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Mathematical Statistics for Finance provides an in-depth discussion of the mathematical topics which lie at foundation of the applications of statistics to finance:

• The roots of the symmetry between the Mean-Covariance versus the Probabilistic Framework;
• The theory to learn from data in both frameworks.

More precisely Mathematical Statistics for Finance consists of the following parts of the "Data Science Map":

• The Probabilistic Framework describes the essential tools to operate in the Probabilistic Ecosystem, where:
  1. Statistical relationships among variables are modeled by probability distributions;
  2. Transformations among variables are non-linear;
  3. Structure is imposed via independence or more generally via conditional independence, which follows from the notion of conditioning.
  This part also covers copulas and respective implementations.
• The Mean-Covariance Framework describes the essential tools to operate in the Mean-Covariance Ecosystem, where:
  1. Statistical relationships among variables are modeled by their mean-covariance equivalence classes;
  2. Transformations among variables are linear or affine;
  3. Structure is imposed via uncorrelation, or more generally partial uncorrelation, which follows from the notion of L² linear projection.
  This part also covers measures of dependence and concordance.
• Decision theory under risk addresses modeling and optimization of decisions under the assumption that the joint mean-covariance classes, or probabilistic distributions, of all random variables are known.
• Estimation leverages decision theory under uncertainty to learn, from data, relevant features of the joint mean-covariance classes, or probabilistic distributions, when these are not known. Key concepts include elicitability, asymptotic/random matrix theory, hypothesis testing.
• Inference covers how to learn not only from data, but also from subjective opinions, both mean-covariance classes (Black-Litterman) and probability distributions (minimum relative entropy).

Mean-Covariance Learning represents the linear blueprint for Probabilistic Machine Learning.

It covers practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Mean-Covariance Framework.

The key ingredients are linear factor models, which model all mean-covariance structures: supervised (linear regression); unsupervised (principal component and factor analysis); hybrid (canonical correlation, total least squares); and causal (structural equation models).

This part also covers the estimation of linear factor models, namely mean/loadings and (high-dimensional) covariance matrices, in the context of financial applications.

This part covers the below portion of the "Data Science Map".

Probabilistic Machine Learning discusses machine learning/artificial intelligence models, presented as generalizations of Mean-Covariance Learning.

It covers practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Probabilistic Framework.

The key ingredients are conditional distributions, which model all probabilistic structures: supervised learning (point and probabilistic); unsupervised learning (autoencoders and graphical models); and one-period reinforcement learning (causal Bayesian networks).

This part also covers the estimation of specific conditional distributions in the context of financial applications.

This part covers the below portion of the "Data Science Map".

Time Series and Reinforcement Learning covers the dynamic counterparts of Mean-Covariance Learning and Probabilistic Machine Learning.

It covers practical ways of learning observational and causal models and taking optimal decisions in both the Mean-Covariance Framework and the Probabilistic Framework, when data is not i.i.d.

As such, this part includes multivariate econometrics, continuous time stochastic processes, and optimal sequential decision making.

This part covers the below portion of the "Data Science Map".

Financial Engineering covers Steps 1-4 of the "Checklist".

Step 1 discusses how to price instruments across asset classes by means of the so-called risk-neutral or "Q" measure, as well as variations such as the CAPM or the APT.

Step 2 discusses how to convert raw financial data into well-behaved times series.

Step 3 discusses how to use econometric tools to model and estimate the evolution of such time series in the so-called real world or "P" measure.

Step 4 discusses how to map the future evolution of the time series back into the object of interest, which is joint distribution of the instruments future payoff.

This part covers the below portion of the "Quantitative Finance Checklist".

Portfolio and Enterprise Risk Management covers Steps 5-7 of the "Checklist":

Step 5 discusses how to compute the aggregate value of a given portfolio, based on the portfolio's holdings; and how to aggregate the future payoff of each instrument into the future payoff of the portfolio under regular and stress market conditions.

Step 6 discusses how to assess the overall risk in a given portfolio at the fund, desk, or enterprise level.

Step 7 discusses how to attribute the overall risk to the contribution of different factors.

This part covers the below portion of the "Quantitative Finance Checklist".

Portfolio Construction and Trading covers Steps 8-10 of the "Checklist".

Step 8 discusses how to construct theoretical static portfolios based on mean-variance optimization or more complex algorithms; and to build dynamic investment strategies based on cross sectional heuristics or option based portfolio insurance.

Step 9 discusses how to implement a theoretical allocation in practice by optimally scheduling small orders in an electronic exchange.

Step 10 discusses how to assess past realized performance and attribute profits and losses to different contributors.

This part covers the below portion of the "Quantitative Finance Checklist".