European Mathematical Society - 90 Operations research, mathematical programming
https://euro-math-soc.eu/msc/90-operations-research-mathematical-programming
enExamples in Markov Decision Processes
https://euro-math-soc.eu/review/examples-markov-decision-processes
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This excellent book provides approximately 100 examples, illustrating the theory of controlled discrete-time Markov processes. The main attention is paid to counter-intuitive, unexpected properties of optimization problems. Such examples illustrate the importance of conditions imposed in the known theorems on Markov decision processes. The aim was to collect them together in one reference book which should be considered as a complement to existing monographs on Markov decision processes. The book is self-contained and unified in presentation.<br />
The main theoretical statements and constructions are provided, and particular examples can be read independently of others. Examples in Markov Decision Problems, is an essential source of reference for mathematicians and all those who apply the optimal control theory for practical purposes. This book brings together examples based upon such sources, along with several new ones. In addition, it indicates the areas where Markov Decision Processes can be used. Active researchers can refer to this book on applicability of mathematical methods and theorems.<br />
The examples presented either lead to counter-intuitive solutions or illustrate the importance of conditions in the known theorems. Not all examples are equally simple or complicated. Several examples are aimed at undergraduate students, whilst others will be of interest to professional or amateur researchers.<br />
This book has four chapters in line with the four main different types of MDP; the finite-horizon, infinite horizon with total or discounted loss, and average loss over an infinite time interval. Some basic theoretical statements and proofs of auxiliary assertions are included in the two appendixes.<br />
We consider only minimization problems. When formulating theorems and examples published in books, journals, articles or papers devoted to maximization, we always adjust the statements for our case without any special remarks. It should be emphasized that the terminology in MDP is not entirely fixed.<br />
Markov Decision Processes (MDP) is a branch of mathematics based on probability theory, optimal control and mathematical analysis. Many books on the subject with counterexamples/paradoxes in probability are in the literature; it is therefore not surprising that Markov Decision Processes is also replete, with unexpected counter-intuitive examples.<br />
The main goal of this intriguing book is to collect together such examples. Fascinating task! This book should be considered as a complement to scientific textbooks and monographs on MDP. The book can also serve as a reference book to which one can turn for answers to curiosities that arise which studying or teaching MDP. All the examples are self-contained and can be read independently of each other. Concerning uncontrolled Markov chains, we mention the illuminating collection of examples in Suhov and Kelbert (2008).<br />
This book is high recommended. Some examples are aimed at undergraduate students, whilst others will be of interest to advanced undergraduates, graduates and research students in probability theory, optimal control and applied mathematics, looking for a better understanding of the theory; experts in Markov decision processes, professional or amateur researchers.<br />
Chapter 1: Finite-Horizon Models (pp. 1-50)<br />
In this chapter on analyzed the definition and basics of finite-horizon models. The aim is to find an optimal control strategy solving the problem related as performance functional using dynamic programming approach because the probabilistic system evolves through time. If loss functions are not bounded, the situation respect dynamic programming approach becomes more complicated. The Lemma 1, 1, and the corollaries 1.1., and the 1.2, can be helpful in the solution, because the lemma and the corollaries provide sufficient conditions of optimality. The examples presented, in this chapter, are 16 interesting problems, which allows, one extensive and complete illustration of finite-horizon models. All the examples considered are self-contained.<br />
The examples are the following:<br />
1. Non-transitivity of the correlation. The property of being positively correlated is not necessarily transitive. This question was studied widely in the literature.<br />
2. Te more frequently used control is not better. The counterexample proposed is clear.<br />
3. Voting. This is an interesting problem related with the final decision in accordance with the majority among the three opinions of three magistrates which investigate an accused person who is actually guilty.<br />
4. The secretary problem. Here on considered only a very simple version of the classical and famous problem studied in depth in the specialized literature.<br />
5. Constrained optimization. This intriguing example showing that the Bellman principle fails to hold and the optimal control strategy can look strange. Note that, very often, the solution to a constrained MDP is given by a randomized Markov control strategy; however, there is still no reason to consider past-dependent strategies. By introducing artificial random variables, the author should modify the final loss, and in this new model, the Bellman principle holds.<br />
6. Equivalent Markov selectors in non-atomic MDPs. In this situation, on presented two different examples showing that, in the non-atomic case, a selector φ does not exist.<br />
7. Strongly equivalent Markov selector in non atomic MDPs. The notion of strategies π and φ strongly equivalent is important in the theory of mass transportation. The author shows in theorem 1.1, associated to this example, that all conditions established are important for the strongly equivalence.<br />
8. Stock exchange. It is worth paying attention to the different loss function employed, because it should cause curious conclusions in real problems.<br />
9. Markov o non Markov strategy? Randomized or not? When is the Bellman principle violated? The example shows that the requirement concerning the infinities is essential. In the proposed example, the optimal control strategy φ is not uniformly optimal, even being the reasoning correct for an arbitrary selector, meaning that non-randomized strategies cannot satisfy all equalities established and cannot be uniformly optimal.<br />
10. Uniformly optimal, but not optimal strategy. This example is the above slightly modified, by ignoring the initial step and putting another distribution of probability.<br />
11. Martingales and the Bellman principle. If the functions C y c, the loss functions are bounded below, then the estimating process is a martingale if and only if π is optimal. The example 9 and the proposed here, show some difficulties in the above assertion, because the estimating process is not a martingale.<br />
12. Conventions on expectation and infinities. These interesting examples on devoted to the use of another conventions on expectations and infinities, as different authors suggest to calculate the performance criterion, accepting the rule (+∞) + (-∞) = +∞.<br />
13. Nowhere-differentiable function vt (x); discontinuous function vt (x). In this example on shows one proposition in analysis: 2It is known that a functional series can converge (absolutely) to a continuous, but nowhere-differentiable function”.<br />
14. The non-measurable Bellman function. The dynamic programming approach is based on the assumption that optimality equation has a measurable solution. The following example shows that the Bellman function may be not Borel-measurable even in the simplest case having T=1, C(x) ≡0 with a measurable loss function.<br />
15. No-one strategy is uniformly ε-optimal. This example here is an old example considered in the literature.<br />
16. Semi-continuous model.MDP is called semi-continuous if the condition 1.1 is satisfied: (Condition 1.1.), 1) the action space is compact, 2) the transition probabilities is a continuous stochastic kernel and 3) the loss functions are lower semi-continuous and bounded below. If the action space is not compact or the transition probability is not continuous, or the loss functions are not lower semi-continuous, then trivial examples show that the desired selector φ* may not exist.<br />
Chapter 2: Homogeneous Infinite-Horizon Models: Expected Total Loss (pp.51-126)</p>
<p>In this chapter on assumed that the time horizon is not finite. The definitions of strategies and selectors are the same as in the case finite horizon. The goal is to find an optimal control strategy solving the performance functional. The following conditions are always satisfied: “for any control strategy, either expected values (positives or negatives) are finite” and also the Putterman (1994), Section 7.2 on positive models. </p>
<p>MDP is called absorbing if there is a state, for which the controlled process is absorbed in at time T. Absorbing models are considered in 2,7,10,13,16,17,19,20,21,24,28. The examples 3,4,9,13,18 are from the area of optimal stopping in which, on each step, there exists the possibility or putting the controlled process in a special absorbing state, with no future loss. </p>
<p>The homogenous infinite-horizon models with the criteria based in the expected total loss. In this chapter on analyzed 28 nice examples with above criteria.</p>
<p>The examples are the following:<br />
1. Mixed Strategies. Definitions and example quasi-trivial.<br />
2. Multiple solutions to the optimality equation. Application to a discrete –time of queuing model.<br />
3. Finite model: multiple solutions to the optimality equation; conserving but not equalizing strategy. The condition Putterman is satisfied, so the Bellman function coincides with the maximal non-positive solution.<br />
4. The single conserving strategy is not equalizing and not optimal. This example shows that there exist no optimal strategies in this model.<br />
5. When strategy iteration is not successful. If the model is positive and the cost function is bounded then the theorem due to Strauch (1966) holds. For negative models, the theorem due to Putterman (1994), said that if the strategy iteration algorithm terminates, then it returns an optimal strategy.<br />
6. When value iteration is not successful. Example when the first condition is violated.<br />
7. When value iteration is not successful: positive model I. The proposed example: value iteration does not converge to the Bellman function.<br />
8. When value iteration is not successful: positive model II. The exampled showed that the statement, the existence of limit, can fail if the model is not negative.<br />
9. Value iteration and stability in optimal stopping problems.<br />
10. A non-equalizing strategy is uniformly optimal. Example when the second condition is violated.<br />
11. A stationary uniformly ε-optimal selector does not exist (positive model). Example of a stationary uniformly ε-optimal selector does not exist.<br />
12. A stationary uniformly ε-optimal selector does not exist (negative model). Example of a stationary uniformly ε-optimal selector does not exist. The proposed example can be called gambling.<br />
13. Finite-action negative model where a stationary uniformly ε-optimal selector does not exist. An example which can be reformulated as a gamble.<br />
14. Nearly uniformly optimal selectors in negative model. The example shows that if the state space is countable the theorem of Ornstein (1969) holds. The example also shows that this theorem cannot be essentially improved.<br />
15. Semi-continuous models and the blackmailer´s dilemma. Many very powerful results are known for semi-continuous models. On required one additional condition: 1) the action space is compact, b) the loss function for each state is a lower semi-continuous in each action, and c) for each state, the function integral is continuous in each action for every (measurable) bounded function u. The example is the blackmailer´s dilemma (Bertsekas, 1987, page 254).<br />
16. Not a semi-continuous model. If the model is not semi-continuous then one cannot guarantee the existence of optimal strategies. The exampled proposed shows that no one stationary selector is ε-optimal.<br />
17. The Bellman function is non-measurable and no one strategy is uniformly ε-optimal<br />
18. A randomized strategy is better than any selector (finite action space)<br />
19. The fluid approximation does not work<br />
20. The fluid approximation: refined model<br />
21. Occupation measures: phantom solutions. For a fixed control strategy, the occupation measure is one particular measure on XxA. The introduction of this measure, convex analytic approach to the MDP is fruitful, especially in constrained problem. In the sequel on analyzed different illustrating examples.<br />
22. Occupation measures in transient models<br />
23. Occupation measures and duality<br />
24. Occupation measures: compactness<br />
25. The bold strategy in gambling is not optimal (house limit). This classical game can be modeled as an MDP.<br />
26. The bold strategy in gambling is not optimal (inflation). Another example of the gambler problem.<br />
27. Search strategy for a moving target. In this example, it seems plausible that the optimal strategy is simply, to search the location that given the highest probability of find the object.<br />
28. The three-way duel (“Truel”). The sequential truel is a game that generalizes the simple duel. Every marksman wants to maximize his probability of winning the game.<br />
Chapter 3: Homogeneous Infinite-Horizon Models: Discounted Loss (pp.127-176)</p>
<p>This chapter is about the equal problem where β € (0, 1) is the discount factor. As usual, vπ is the performance functional. The discount model is a particular case of an MDP with total expected loss. The problem is now equivalent to the investigation of the modified (absorbing) model, with finite, totally bounded expected absorption time.<br />
Nevertheless, discounted models traditionally constitute a special area in MDP. By this reason the following examples, except some cases, are special modifications which introduced the discount factor. 23 different examples contain this chapter.<br />
The examples are the following:<br />
1. Phantom solutions of the optimality equation<br />
2. When value iteration of the optimality equation<br />
3. A non-optimal strategy π for which v π x solves the optimality equation<br />
4. The single conserving strategy is not equalizing and not optimal<br />
5. Value iteration and convergence of strategies<br />
6. Value iteration in countable models<br />
7. The Bellman function is non-measurable and no one strategy is uniformly ε-optimal<br />
8. No one selector is uniformly ε-optimal<br />
9. Myopic strategies. A stationary strategy, uniformly optimal in the homogenous one-step model, T=1 with terminal loss C(x) =0, is called myopic.<br />
10. Stable and unstable controllers for linear systems<br />
11. Incorrect optimal actions in the model with partial information<br />
12. Occupation measures and stationary strategies<br />
13. Constrained optimization and the Bellman principle<br />
14. Constrained optimization and Lagrange multipliers<br />
15. Constrained optimization: multiple solutions<br />
16. Weighted discounted loss and (N,∞)-stationary selectors<br />
17. Non-constant discounting<br />
18. The nearly optimal strategy is not Blackwell optimal<br />
19. Blackwell optimal strategies and opportunity loss<br />
20. Blackwell optimal and n-discount optimal strategies<br />
21. No Blackwell (Maitra) optimal strategies<br />
22. Optimal strategies as β→ 1 - and MDPs with the average loss -I<br />
23. Optimal strategies as β→ 1 - and MDPs with the average loss -II</p>
<p>Chapter 4: Homogeneous Infinite-Horizon Models: Average Loss and Other Criteria (pp.177-252)</p>
<p>Under rather general conditions, problem of optimization, with average loss (or another criteria), of the performance functional is well defined, e.g. if the loss function c is bounded below. In this context, such strategies will be called AC-optimal, i.e. average-cost-optimal. If the model is finite then there exists a stationary AC-optimal selector. The situation becomes more complicated if either space X or A is not finite. The dynamic programming approach is very adequate for this problem.<br />
Some strategies Marko and Semi-Markov are also considered in the context of AC-optimality.<br />
In this chapter four, an interesting and fascinating selection on analyzed. In particular 31 examples are examined with above criteria.</p>
<p>The examples are the following:<br />
1. Why limsup?<br />
2. AC-optimal non-canonical strategies<br />
3. Canonical triplets and canonical equations<br />
4. Multiple solutions to the canonical equations in finite models<br />
5. No AC-optimal strategies<br />
6. Canonical equations have no solutions: the finite action space<br />
7. No AC-optimal stationary strategies in a finite state model<br />
8. No AC-optimal strategies in a finite-state semi-continuous model<br />
9. Semi-continuous models and the sufficiency of stationary selectors<br />
10. No AC-optimal stationary strategies in a unichain model with a finite action space<br />
11. No AC- ε-optimal stationary strategies in a finite action model<br />
12. No AC- ε-optimal Markov strategies<br />
13. Singular perturbation of an MDP<br />
14. Blackwell optimal strategies and AC-optimality<br />
15. Strategy iteration in a unichain model<br />
16. Unichain strategy iteration in a finite communicating model<br />
17. Strategy iteration in a semi-continuous models<br />
18. When value iteration is not successful<br />
19. The finite-horizon approximation does not work<br />
20. The linear programming approach to finite models<br />
21. Linear programming for infinite models. The linear programming proved to be effective in finite models. In the general case, this approach was developed in the literature, but under special conditions.<br />
22. Linear programs and expected frequencies in finite models<br />
23. Constrained optimization<br />
24. AC-optimal, bias optimal, overtaking optimal and opportunity-cost optimal strategies: periodic model<br />
25. AC-optimal and average-overtaking optimal strategies<br />
26. Blackwell optimal, bias optimal, overtaking optimal and AC-optimal strategies<br />
27. Nearly optimal and average-overtaking optimal strategies<br />
28. Strong-overtaking/average optimal, overtaking optimal, AC-optimal strategies and minimal opportunity loss<br />
29. Strong-overtaking optimal and strong*-overtaking optimal strategies<br />
30. Parrondo´s paradox<br />
31. An optimal service strategy in a queuing system<br />
Afterword<br />
Briefly mention several real-life applications of MDP<br />
- Control of a moving object. The objective can be, for example, reaching the goal with the minimal expected energy. The state is the position of the object subject to random disturbances and the action corresponds to the power of an engine.<br />
- Control of water resources. The performance to be maximized corresponds to the expected utility of the water consumed. The state is the amount of water in a reservoir and on decisions about using the water.<br />
- Consumption-investment problems. The objective is to minimize the total expected consumption over the planning interval.<br />
- Inventory control. The goal is to maximize the total expected profit from selling the product.<br />
- Reliability. In this case, to minimize the total expected loss resulting from failures and from the maintenance cost.<br />
- Financial mathematics. The state is the current wealth along the vector of stock prices in a random market. The action represents the restructuring of the self-financing portfolio. The aim is the maximization of the expected utility associated with the final wealth.<br />
- Selling an asset. The objective is to maximize the total expected profit.<br />
- Gambling. The objective is to maximize the probability of reaching the goal. Gambling examples are given in chapter two, examples 14, 25 and 26.<br />
They are, in the literature other meaningful examples, for example: quality control in a production line; forest management; controlled populations; participating in a quiz show; organizing of teaching and examinations; optimization of publicity efforts; insurance, and so on.</p>
<p>Appendix A: Borel Spaces and Other Theoretical Issues (pp.257-266)</p>
<p>A.1. Main concepts: Definitions, theorems: Tychonoff, Urysohn, etc., metrizable compact spaces, Hilbert cube and so on.</p>
<p>A.2. Probability Measures on Borel Spaces: Definitions, the existence of measurable stochastic kernels, concepts on a metric space, etc. </p>
<p> A.3. Semicontinuous Functions and Measurable Selection: Basic definitions, properties of functions lower and upper semi-continuous and bounded below on metrizable and separable metrizable spaces. </p>
<p> A.4. Abelian (Tauberian) Theorem</p>
<p>Appendix B: Proofs of Auxiliary Statements (pp. 267-280)</p>
<p> In this appendix devoted to the Proofs of auxiliary statements on given the following: Lemmas 2.1 and 3.2., and the propositions 4, 1, 4.2 and 4.4. </p>
<p>Notation (pp. 281-282)</p>
<p>List of the Main Statements (pp. 283-284)</p>
<p>Bibliography (pp. 285-290). They are about 69 interesting references.</p>
<p>Index (pp. 291-293)</p>
<p>CONCLUSION</p>
<p>This book should be considered as a complement to scientific textbooks and monographs on MDP. The book can also serve as a reference book to which one can turn for answers to curiosities that arise which studying or teaching MDP. All the examples are self-contained and can be read independently of each other. When studying or using mathematical methods, the advanced student or researcher must understand what can happen if some of the conditions imposed in rigorous theorems are not satisfied. Many examples confirming the importance of such conditions were published in different journal articles which are often difficult to find.</p>
<p>In my opinion, this remarkable and intriguing book is high recommended. Some examples are aimed at undergraduate students, whilst others will be of interest to advanced undergraduates, graduates and research students in probability theory, optimal control and applied mathematics, looking for a better understanding of the theory; experts in Markov decision processes, professional or amateur researchers. Active researchers can refer to this book on applicability of mathematical methods and theorems.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Francisco José Cano Sevilla</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Profesor Universidad Complutense</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This excellent book provides approximately 100 examples, illustrating the theory of controlled discrete-time Markov processes. The main attention is paid to counter-intuitive, unexpected properties of optimization problems.The main goal of this intriguing book is to collect together such examples. Fascinating task! . The aim was to collect them together in one reference book which should be considered as a complement to existing monographs on Markov decision processes. The book is self-contained and unified in presentation.This book is high recommended. Some examples are aimed at undergraduate students, whilst others will be of interest to advanced undergraduates, graduates and research students in probability theory, optimal control and applied mathematics, looking for a better understanding of the theory; experts in Markov decision processes, professional or amateur researchers. In my opinion, this remarkable and intriguing book is high recommended. Some examples are aimed at undergraduate students, whilst others will be of interest to advanced undergraduates, graduates and research students in probability theory, optimal control and applied mathematics, looking for a better understanding of the theory; experts in Markov decision processes, professional or amateur researchers.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/b-piunovskiy" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. b. piunovskiy</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/london-imperial-college-press-optimization-series-vol-2-series-optimization-series-vol-2" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">london: imperial college press optimization series: vol. 2. series on optimization series vol 2.</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-84816-793-3 (Hardcover); ISBN 978-1908979-66-7(ebook); 978184816794-0 (ebook -institutions only); ISSN 2041-1677</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">From 41€ to 50€</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/www.icpress.co.uk" title="Link to web page">www.icpress.co.uk</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/90c40-90c39-6ojxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90c40, 90c39, 6ojxx</a></li></ul></span>Mon, 01 Apr 2013 09:36:01 +0000Anonymous45498 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/examples-markov-decision-processes#commentsIn Pursuit of the Traveling Salesman
https://euro-math-soc.eu/review/pursuit-traveling-salesman
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>There are many mathematical problems whose formulation is so easy that it can be understood by anybody, even without any mathematical background, but, however, are extremely difficult to solve. The book under review is focused on one of these problems, called the Traveling Salesman Problem (TSP). It consists on finding the shortest possible route for a traveling salesman to visit each of his customers exactly once and go back home. The formulation is simple, but after several decades of intense research, it still remains unsolved.</p>
<p>The book begins with an amusing description of the problem, focusing on its undeniable practical origins and presenting innumerable real applications. The complexity issues related to solving the TSP are also introduced, and then a historical tour through the main approaches to the problem is performed, revising the most important mathematical techniques used and highlighting the research lines joining different solution methods. The technical details are described with precision, but the inherent mathematical concepts are explained in an informal way so that readers without a deep mathematical background can also follow the story. </p>
<p>The book is full of examples, real applications and historical anecdotes, making it really enjoyable to read. At the end there is a section devoted to the relation of the TSP with several works of art, showing how the complexity of the problem is so attractive to researchers and artists and highlighting the beauty of its geometry and mathematical properties. The book finishes with a discussion on the future research on the problem, commenting on several possible lines and emphasizing that there is still a lot to be done.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Gregorio Tirado Domínguez</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book describes the history around one of the most important mathematical problems approached during the last century: the traveling salesman problem. It goes from the origins of the problem to some of the most sophisticated approaches, illustrating this trip with examples, real applications and historical anecdotes.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/william-j-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">william j. cook</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15270-7</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">27.95 dollars</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/90b06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90b06</a></li></ul></span>Wed, 21 Nov 2012 13:56:42 +0000Anonymous45471 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/pursuit-traveling-salesman#commentsIntroduction a l'optimisation différentiable
https://euro-math-soc.eu/review/introduction-loptimisation-diff%C3%A9rentiable
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Optimization theory on finite-dimensional linear spaces involving smooth (i.e. differentiable) cost functions and possibly also functional constraints is a fundamental discipline in solving various applied optimization problems with a finite number of design variables and a finite number of constraints. After introducing the general area and basic concepts in part 1, the textbook lucidly presents all major aspects of this theory. Part 2 deals with necessary and sufficient optimality conditions for problems without (or with) constraints, the Lagrange multiplier method, Karush-Kuhn-Tucker conditions, and sensibility analysis. Special attention is paid to linear and quadratic mathematical programming. Part 3 presents the Newton and quasi-Newton methods for solving systems of nonlinear equations. Part 4 then treats unconstrained optimization, starting by solving a quadratic case with the direct method or the conjugate-gradient method and continuing for a general case with the Newton method applied to optimality conditions, the descent method including line-search strategies, trust-region methods, and the quasi-Newton method with the Broyden-Fletcher-Goldfard-Shanno (BFGS) method. </p>
<p>Eventually, part 5 addresses constrained problems: the simplex method for linear programming, the Newton method with projected gradients, interior-point methods, the augmented-Lagrangian method and sequential quadratic programming. The textbook is equipped with many numerical examples illustrating the efficiency of the expounded methods and algorithms. It is also augmented with attractive biographical sketches and photos of mathematicians who have contributed substantially to the field. It ends with lists and appendices of definitions, theorems, examples and algorithms, and an index. All this makes it an excellent textbook for French-reading students and also for researchers interested in optimization theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">troub</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/m-bierlaire" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">m. bierlaire</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/presses-polytechniques-et-univ-romandes-lausanne-enseignement-des-math%C3%A9matiques" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">presses polytechniques et univ. romandes, lausanne: enseignement des mathématiques</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-88074-669-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 54.95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Sun, 23 Oct 2011 14:38:29 +0000Anonymous40067 at https://euro-math-soc.euThe Traveling Salesman Problem. A Computational Study
https://euro-math-soc.eu/review/traveling-salesman-problem-computational-study
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The traveling salesman problem is probably one of the best known problems in combinatorial optimization: given a collection of cities, find the shortest round-trip route that visits each of the cities exactly once. Though simple to state, finding an optimal solution is far from trivial. The authors of the book have led investigations of the problem for almost two decades and this book presents their findings. The main aim of the book is to explain the theory and algorithms utilized in a computer code (called Concorde), which has successfully solved a number of large scale instances of the problem. Having said that, it must be stressed that programming skills are not prerequisites for reading the book and that the book is written in a readable style. </p>
<p>It opens with an overview of the rich history of the problem, complemented with many illustrations. Chapter 2 contains a description of the main applications of the traveling salesman problem. The next two chapters survey development in the 50s on the traveling salesman problem by Dantzig, Fulkerson and Johnson and later by Grötschel and Padberg. Development on the problem was closely bound up with development in linear programming, and the cutting plane method for linear programming is at the core of chapters 3 and 4. In chapters 5-11, the authors describe their own techniques for finding cutting planes for the problem; these techniques form the core of the Concorde code. This part of the book requires a solid background in linear programming but reading it carefully will pay off. The subject of chapters 12 and 13 is how to manage and solve the large linear programs derived by the techniques described in previous chapters. Chapters 14 and 15 deal with enumeration schemes and with heuristics complementing the cutting plane method. Finally, chapter 16 provides results of the computational tests with Concorde and chapter 17 concludes the book with a sketch of possible future research directions. To summarize, the book provides a comprehensive treatment of the travelling salesman problem and I highly recommend it not only to specialists in the area but to anyone interested in combinatorial optimization.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">pkol</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/d-l-applegate" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. l. applegate</a></li><li class="vocabulary-links field-item odd"><a href="/author/r-e-bixby" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r. e. bixby</a></li><li class="vocabulary-links field-item even"><a href="/author/v-chv%C3%A1tal" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">v. chvátal</a></li><li class="vocabulary-links field-item odd"><a href="/author/w-j-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">w. j. cook</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press-princeton-princeton-series-applied-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton: princeton series in applied mathematics</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-12993-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 45</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Sat, 01 Oct 2011 09:09:14 +0000Anonymous39846 at https://euro-math-soc.euAlgorithms for Worst-Case Design and Applications to Risk Management
https://euro-math-soc.eu/review/algorithms-worst-case-design-and-applications-risk-management
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Risk management topics are treated in this book in a way that is different from what the reader might think of first. World-wide experience during the last decade shows, that it is perhaps more realistic to consider worst possible scenarios in the financial world than just to perform inference based on more optimistic criteria. The roots of the presented ideas and methods arise in part from the theory of games, in particular from the minimax principle. This seems to be the leitmotif. The title of the book does not really describe the content of the book and may be misleading. There is much more in the book than just algorithms. Most of the key concepts of financial mathematics are clearly explained and discussed in the text. Nevertheless, the fundamental idea of “worst-case” is thoroughly kept through the whole text. Let us describe just a sample from a broad area of topics covered in the book’s eleven chapters: computing of saddle points, numerical experiments with minimax algorithms, strategies for securities’ hedging, simulation studies, asset allocation problems, asset-liability management including immunization, and currency management. In the bonanza of books on mathematical finance and/or financial mathematics, the book under review is surely an exception. The presented ideas may be exploited both in theory and practice. Strongly recommended to anyone with interest in financial mathematics and keen to acquire non-standard pieces of knowledge of standard and other financial problems. The book is also recommended to practitioners.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jh</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/b-rustem" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">b. rustem</a></li><li class="vocabulary-links field-item odd"><a href="/author/m-howe" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">m. howe</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2002</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">ISBN 0-691-09154-4</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£52</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Fri, 17 Jun 2011 08:39:39 +0000Anonymous39605 at https://euro-math-soc.euA Java Library of Graph Algorithms and Optimization + CD
https://euro-math-soc.eu/review/java-library-graph-algorithms-and-optimization-cd
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The series ‘Discrete Mathematics and Its Applications’ has delivered an unusual fruit this time: a library of algorithms for various combinatorial problems programmed in Java. The book covers all the usual areas of algorithmic graph theory and optimization ranging from the random generation of graphs, connectivity, network flows and graph embedding, to linear and quadratic optimization, each area being discussed in a separate chapter. The chapters share a common structure and each of them lists a bunch of related problems. All problems are first described and then an outline of the algorithm is presented, followed by a full program. The program is accompanied by a description of inputs and outputs and a simple example of its use. The code of the programs is also available on a CD included in the book. </p>
<p>However, serious users of the book will notice several drawbacks: the description of the algorithms sometimes lacks important details such as the time complexity (which is surprising, especially in exponential algorithms for NP-complete problems) and the program code is often hard to understand because the aspiration to make all programs self-contained has led to avoiding all abstractions and repeating the same code patterns in many places. For some problems, faster and more easily implemented algorithms are already known. In short, it is a good book for anybody who needs to solve an isolated problem but more experienced programmers will probably still prefer the existing open-source program libraries like LEDA or Pigale.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">mmar</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/h-t-lau" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">h. t. lau</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-718-4</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 99,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Wed, 01 Jun 2011 16:45:05 +0000Anonymous39325 at https://euro-math-soc.euTraffic Flow on Networks
https://euro-math-soc.eu/review/traffic-flow-networks
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is devoted to a detailed exposition of the nowadays classical approach to modelling traffic flow by first order hyperbolic conservation laws. Networks can be thought of as oriented graphs, where the flow on adjacent edges is related by suitable conditions on the common vertices (= junctions). In this setting, one can model a more complicated traffic setup, including multilanes and junctions with lights or circles. After reviewing the basic theory of conservation laws, the book discusses the background of several models (chapter 3), in particular the Lighthill-Whitham model and the Aw-Rascle model. Some higher order models are mentioned as well. The core of the book (chapters 5-7) is devoted to a study of the dynamics of these models on networks. An interesting case study comparing the regulation using traffic lights with the traffic on circles is presented (chapter 8). The related problem of flow on telecommunications networks is addressed in chapter 9, while numerical results are discussed in the concluding chapter 10. The presentation is clear and readable, accompanied with a lot of graphs and diagrams as well as exercises concluding each chapter. The book is accessible with a mild background in partial differential equations.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">dpr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/b-piccoli" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">b. piccoli</a></li><li class="vocabulary-links field-item odd"><a href="/author/m-garavello" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">m. garavello</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-institute-mathematical-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american institute of mathematical sciences</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-60133-000-0 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 36.50</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Sat, 28 May 2011 09:15:13 +0000Anonymous39201 at https://euro-math-soc.euMathematics for Finance: An Introduction to Financial Engineering
https://euro-math-soc.eu/review/mathematics-finance-introduction-financial-engineering
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>1973 was a key year in the development of financial mathematics. The Chicago Board Options Exchange was founded and at the same time Fischer Black and Myron Scholes published the paper, “The Pricing of Options and Corporate Liabilities” [1], while Robert Merton published the article, “Theory of Rational Option Pricing” [2].</p>
<p>These works contained a new methodology for the valuation of derivatives (financial instruments whose payoffs depend upon the value of other instruments) and developed the famous Black-Scholes pricing formulae for calls and puts.</p>
<p>In 1997 the Royal Swedish Academy of Sciences awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (the Nobel Prize for Economics) to Merton and Scholes “for a new method to determine the value of derivatives” (Black had passed away some years before).</p>
<p>Black, Merton and Scholes’ framework rests upon two main pillars. The first is the modelling of stock prices with geometric Brownian motion (as Mandelbrojt and Nobel Prize winner Samuelson had done some years before). The second, and here lies their contribution, is a principle of equilibrium: the absence of arbitrage opportunities (and the corresponding replicating portfolio arguments). Nowadays, their arguments might sound natural, yet it is worth noting that Black and Scholes’ paper, which has become one of the most cited papers in the scientific literature of this field, was rejected by some prestigious economics journals for some years. </p>
<p>The history of derivatives is not as recent a topic as one might think. There are references to the use of these financial contracts (mainly futures) throughout history. It is said that Thales of Miletus made a fortune trading with the rights of use of oil mills. In the 17th century there was a speculative boom in the tulip futures market in Amsterdam. However, the first mathematical attempt to model the financial markets was made in 1900, in the doctoral thesis “Théorie de la Spéculation” by Louis Bachelier, one of Poincaré’s students. </p>
<p>Black, Merton and Scholes’ ideas and valuation formulae were immediately understood and accepted by the markets. This is an amazing example of vertiginous transference of knowledge between the academic world and the ‘real’ world; as they allowed risks to be translated into prices and different derivatives to be compared, the formulae actually worked!</p>
<p>Since then, the range of applications of stochastic processes and partial differential equations in finance (modelling and designing of financial instruments, pricing, hedging, risk management, etc.) has been increasing day by day. Thus, it has become a standard topic in undergraduate courses in mathematics and economics. The book, “Mathematics for Finance: An Introduction to Financial Engineering”, is a textbook for an introductory course on three basic questions: the non-arbitrage option pricing theory, the Markowitz portfolio theory and the modelling of interest rates.</p>
<p>The book<br />
The book is divided into eleven chapters. The first four serve as an initial approach to financial markets. The first chapter describes some of the basic hypotheses, illustrated with the one-step binomial model. A detailed account of riskless assets (time value of the money, bonds, etc.) is included in the second chapter. The third is devoted to the dynamics of risky assets, with particular emphasis on the binomial tree model, although the trinomial tree model is also analysed and a brief discussion on the continuous time limit is included as well. The fourth chapter ties together the previous concepts in the general framework of discrete models. Questions relating to portfolio management (Markowitz theory, efficient frontiers, CAPM) are treated in detail in the fifth chapter. Chapters six and seven are dedicated to explain some basic characteristics of derivatives such as futures, forward contracts and options.</p>
<p>Chapter eight deals with option pricing. The Cox-Ross-Rubinstein binomial model is used to price European and American options. A brief outline of the arguments that lead to the Black-Scholes formula is also included. Issues dealing with the use of derivatives in risk management are described in the ninth chapter: hedging, risk measures as value-at-risk (VaR), speculating strategies with options, etc. These questions are treated by making use of a case study approach. Finally, chapters ten and eleven are devoted to interest rates including term structure, the binomial model and a brief account of interest rates derivatives (swaps, caps, floors).</p>
<p>A textbook on financial mathematics like the one we are reviewing unavoidably faces a dilemma, namely the balance between mathematical rigor and financial concepts and their practical implementation. </p>
<p>In most chapters, Capiński and Zastawniak’s work succeeds in moving away from some advanced mathematical language, such as stochastic calculus and partial differential equations, and focuses on discrete time models, so that only basic notions of calculus, linear algebra and probability are needed to read it. At the same time, despite its mathematically formal prose, the text is very readable. The relevant concepts are introduced gradually; often these concepts are preceded by numerical examples and in the end they are all given their corresponding formal definitions. Almost all the main results are accompanied by corresponding proofs. A good deal of worked examples and remarks elucidate the associated financial concepts. Each chapter contains a collection of useful exercises (interspersed in the text). Some of the quite detailed solutions are included in an appendix. The reader will find other solutions (in Excel files) on the web page www-users.york.ac.uk/~tz506/m4f/index.html (see also <a href="http://www.springeronline.com/1-85233-330-8">www.springeronline.com/1-85233-330-8</a>). Some typos are listed on the aforementioned web page.</p>
<p>Thus, we can say that the book strikes a balance between mathematical rigor and practical application and we recommend it as a textbook for an introductory course on financial mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Pablo Fernández Gallardo (Madrid, Spain)</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/marek-capi%C5%84ski" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">marek capiński</a></li><li class="vocabulary-links field-item odd"><a href="/author/tomasz-zastawniak" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">tomasz zastawniak</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-undergraduate-mathematical-series" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer undergraduate mathematical series</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2003</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-85233-330-8</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/90-operations-research-mathematical-programming" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90 Operations research, mathematical programming</a></li></ul></span>Sun, 08 May 2011 15:19:59 +0000Anonymous39051 at https://euro-math-soc.eu