Theorem

theorem

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem

Table of Links Abstract and 1 Introduction 2 Preliminaries 3 Temporal Hierarchies 4 Rating Maps 5 Optimal Imprints for TL(AT) 6 Conclusion and References Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 5. Optimal imprints for TL(AT) This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Thomas Place;
(2) Marc Zaitoun. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 3 Temporal Hierarchies 3 Temporal Hierarchies 4 Rating Maps 4 Rating Maps 5 Optimal Imprints for TL(AT) 5 Optimal Imprints for TL(AT) 6 Conclusion and References 6 Conclusion and References Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 5. Optimal imprints for TL(AT) This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Thomas Place; (2) Marc Zaitoun. Authors: Authors: (1) Thomas Place; (2) Marc Zaitoun.

Optimal Imprints for TL(AT): Explaining Our Main Theorem

Author:
(1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Table of Links 1 Introduction 2 Dancing Links 3 The Rust Programming Language 4 RAC: Hardware/Software Co-Assurance at Scale 5 Rust and RAR 5.1 Restricted Algorithmic Rust 6 Dancing Links in Rust and 6.1 Definitions 6.2 Translation to ACL2 6.3 Dancing Links Theorems 7 Related Work 8 Conclusion 9 Acknowledgments and References 6.3 Dancing Links Theorems Once we have translated the circular doubly-linked list functions into ACL2, we can begin to prove theorems about the data structure implementation. We begin by defining a “well-formedness” predicate for CDLLs. Given this definition of a good CDLL state, we can prove functional correctness theorems for Dancing Links operations, of the sort stated below. Note that this proof requires some detailed well-formedness hypotheses related to the prev and next indices for the nth element: ACL2 performs the correctness proof for this cdll_restore of cdll_remove theorem automatically. In addition to the Dancing Links operator proofs, we have proved approximately 160 theorems related to the CDLL data structure, including theorems about cdll_cns() (cons equivalent), cdll_- rst() (cdr equivalent), cdll_snc() (add to end of data structure), cdll_tsr() (delete from end of data structure), cdll_nth(), etc. All of these proofs will be made publicly available in the ACL2 workshop books repository. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Author: Author: (1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Table of Links 1 Introduction 1 Introduction 2 Dancing Links 2 Dancing Links 3 The Rust Programming Language 3 The Rust Programming Language 4 RAC: Hardware/Software Co-Assurance at Scale 4 RAC: Hardware/Software Co-Assurance at Scale 5 Rust and RAR 5 Rust and RAR 5.1 Restricted Algorithmic Rust 5.1 Restricted Algorithmic Rust 6 Dancing Links in Rust and 6.1 Definitions 6 Dancing Links in Rust and 6.1 Definitions 6.2 Translation to ACL2 6.2 Translation to ACL2 6.3 Dancing Links Theorems 6.3 Dancing Links Theorems 7 Related Work 7 Related Work 8 Conclusion 8 Conclusion 9 Acknowledgments and References 9 Acknowledgments and References 6.3 Dancing Links Theorems Once we have translated the circular doubly-linked list functions into ACL2, we can begin to prove theorems about the data structure implementation. We begin by defining a “well-formedness” predicate for CDLLs. Given this definition of a good CDLL state, we can prove functional correctness theorems for Dancing Links operations, of the sort stated below. Note that this proof requires some detailed well-formedness hypotheses related to the prev and next indices for the nth element: ACL2 performs the correctness proof for this cdll_restore of cdll_remove theorem automatically. In addition to the Dancing Links operator proofs, we have proved approximately 160 theorems related to the CDLL data structure, including theorems about cdll_cns() (cons equivalent), cdll_- rst() (cdr equivalent), cdll_snc() (add to end of data structure), cdll_tsr() (delete from end of data structure), cdll_nth(), etc. All of these proofs will be made publicly available in the ACL2 workshop books repository. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Proving Theorems for Rust’s Dancing Links in ACL2

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility E.4 Rouche’s Theorem Conditions This part augments Proposition 17 with more cases derived from Rouche’s theorem. Proposition 31. Any of the following conditions would prevent the existence of a recursive equilibrium. Proof. Following Proposition 17, it is sufficient to note that the cases (I), (II), (III) and (IV) correspond respectively to the conditions guaranteeing zero, one, three and four roots outside the unit circle. If price dispersion were jettisoned the system would be governed by (163), in which case the sufficient condition would be whilst the following would imply non-existence This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility E.4 Rouche’s Theorem Conditions This part augments Proposition 17 with more cases derived from Rouche’s theorem. Proposition 31. Any of the following conditions would prevent the existence of a recursive equilibrium. Proposition 31. Proof. Following Proposition 17, it is sufficient to note that the cases (I), (II), (III) and (IV) correspond respectively to the conditions guaranteeing zero, one, three and four roots outside the unit circle. If price dispersion were jettisoned the system would be governed by (163), in which case the sufficient condition would be whilst the following would imply non-existence This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Rouche’s Theorem: Conditions for Preventing Recursive Equilibrium

Table of Links Abstract and 1. Introduction 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 3.2 THEOREMS AND MODEL DISCUSSION The three steps we take are outlined in Theorems 1 through 3. We begin by deriving an equation for the hedge error variance in the case when each call position in the portfolio has a standard delta hedge applied to it [22]. Following that, To protect against changes in the value of the option portfolio, we create an analogous formula for the scenario in which an arbitrary portfolio in the underlying values is retained [23]. As a result, we are able to show how selecting the right hedge portfolio helps lower hedge error variances. We last provide our major theoretical finding, which demonstrates that by focusing only on the linear and higher order exposures to the systematic risk factor [24], it becomes possible to design a perfect static hedge portfolio in finite time, provided that the portfolio size grows infinitely [25]. Hedge error is defined by ∆𝐻 as The hedge portfolio is the total of the hedge portfolios for each individual position when using the hedging approach described above [26]. We create a power series by expanding the hedge mistake (14) [27]. Regarding the duration of the hedging period ∆𝑡, See Leland (1985) as well as Mello and Neuhaus (1998). The predicted hedging error and its variation under the current conventional delta hedging method are as follows [28]. Theorem 1: The hedging error ∆𝐻 in (14), which is obtained through delta hedging of the individual option holdings, satisfies Essentially, for N = 1, we obtain the square of the option's gamma, which is the well-known calculation for the hedge error variance [3]. This variance's explicit representation clearly indicates that the option portfolio's return is no longer reproduced riskfree in a defined amount of time using the Black-Scholes hedging strategy. Thus, it appears that the option portfolio does not have a unique preference free pricing. With this unconventional approach to hedging, we get the following outcome for the hedge error variance. There is a trade-off between insuring all idiosyncratic risk (in the limit 𝑛∆𝑡 → 0 , the traditional Black-Scholes technique) and hedging market risk exclusively (in the limit 𝑛∆𝑡 →∞) for finite portfolio size 𝑛 and finite revision time ∆𝑡. Clearly, with bigger portfolio sizes and revision intervals, there are more departures from the traditional hedging technique in terms of hedge ratios and variance reduction. Now that we have established the general conclusion, we may argue that it is better to hedge higher order systematic risk rather than linear idiosyncratic risk in big option portfolios. The subsequent theorem does this. One can demonstrate a similar outcome if idiosyncratic risk is valued [3]. In that scenario, the quantity of securities required to build the hedging portfolio grows quadratically with 𝑎. Furthermore, there are now prospects for arbitrage since the projected return of this hedging approach need not be zero. This is to be expected because arbitrage is possible in a market structure that doesn't even have options [24]. Theorem 3's result shows that if the 𝜑𝑖 s are different, we can create a riskless hedging strategy for finite time ∆𝑡 in the limit 𝑛 →∞. Stated otherwise, the risk incurred by switching to a discrete time setup entirely disappears: by selecting the suitable non-standard hedging approach, the systematic risk component can be eliminated to any arbitrary order of ∆𝑡. Diversification causes the idiosyncratic risk component to vanish at the same time. The hedge portfolio is selected so that, up to a high enough order in ∆t, its expectation conditional on the systematic risk component corresponds with its unconditional expectation. This is the key to the demonstration. By matching the higher order properties of the systematic risk exposure, this may be proven. The portfolio loadings 𝜓 𝑖 must be subject to a series of restrictions in order to meet the two sorts of expectations. In the 𝜓 𝑖 s, each of these restrictions is linear. The coefficients of these constraints entail powers of the systematic volatility 𝜑𝑖 and higher order derivatives of the Black-Scholes price (to capture the proper curvature). The number of securities, 𝑛 ≥ 𝑎, must be sufficiently large in order for this set of constraints to have a solution. Secondly, the limitations system must not be unique. Requiring that at least 𝑎 of the 𝜑𝑖 s be different and not equal to zero ensures the latter. In the current discrete time framework, one can question if the Black-Scholes pricing remain accurate, given the stark differences between the behavior of the conventional and portfolio hedging approaches. They are, according to the next corollary. Corollary 1 (the price is right): If only market risk is priced and the number of securities diverges (𝑛 →∞), and if the 𝜑𝑖 s are different such that we can set the approximation order n arbitrarily high (𝑎 →∞), then the only arbitrage-free price of the options is equal to their Black-Scholes price, except for a set of measure zero. According to Corollary 1, arbitrage opportunities are those in which there is an almost certain chance of earning a return greater than the risk-free rate [3]. The hedge portfolio is priced the same as Black-Scholes by construction. This suggests that the total of the Black-Scholes prices and the options prices are equal, based on the outcome of Theorem 3. However, Theorem 3 also holds for all call option subseries and their underlying values, meaning that the set of options with prices that deviate from the Black-Scholes price has measured zero. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 3.2 THEOREMS AND MODEL DISCUSSION The three steps we take are outlined in Theorems 1 through 3. We begin by deriving an equation for the hedge error variance in the case when each call position in the portfolio has a standard delta hedge applied to it [22]. Following that, To protect against changes in the value of the option portfolio, we create an analogous formula for the scenario in which an arbitrary portfolio in the underlying values is retained [23]. As a result, we are able to show how selecting the right hedge portfolio helps lower hedge error variances. We last provide our major theoretical finding, which demonstrates that by focusing only on the linear and higher order exposures to the systematic risk factor [24], it becomes possible to design a perfect static hedge portfolio in finite time, provided that the portfolio size grows infinitely [25]. Hedge error is defined by ∆𝐻 as The hedge portfolio is the total of the hedge portfolios for each individual position when using the hedging approach described above [26]. We create a power series by expanding the hedge mistake (14) [27]. Regarding the duration of the hedging period ∆𝑡, See Leland (1985) as well as Mello and Neuhaus (1998). The predicted hedging error and its variation under the current conventional delta hedging method are as follows [28]. Theorem 1: The hedging error ∆𝐻 in (14), which is obtained through delta hedging of the individual option holdings, satisfies Theorem 1: Theorem 1: The hedging error ∆𝐻 in (14), which is obtained through delta hedging of the individual option holdings, satisfies Essentially, for N = 1, we obtain the square of the option's gamma, which is the well-known calculation for the hedge error variance [3]. This variance's explicit representation clearly indicates that the option portfolio's return is no longer reproduced riskfree in a defined amount of time using the Black-Scholes hedging strategy. Thus, it appears that the option portfolio does not have a unique preference free pricing. With this unconventional approach to hedging, we get the following outcome for the hedge error variance. There is a trade-off between insuring all idiosyncratic risk (in the limit 𝑛∆𝑡 → 0 , the traditional Black-Scholes technique) and hedging market risk exclusively (in the limit 𝑛∆𝑡 →∞) for finite portfolio size 𝑛 and finite revision time ∆𝑡. Clearly, with bigger portfolio sizes and revision intervals, there are more departures from the traditional hedging technique in terms of hedge ratios and variance reduction. Now that we have established the general conclusion, we may argue that it is better to hedge higher order systematic risk rather than linear idiosyncratic risk in big option portfolios. The subsequent theorem does this. One can demonstrate a similar outcome if idiosyncratic risk is valued [3]. In that scenario, the quantity of securities required to build the hedging portfolio grows quadratically with 𝑎. Furthermore, there are now prospects for arbitrage since the projected return of this hedging approach need not be zero. This is to be expected because arbitrage is possible in a market structure that doesn't even have options [24]. Theorem 3's result shows that if the 𝜑𝑖 s are different, we can create a riskless hedging strategy for finite time ∆𝑡 in the limit 𝑛 →∞. Stated otherwise, the risk incurred by switching to a discrete time setup entirely disappears: by selecting the suitable non-standard hedging approach, the systematic risk component can be eliminated to any arbitrary order of ∆𝑡. Diversification causes the idiosyncratic risk component to vanish at the same time. The hedge portfolio is selected so that, up to a high enough order in ∆t, its expectation conditional on the systematic risk component corresponds with its unconditional expectation. This is the key to the demonstration. By matching the higher order properties of the systematic risk exposure, this may be proven. The portfolio loadings 𝜓 𝑖 must be subject to a series of restrictions in order to meet the two sorts of expectations. In the 𝜓 𝑖 s, each of these restrictions is linear. The coefficients of these constraints entail powers of the systematic volatility 𝜑𝑖 and higher order derivatives of the Black-Scholes price (to capture the proper curvature). The number of securities, 𝑛 ≥ 𝑎, must be sufficiently large in order for this set of constraints to have a solution. Secondly, the limitations system must not be unique. Requiring that at least 𝑎 of the 𝜑𝑖 s be different and not equal to zero ensures the latter. In the current discrete time framework, one can question if the Black-Scholes pricing remain accurate, given the stark differences between the behavior of the conventional and portfolio hedging approaches. They are, according to the next corollary. Corollary 1 (the price is right): If only market risk is priced and the number of securities diverges (𝑛 →∞), and if the 𝜑𝑖 s are different such that we can set the approximation order n arbitrarily high (𝑎 →∞), then the only arbitrage-free price of the options is equal to their Black-Scholes price, except for a set of measure zero. Corollary 1 (the price is right): Corollary 1 (the price is right): If only market risk is priced and the number of securities diverges (𝑛 →∞), and if the 𝜑𝑖 s are different such that we can set the approximation order n arbitrarily high (𝑎 →∞), then the only arbitrage-free price of the options is equal to their Black-Scholes price, except for a set of measure zero. According to Corollary 1, arbitrage opportunities are those in which there is an almost certain chance of earning a return greater than the risk-free rate [3]. The hedge portfolio is priced the same as Black-Scholes by construction. This suggests that the total of the Black-Scholes prices and the options prices are equal, based on the outcome of Theorem 3. However, Theorem 3 also holds for all call option subseries and their underlying values, meaning that the set of options with prices that deviate from the Black-Scholes price has measured zero. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Theorems on Error Variance in the Black-Scholes Framework

Authors:
(1) Edward Crane, School of Mathematics, University of Bristol, BS8 1TH, UK;
(2) Stanislav Volkov, Centre for Mathematical Sciences, Lund University, Box 118 SE-22100, Lund, Sweden. Table of Links Abstract and Introduction Preliminaries Reduction to the case of uniform geometry All original points are eventually removed, a. s. Proof of Theorem 1 Coupling Y (⋅) and Z(⋅) Acknowledgements and References Appendix 5 Proof of Theorem 1 Without loss of generality, we assume that the initial state Z(0) is deterministic. This is harmless since if for each deterministic choice of Z(0) the limit z∞ exists a.s. and has an absolutely continuous distribution, then if instead Z(0) is random, z∞ still exists a.s. and its distribution is a mixture of absolutely continuous distributions, which is necessarily absolutely continuous. This paper is available on arxiv under CC 4.0 license. 7 i.e., a set defined by a number of polynomial inequalities and equalities; in our case, a.s. these will be just inequalities. Authors: (1) Edward Crane, School of Mathematics, University of Bristol, BS8 1TH, UK; (2) Stanislav Volkov, Centre for Mathematical Sciences, Lund University, Box 118 SE-22100, Lund, Sweden. Authors: Authors: (1) Edward Crane, School of Mathematics, University of Bristol, BS8 1TH, UK; (2) Stanislav Volkov, Centre for Mathematical Sciences, Lund University, Box 118 SE-22100, Lund, Sweden. Table of Links Abstract and Introduction Abstract and Introduction Preliminaries Preliminaries Reduction to the case of uniform geometry Reduction to the case of uniform geometry All original points are eventually removed, a. s. All original points are eventually removed, a. s. Proof of Theorem 1 Proof of Theorem 1 Coupling Y (⋅) and Z(⋅) Coupling Y (⋅) and Z(⋅) Acknowledgements and References Acknowledgements and References Appendix Appendix 5 Proof of Theorem 1 Without loss of generality, we assume that the initial state Z(0) is deterministic. This is harmless since if for each deterministic choice of Z(0) the limit z∞ exists a.s. and has an absolutely continuous distribution, then if instead Z(0) is random, z∞ still exists a.s. and its distribution is a mixture of absolutely continuous distributions, which is necessarily absolutely continuous. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv 7 i.e., a set defined by a number of polynomial inequalities and equalities; in our case, a.s. these will be just inequalities.

Analysis of the Jante’s Law Process and Proof of Conjecture: Proof of Theorem 1

Author:
(1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 2. Proof of Results In the proof of both theorems, we use induction. using properties of Pascal triangle, we get The statement for k + 1 is also true, and the proof is completed. using properties of the trinomial coefficients, we get The statement for k + 1 is also true, and the proof is completed. The hockey stick theorem in the trinomial triangles has been proved. This theorem can be translated in Pascal pyramid as follows : Other similar theorems might be obtained for Pascal’s four dimensional and even n-dimensional pyramid. References 1] G. Andrews, Euler’s ’Exemplum Memorabile Inductionis Fallacis’ and Trinomial Coefficients J. Amer. Math. Soc. 3 (1990), 653-669. [2] P. Hilton and J. Pedersen, Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry Mathematics Magazine, Vol. 60, No. 5 (Dec., 1987), 305-316. [3] Eric W.Weisstein, Trinomial Coefficient From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html [4] Eric W.Weisstein, Trinomial Triangle From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) Sima Mehri, Farzanegan High School. Author: Author: (1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 2. Proof of Results and References 2. Proof of Results In the proof of both theorems, we use induction. using properties of Pascal triangle, we get The statement for k + 1 is also true, and the proof is completed. using properties of the trinomial coefficients, we get The statement for k + 1 is also true, and the proof is completed. k The hockey stick theorem in the trinomial triangles has been proved. This theorem can be translated in Pascal pyramid as follows : Other similar theorems might be obtained for Pascal’s four dimensional and even n-dimensional pyramid. References 1] G. Andrews, Euler’s ’Exemplum Memorabile Inductionis Fallacis’ and Trinomial Coefficients J. Amer. Math. Soc. 3 (1990), 653-669. [2] P. Hilton and J. Pedersen, Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry Mathematics Magazine, Vol. 60, No. 5 (Dec., 1987), 305-316. [3] Eric W.Weisstein, Trinomial Coefficient From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html [4] Eric W.Weisstein, Trinomial Triangle From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring Hockey Stick Theorems: Proof of Results and References

Author:
(1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results 2. Proof of Results and References Abstract There are some theorems in the Pascal’s triangle which their figures resemble to shoot a ball by hockey stick, so they are called hockey stick theorems. P. Hilton and J. Pedersen, in the article ”Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry”,[2], have stated the little and big hockey stick and puck theorems in the Pascal’s triangle. The big hockey stick theorem is a special case of a general theorem which our goal is to introduce it. We state a hockey stick theorem in the trinomial triangle too. 1 Introduction and Description of Results The big hockey stick and puck theorem, stated in [2] is: Theorem 1.1. [2] (The Big Hockey Stick and Puck Theorem) In [2], this theorem is also demonstrated by Figure 1. We have found the general form of above theorem in Pascal triangle as below. Theorem 1.2. (The Hockey Stick Theorem in Pascal Triangle) An example of this theorem is illustrated in Figure 2. Now we wish to state the hockey stick theorem in trinomial triangle. First using [3] and [4], we explain what is the trinomial triangle. Equivalently, the trinomial coefficients are defined by We have proven the following theorem in this triangle: Theorem 1.3. (The Hockey Stick Theorem in The Trinomial Triangle) For Example see Figure 4. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) Sima Mehri, Farzanegan High School. Author: Author: (1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 2. Proof of Results and References Abstract There are some theorems in the Pascal’s triangle which their figures resemble to shoot a ball by hockey stick, so they are called hockey stick theorems. P. Hilton and J. Pedersen, in the article ”Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry”,[2], have stated the little and big hockey stick and puck theorems in the Pascal’s triangle. The big hockey stick theorem is a special case of a general theorem which our goal is to introduce it. We state a hockey stick theorem in the trinomial triangle too. 1 Introduction and Description of Results The big hockey stick and puck theorem, stated in [2] is: Theorem 1.1. [2] (The Big Hockey Stick and Puck Theorem) Theorem 1.1. [2] (The Big Hockey Stick and Puck Theorem) In [2], this theorem is also demonstrated by Figure 1. We have found the general form of above theorem in Pascal triangle as below. Theorem 1.2. (The Hockey Stick Theorem in Pascal Triangle) Theorem 1.2. (The Hockey Stick Theorem in Pascal Triangle) An example of this theorem is illustrated in Figure 2. Now we wish to state the hockey stick theorem in trinomial triangle. First using [3] and [4], we explain what is the trinomial triangle. Equivalently, the trinomial coefficients are defined by We have proven the following theorem in this triangle: Theorem 1.3. (The Hockey Stick Theorem in The Trinomial Triangle) Theorem 1.3. (The Hockey Stick Theorem in The Trinomial Triangle) For Example see Figure 4. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring Hockey Stick Theorems: Abstract, Introduction and Description of Results

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)-type graphs. The contribution of the (α, 2)-type graphs can be computed as follows. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)- type graphs . The contribution of the (α, 2)-type graphs can be computed as follows. Contribution of the type graphs This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Lagrangian Cones of Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের ল্যাগ্রাঞ্জিয়ান শঙ্কু

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Lagrangesche Kegel torischer Bündel

Un teorema del espejo para haces tóricos no divididos: conos lagrangianos de haces tóricos

Un théorème du miroir pour les faisceaux toriques non divisés : les cônes lagrangiens des faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लैग्रेंजियन शंकु

非分割トーリックバンドルのミラー定理: トーリックバンドルのラグランジアン円錐

비분할 토릭 번들에 대한 거울 정리: 토릭 번들의 라그랑주 원뿔

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Cones Lagrangianos de Pacotes Tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: лагранжевы конусы торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetlerin Lagrange Konileri

Định lý gương cho các bó Toric không phân chia: Nón Lagrange của các bó Toric

非分裂环面束的镜像定理：环面束的拉格朗日锥

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants. We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants . We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. Gromov-Witten invariant and its variants 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. Quantum Riemann-Roch theorem and twisted theory. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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