3} \pi (3r^2 \frac{dr}{dt})$, which simplifies to $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$.
• Step 5: $100 = 4 \pi (25)^2 \frac{dr}{dt}$ then $\frac{dr}{dt} = \frac{100}{4 \pi (625)}$.
• So, $\frac{dr}{dt} = \frac{1}{25 \pi} \text{ cm/s}$.

Example 2

• A 10 ft ladder rests against a wall. The bottom slides at 1 ft/s.
• How fast is the top sliding down when the bottom is 6 ft from the wall?

Solution:

• Step 1: Imagine a right triangle with the ladder (10 ft), wall, and ground and make a diagram.
  • Distance from wall to base of ladder: $x$.
  • Distance from ground to top of ladder: $y$.
• Step 2: $\frac{dx}{dt} = 1 \text{ ft/s}$ (base sliding rate). We want to find $\frac{dy}{dt}$ when $x = 6 \text{ ft}$.
• Step 3: Use Pythagorean Theorem: $x^2 + y^2 = 10^2$, so $x^2 + y^2 = 100$.
• Step 4: $\frac{d}{dt} (x^2 + y^2) = \frac{d}{dt} (100)$ becomes $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$.
• Step 5: We know $x = 6$, so find $y$: $6^2 + y^2 = 100$, then $y^2 = 64$, so $y = 8$.
  • $2(6)(1) + 2(8) \frac{dy}{dt} = 0$ becomes $12 + 16 \frac{dy}{dt} = 0$, so $16 \frac{dy}{dt} = -12$.
  • Thus, $\frac{dy}{dt} = -\frac{12}{16} = -\frac{3}{4} \text{ ft/s}$.

Algorithmic Trading and Order Execution

• Instructor: Prof. Xin Guo (Email: [email protected])
• Teaching Assistant: TBA
• Class Time: Tuesday & Thursday 14:30 - 15:45
• Venue: ERB 705
• Office Hours: Tuesday & Thursday 16:00 - 17:00 @ SHB 918 or by appointment

Course Description

• This course covers algorithmic trading and order execution.
• Topics include market microstructure, optimal trading strategies, and practical concerns in implementing algorithms.

Learning Outcomes

• Understand market microstructure and its impact on trading.
• Design and implement optimal trading strategies.
• Evaluate the performance of trading algorithms.
• Understand practical implementation issues.

Prerequisites

• Basic knowledge of probability and statistics.
• Basic knowledge of financial markets.
• Basic programming skills, particularly Python.

Grading Policy

• Homework: 30%
• Midterm Exam: 30%
• Final Project: 40%

Textbook

• Algorithmic Trading: Winning Strategies and Their Rationale, by Ernest P. Chan.
• Quantitative Trading: How to Build Your Own Algorithmic Trading Business, by Ernest P. Chan.

Reference Books

• Trading and Exchanges: Market Microstructure for Practitioners, by Larry Harris.
• Algorithmic Trading and DMA: An introduction to direct access trading, by Barry Johnson.

Course Schedule

• This schedule is subject to change.