Stackelberg Competition Model

Dynamic Games with Complete Information

• Dynamic games involve sequential moves by players, allowing for information about previous actions.
• Stackelberg's model of duopoly exemplifies a dynamic game.

Stackelberg's Model

• Two firms choose quantities $q_1$ and $q_2$ of a homogeneous product.
• Market-clearing price: $P(Q) = a - Q$, where $Q = q_1 + q_2$.
• Cost functions: $C_i(q_i) = cq_i$, where $0 < c < a$.
• Firm 1 (leader) chooses quantity $q_1$ first; firm 2 (follower) observes $q_1$ and then chooses $q_2$.
• Payoffs for the firms: $\pi_i(q_1, q_2) = q_i[P(q_1 + q_2) - c] = q_i[a - q_1 - q_2 - c]$.

Firm 2's Problem

• Firm 2 maximizes profit $\pi_2(q_1, q_2) = q_2[a - q_1 - q_2 - c]$.
• First-order condition: $\frac{\partial \pi_2}{\partial q_2} = a - q_1 - 2q_2 - c = 0$.
• Firm 2's reaction function: $q_2^*(q_1) = \frac{a - q_1 - c}{2}$.

Firm 1's Problem

• Firm 1 maximizes profit $\pi_1(q_1, q_2^(q_1)) = q_1[a - q_1 - q_2^(q_1) - c]$.
• Substituting $q_2^(q_1)$, Firm 1's profit function becomes: $\pi_1(q_1, q_2^(q_1)) = q_1[\frac{a - q_1 - c}{2}]$.
• First-order condition: $\frac{\partial \pi_1}{\partial q_1} = \frac{a - 2q_1 - c}{2} = 0$.
• Firm 1's optimal quantity: $q_1^* = \frac{a - c}{2}$.
• Substituting $q_1^$ into Firm 2's reaction function gives $q_2^ = \frac{a - c}{4}$.

Equilibrium

• Equilibrium quantities: $q_1^* = \frac{a - c}{2}$, $q_2^* = \frac{a - c}{4}$.
• Total quantity: $Q^* = q_1^* + q_2^* = \frac{3(a - c)}{4}$.
• Equilibrium price: $P^* = a - Q^* = \frac{a + 3c}{4}$.
• Equilibrium profits: $\pi_1^* = q_1^* (P^* - c) = \frac{(a - c)^2}{8}$ and $\pi_2^* = q_2^* (P^* - c) = \frac{(a - c)^2}{16}$.

Comparison with Cournot Equilibrium

• Quantities chosen simultaneously
• Equilibrium quantities in Cournot are $q_1^C = q_2^C = \frac{a - c}{3}$.
• Total quantity: $Q^C = q_1^C + q_2^C = \frac{2(a - c)}{3}$.
• Equilibrium price: $P^C = a - Q^C = \frac{a + 2c}{3}$.
• Equilibrium profits: $\pi_1^C = \pi_2^C = \frac{(a - c)^2}{9}$.
• In Stackelberg: The leader produces more than the follower ($q_1^* > q_2^$). The leader produces more than in Cournot ($q_1^ > q_1^C$). The follower produces less than in Cournot ($q_2^* < q_2^C$).
• Total quantity is greater in Stackelberg ($Q^* > Q^C$) and price is lower ($P^* < P^C$).
• The leader earns more profit than the follower ($\pi_1^* > \pi_2^$), and more profit than in Cournot ($\pi_1^ > \pi_1^C$).
• The follower earns less profit than in Cournot ($\pi_2^* < \pi_2^C$).

Backward Induction

• This method identifies the subgame perfect Nash equilibrium (SPNE) in dynamic games.
• Steps involve determining optimal actions by working backward from the end, considering subsequent players' actions at each stage.

Fonction valeur absolue

Définition

• Expressed as $f(x)=|x|$, reveals the magnitude of a real number x.
• For $x \geq 0$, $|x| = x$, but if $x < 0$, then $|x| = -x$.

Properties

• The absolute value is greater than or equal to zero: $|a| \geq 0$.
• Absolute negation equals absolute original: $|-a| = |a|$.
• Absolute product equals absolute product: $|ab| = |a||b|$.
• Absolute quotient equals absolute quotient, provided the denominator doesn't equal zero: $|\frac{a}{b}| = \frac{|a|}{|b|}$, if $b \neq 0$.
• The Inequality of the Triangle states: $|a + b| \leq |a| + |b|$.
• $|a - b| \geq ||a| - |b||$

Graph

• Forms a "V" shape with its vertex at the origin in a Cartesian coordinate system.
• It is symmetrical around the y-axis, which means that it is an even function.

Statistical Inference

Point Estimation

• Estimating values of unknown parameters in a population based on a random sample $X_1,..., X_n$ with pdf $f(x;\theta)$.
• A point estimator, $\hat{\theta} = h(X_1,..., X_n)$, is a statistic that estimates the value of $\theta$.
• A point estimate, $\hat{\theta}(x_1,..., x_n)$, is the observed value of the estimator.

Methods for Finding Estimators

Method of Moments

• Equate population moments to sample moments and solve for unknown parameters $\theta_1,..., \theta_k$.
• Population moments are represented by $\mu_j' = E(X^j)$.
• Sample moments are calculated as $m_j' = \frac{1}{n}\sum_{i=1}^{n} X_i^j$.
• Solve the system of equations $\mu_1' = m_1'$, $\mu_2' = m_2'$, ..., $\mu_k' = m_k'$ to find estimators $\hat{\theta_1},..., \hat{\theta_k}$.

Maximum Likelihood Estimation (MLE)

• MLE of $\theta$ is the value $\hat{\theta}$ that maximizes the likelihood function $L(\theta) = \prod_{i=1}^{n} f(x_i;\theta)$.
• Maximize the log-likelihood function $\ell(\theta) = \log L(\theta)$ by solving $\frac{d\ell(\theta)}{d\theta} = 0$, ensuring $\frac{d^2\ell(\theta)}{d\theta^2} < 0$ for a maximum.

Properties of Estimators

Bias

• The bias of an estimator $\hat{\theta}$ is $B(\hat{\theta}) = E(\hat{\theta}) - \theta$.
• An unbiased estimator has $B(\hat{\theta}) = 0$.

Variance

• The variance of an estimator $\hat{\theta}$ is $Var(\hat{\theta}) = E[(\hat{\theta} - E(\hat{\theta}))^2]$.

Mean Squared Error (MSE)

• The MSE of an estimator $\hat{\theta}$ is $MSE(\hat{\theta}) = E[(\hat{\theta} - \theta)^2] = Var(\hat{\theta}) + [B(\hat{\theta})]^2$.

Efficiency

• An unbiased estimator $\hat{\theta_1}$ is more efficient than $\hat{\theta_2}$ if $Var(\hat{\theta_1}) < Var(\hat{\theta_2})$.
• Relative efficiency of $\hat{\theta_1}$ to $\hat{\theta_2}$ is $\frac{Var(\hat{\theta_2})}{Var(\hat{\theta_1})}$.

Consistency

• An estimator $\hat{\theta}$ is consistent for $\theta$ if $\hat{\theta}$ converges to $\theta$ in probability as $n \to \infty$.
• If $E(\hat{\theta}) \to \theta$ and $Var(\hat{\theta}) \to 0$ as $n \to \infty$., then it is a suffiecient condition for consistency

Cramér-Rao Lower Bound (CRLB)

• Provides a lower bound on the variance of any unbiased estimator $\hat{\theta}$: $Var(\hat{\theta}) \geq \frac{1}{nE[(\frac{\partial}{\partial\theta} \log f(X;\theta))^2]} = \frac{-1}{nE[\frac{\partial^2}{\partial\theta^2} \log f(X;\theta)]}$.

Sufficiency

• A statistic $T(X_1,..., X_n)$ is sufficient for $\theta$ if the conditional distribution of $X_1,..., X_n$ given $T(X_1,..., X_n) = t$ does not depend on $\theta$.
• Neyman-Fisher Factorization Theorem states that the joint pdf can be factored as $f(x_1,..., x_n;\theta) = g(T(x_1,..., x_n);\theta)h(x_1,..., x_n)$.

Exponential Family

• A family of pdfs with the form $f(x;\theta) = h(x)c(\theta)exp[w_1(\theta)t_1(x) +... + w_k(\theta)t_k(x)]$.
• $T(X_1,..., X_n) = (\sum_{i=1}^{n} t_1(X_i),..., \sum_{i=1}^{n} t_k(X_i))$ is a sufficient statistic for $\theta$.

Completeness

• A statistic T is complete if $E[g(T)] = 0$ for all $\theta$ implies that $P(g(T) = 0) = 1$ for all $\theta$.

Lehmann-Scheffé Theorem

• If $g(T)$ is an unbiased estimator of $\theta$ and $T$ is a complete sufficient statistic for $\theta$, then $g(T)$ is the UMVUE of $\theta$.

Rao-Blackwell Theorem

• $E[\hat{\theta}|T]$ is an estimator of $\theta$ and $Var(E[\hat{\theta}|T]) \leq Var(\hat{\theta})$.

Algorithmic Trading

• Employs computer programs following algorithms to automatically execute trades.
• Can produce profits more rapidly and frequently than human traders.
• Also known as automated trading, black-box trading, or system trading.

Process

• Involves strategy identification, algorithm development, backtesting, deployment into live market, and ongoing monitoring.

Example Scenario

• Algorithm could monitor prices of a specified asset on Exchange A and Exchange B.
• Execute trades by Buying on Exchange B while also Selling on Exchange A to profit.

Advantages

• Greater trading speed than humans.
• Reduction of trading costs.
• Eliminates emotional biases from trading activity.

Disadvantages

• Risk of system malfunctions that cause financial harm.
• Over-optimization on training data leads to poor performance on live trading.
• Requires constant and ongoing attention by a dev or risk management team.

Algorithmic Trading Strategies

• Trend Following: Identifies trends using moving averags or indicators and directs orders.
• Mean Reversion: Bet the price will return to its average, buy when average price is below certain mark, and sell above it.
• Arbitrage: Buying an asset at a cheaper market while simultaneously selling it at a higher market on another market.
• Index Fund Rebalancing: Making portfolios match the composition of an index quickly and automatcially.
• Mathematical Model-Based Strategies: Taking the market price when it deviates and use pricing models like Black-Scholes.

Types of Algorithms

Execution Algorithms

• VWAP (Volume Weighted Average Price): Uses volume profiles that distributes small chunks dynamically to the market.
• TWAP (Time Weighted Average Price): Uses evenly divided time slots to distribute orders.
• Percentage of Volume (POV): Releases orders as a function of the volumes traded

Statistical Algorithms

• Pairs Trading: Exploits stocks correlated overtime, when they lose the correlation, take a long on underperforming stock and short on higher performing stock.
• Arbitrage: Works on price differences on similar or exact same assets on different forms.

Future Trends

Machine Learning

• Increases in use for trading to identify patterns and adjustments that traditional algorithms cannot.

Regulation

• Concerned parties of regulators of algorithms leading to market instability.

Algorithmic Trading and Order Execution

Algorithmic Trading

• Employs computer programs to automate trading, execute large orders, and exploit arbitrage opportunities.

Advantages

• Provides faster execution speed, lower transaction costs, improved order prices.

Order Types

Market Order

• Executes trades immediately at best price available; execution guaranteed but price not.

Limit Order

• Executed only at specified price (or better); price guaranteed but execution not.

Stop Order

• Converts to market order upon reaching specified price; limits losses or protects profits.

Stop-Limit Order

• Converts to limit order upon reaching specified price; similar to stop and limit orders.

Other Order Types

• Day order is valid until end of trading day, a Good-Till-Cancelled (GTC) is valid until executed or cancelled, Fill or Kill (FOK) requires complete, immediate execution (else cancel), Immediate or Cancel (IOC) order executes any portion immediately and cancels the remainder, and All or None (AON) executes entire order all at once.

Order Execution Algorithms

Volume-Weighted Average Price (VWAP)

• Executes orders to match day’s VWAP; suitable for large orders.
• Formula: $VWAP = \frac{\sum_{i=1}^{n} P_i * V_i}{\sum_{i=1}^{n} V_i}$.

Time-Weighted Average Price (TWAP)

• Executes orders evenly over a specified time range.

Implementation Shortfall

• Aims to keep down costs of high execution orders to balance execution speed and market impact.

Percentage of Volume (POV)

• Involves trading fixed percentage of market volume; passive stance.

Dark Strategies

• Execute orders in dark pools, where there is no displayed market information.

Smart Order Routing

• Routes orders to the best market for price, liquidity and fees.

High-Frequency Trading (HFT)

• Subset of algorithmic trading that used extremely high turnover, short term markets, and high speeds.

Strategies

• Market Making is liquidity by bid and offer, statitstical arbitrage, event arbitrage, and index arbitrage.

Regulation of Gene Expression

Regulatory Sequences and Regulatory Proteins

• Genes controlled through Regulatory Sequences on DNA controlling gene transcription.

Prokaryotic Regulation

• Operon: includes a promoter, operator, and structural genes

lac Operon

• Encodes genes for lactose metabolism that consists of the lacZ, lacY, lacA, and lacI
• Absence of Lactose: The lac repressor binds to the operator, preventing transcription.
• Presence of Lactose: Lactose is converted to allolactose, which binds to the repressor, causing it to detach from the operator, allowing transcription.
• Catabolite Repression: Glucose being present means cAMP levels are low. Glucose being absent means high camp levels, which increases transcription.

trp Operon

• Encodes genes for tryptophan biosynthesis that regulates its transcription.
• Absence of Tryptophan: The trp repressor is inactive, and transcription occurs.
  • Presence of Tryptophan: Causes the activator to bind with tryptophan and inhibit transcription.
• Trp level fine tunes by Attenuation.

Eukaryotic Regulation

Chromatin Structure

• Chromatin makes up chromosomes that consists of Euchromatin (loosely packed chromatin) and Heterochromatin (Densely packed).
• Histone Modification and DNA Methylation occurs

Transcription Factors

• Activators increase transcription.
• Repressors decrease transcription.

Enhancers and Silencers

• Enhancers increase the rate of transcription when bound by activators
• Silencers Decrease transcription when bound by activators

Post-Transcriptional Regulation

• Alternative Splicing: Different mRNA molecules are produced from the same pre-mRNA molecule.
• mRNA Degradation: Lifespan of mRNA molecules regulated.
• Translation Initiation: Factors that control this translation can be regulated.
• Protein processing and degradation: Protein folding, modification, and degradation regulated by Ubiquitination.

Non-coding RNAs

• MicroRNAs (miRNAs): Block translation or cause degradation due to binding.
• Small interfering RNAs (siRNAs): Involved in interfering RNA.

Chapitre 1. Calcul numérique

Définitions

• Le calcul numérique est l'ensemble des méthodes permettant de résoudre par des opérations arithmétiques des problèmes mathématiques.
• L'analyse numérique est la branche des mathématiques qui s'intéresse à la conception et à l'étude des algorithmes de résolution de problèmes de mathématiques continues.
• Un algorithme est une suite finie et non ambiguë d'opérations ou d'instructions permettant de résoudre un problème.

Types d'erreurs

• Les erreurs inhérentes sont liées aux données du problème (mesures physiques).
• Les erreurs d'arrondi sont liées à la représentation des nombres en machine (nombre fini de chiffres).
• Les erreurs de troncature sont liées à l'approximation de fonctions ou de processus infinis par des processus finis.
• L'erreur humaine est une erreur de programmation, de manipulation des données.

Représentation des nombres

Représentation en base b

• $x = \pm (a_n b^n + a_{n-1} b^{n-1} +... + a_0 b^0 + a_{-1} b^{-1} +... + a_{-m} b^{-m})$, où $a_i \in {0, 1,..., b-1}$.

Représentation en virgule flottante

• $x = \pm (0.a_1 a_2... a_t)b \times b^e$, où b est la base, t est la précision (nombre de chiffres significatifs), e est l'exposant ($e{min} \le e \le e_{max}$), et $(0.a_1 a_2... a_t)_b$ est la mantisse normalisée ($a_1 \neq 0$).

Norme IEEE 754

• Définit les formats de représentation des nombres en virgule flottante.
  • Simple précision (32 bits): 1 bit pour le signe, 8 bits pour l'exposant (décalé de 127), et 23 bits pour la mantisse.
  • Double précision (64 bits): 1 bit pour le signe, 11 bits pour l'exposant (décalé de 1023), et 52 bits pour la mantisse.

Erreurs d'arrondi

• Définition: L'erreur d'arrondi est la différence entre la valeur exacte d'un nombre et sa représentation en virgule flottante.
• Types d'arrondi :
  • Arrondi par défaut (truncation) : On supprime les chiffres au-delà de la précision t.
  • Arrondi au plus près : On choisit la représentation la plus proche de la valeur exacte.
• Erreur relative d'arrondi: $\epsilon = \frac{|x - fl(x)|}{|x|}$ où $fl(x)$ est la représentation en virgule flottante de $x$ et l'unité d'arrondi (machine epsilon) $\mu = \frac{1}{2} b^{1-t}$.

Annulation et absorption

• L'annulation est un type of error that occurs with too many digits when subtracting numbers that are close together.
• L’absorption a type of error that occurs with too many digits that ignores a higher digit number being added.

Chemical Engineering Thermodynamics

Fugacity

• Describes pure gas i with Gibbs free energy.
• Equation: $dG_i = RTd(lnf_i)$.
  • $G_i$ is the Gibbs free energy and $f_i$ is fugacity.
• fugacity coefficient: $\varPhi_i = \frac{f_i}{P}$.

Generalized Virial Coefficient Charts

• Uses rediced temperature and pressure.
  • $T_r = \frac{T}{T_c}$ $P_r = \frac{P}{P_C}$
  • $T_c$ is critical temperature and $P_c$ is critical pressure.

To estimate the fugacity coefficient, use the Virial equation: $Z = 1 + B'p$

• Z is the compressibility factor, B' is the Virial coefficient, p is the molar density.
• to estimate this coefficient $B = \frac{RT_c}{P_c}$
  • Where $B_0=0.083 - \frac{0.422}{T_r^{1.6}}$.
  • $\varPhi = exp(\frac{BP}{RT}$.

Resumen del artículo

Titulo

Efecto del entrenamiento de fuerza en la fuerza y potencia muscular en adultos mayores: una revisión sistemática

Objetivo

El estudio analizó los efectos del entrenamiento de fuerza (EF) sobre la fuerza y potencia muscular en adultos mayores, así como identificar las variables de entrenamiento más efectivas.

Metodo

Publicaciones hasta marzo de 2022 en las bases de datos PubMed, Scopus y Web of Science muestran una revisión sistemática de ensayos controlados aleatorios (ECA) en adultos mayores (≥ 60 años.)

Resultados

43 ECA con un total de 1597 participantes es eficaz para mejorar la fuerza y la potencia muscular debido a las variables EF en la intensidad, duración, ejercicios, volumen, y frecuencia.

Conclusion

El EF es una intervención eficaz para mejorar la fuerza y la potencia muscular en adultos mayores que se recomiendan con intensidad, volumen, frecuencia, duración y ejercicios.

Algèbre linéaire

Vecteurs

• A vector is math entity defined by, direction, sense and magnitude.

Représentation

• Geometrically with an arrow
• Analytically with coordinate components
• Algebracally with symbolic representation

Opérations

• Vector addition
• Scalar multiplication of vectors
• Scalar Product: $\qquad \vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos(\theta)$
• Vector Product: $\qquad ||\vec{u} \times \vec{v}|| = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \sin(\theta)$

Espace Vectoriel

• vector addition
• scalar multiplcation

Base

• A space defined by linear independant vectors and combined linear to create the original vector of the space.

Dimension

• The number of vectors in space

Matrices

Définition

• Matrices are math table with rectanglar numbers.

Opérations

• Addition, scalar multiplication, and matric multiplication of matrices

Types

• There are square matrices, inverse matrices, transposée matrices, and identity matrices.

Déterminant

The determinant that can be determined by whether not a matrix is reversie.

Valeurs propres et vecteurs propres

$\qquad A\vec{v} = \lambda\vec{v}$

Transformations linéaires

Définition

Transforms vector preserving additional scalar operations.

Représentation matricielle

Equivalent application applying lineal transforme equivalent to matrix multication.

Exemples

rotations, réflexions, projections and dilatations

Applications

Informatique, ingénierie, physique, économie et mathématiques