Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal Advanced mathematical methods for finance, nunno oksendal
Advanced Mathematical Methods for Finance Giulia Di Nunno Bernt Øksendal Editors Advanced Mathematical Methods for Finance Editors Giulia Di Nunno Bernt Øksendal CMA, Department of Mathematics University of Oslo P.O Box 1053, Blindern 0316 Oslo, Norway and Norwegian School of Economics and Business Administration Helleveien 30 5045 Bergen, Norway giulian@math.uio.no oksendal@math.uio.no ISBN 978-3-642-18411-6 e-ISBN 978-3-642-18412-3 DOI 10.1007/978-3-642-18412-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011925381 Mathematics Subject Classification (2010): 91Gxx, 91G10, 91G20, 91G40, 91G70, 91G80, 91B16, 91B30, 91B70, 93E11, 93E20, 60E15, 60G15, 60G22, 60G40, 60G44, 60G51, 60G57, 60G60, 60H05, 60H07, 60H10, 60H15, 60H20, 60H30, 60H40, 60J65, 60K15, 62G07, 62G08, 62M07, 62P20, 41A25, 46B70, 94Axx, 35F20, 35Q35 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The title of this volume “Advanced Mathematical Methods for Finance,” AMaMeF for short, originates from the European network of the European Science Foundation with the same name that started its activity in 2005 The goals of its program have been the development and the use of advanced mathematical tools for finance, from theory to practice This book was born in the same spirit of the program It presents innovations in the mathematical methods in various research areas representing the broad spectrum of AMaMeF itself It covers the mathematical foundations of financial analysis, numerical methods, and the modeling of risk The topics selected include measures of risk, credit contagion, insider trading, information in finance, stochastic control and its applications to portfolio choices and liquidation, models of liquidity, pricing, and hedging The models presented are based on the use of Brownian motion, Lévy processes and jump diffusions Moreover, fractional Brownian motion and ambit processes are also introduced at various levels The chosen blending of topics gives a large view of the up-to-date frontiers of the mathematics for finance This volume represents the joint work of European experts in the various fields and linked to the program AMaMeF After five years of activity, AMaMeF has reached many of its goals, among which the creation and enhancement of the relationships among European research teams in the sixteen participating countries: Austria, Belgium, Denmark, Finland, France, Germany, Italy, The Netherlands, Norway, Poland, Romania, Slovenia, Sweden, Switzerland, Turkey, and United Kingdom We are grateful to all the researchers and practitioners in the financial industry for their valuable input to the program and for having participated to the proposed activities, either conferences, or workshops, or exchange research visits these may have been We are also grateful to Carole Mabrouk for her administrative assistance It was an honor to be chairing this program during these years and to have worked together with an engaged team as the AMaMeF Steering Committee, whose members, in addition to ourselves, have been (in alphabetic order): Ole BarndorffNielsen, Tomas Björk, Vasili Brinzanescu, Mark Davis, Arnoldo Frigessi, Lane Hughston, Hayri Körezlioglu, Claudia Klüppelberg, Damien Lamberton, Marco v vi Preface Papi, Benedetto Piccoli, Uwe Schmock, Christoph Schwab, Mete Soner, Peter Spreij, Lukasz Stettner, Johan Tysk, Esko Valkeila, and Michèle Vanmaele We thank them all for the important work done together and the cooperative and friendly atmosphere Oslo 30th August 2010 Giulia Di Nunno Bernt Øksendal Contents Dynamic Risk Measures Beatrice Acciaio and Irina Penner Ambit Processes and Stochastic Partial Differential Equations Ole E Barndorff-Nielsen, Fred Espen Benth, and Almut E.D Veraart 35 Fractional Processes as Models in Stochastic Finance Christian Bender, Tommi Sottinen, and Esko Valkeila 75 Credit Contagion in a Long Range Dependent Macroeconomic Factor Model 105 Francesca Biagini, Serena Fuschini, and Claudia Klüppelberg Modelling Information Flows in Financial Markets 133 Dorje C Brody, Lane P Hughston, and Andrea Macrina An Overview of Comonotonicity and Its Applications in Finance and Insurance 155 Griselda Deelstra, Jan Dhaene, and Michèle Vanmaele A General Maximum Principle for Anticipative Stochastic Control and Applications to Insider Trading 181 Giulia Di Nunno, Olivier Menoukeu Pamen, Bernt Øksendal, and Frank Proske Analyticity of the Wiener–Hopf Factors and Valuation of Exotic Options in Lévy Models 223 Ernst Eberlein, Kathrin Glau, and Antonis Papapantoleon Optimal Liquidation of a Pairs Trade 247 Erik Ekström, Carl Lindberg, and Johan Tysk 10 A PDE-Based Approach for Pricing Mortgage-Backed Securities 257 Marco Papi and Maya Briani vii viii Contents 11 Nonparametric Methods for Volatility Density Estimation 293 Bert van Es, Peter Spreij, and Harry van Zanten 12 Fractional Smoothness and Applications in Finance 313 Stefan Geiss and Emmanuel Gobet 13 Liquidity Models in Continuous and Discrete Time 333 Selim Gökay, Alexandre F Roch, and H Mete Soner 14 Some New BSDE Results for an Infinite-Horizon Stochastic Control Problem 367 Ying Hu and Martin Schweizer 15 Functionals Associated with Gradient Stochastic Flows and Nonlinear SPDEs 397 B Iftimie, M Marinescu, and C Vârsan 16 Pricing and Hedging of Rating-Sensitive Claims Modeled by F-doubly Stochastic Markov Chains 417 Jacek Jakubowski and Mariusz Niew˛egłowski 17 Exotic Derivatives under Stochastic Volatility Models with Jumps 455 Aleksandar Mijatovi´c and Martijn Pistorius 18 Asymptotics of HARA Utility from Terminal Wealth under Proportional Transaction Costs with Decision Lag or Execution Delay and Obligatory Diversification 509 Lukasz Stettner 522 L Stettner a discounted discrete-time problem Assume that decisions are made in discrete times {0, Δ, 2Δ, , iΔ, } The decision is executed with a delay h, which is a multiplicity of Δ Since delay h is fixed, we consider only such Δ for which its multiplicity forms h Denote by TΔ the family of stopping times with values in {0, Δ, 2Δ, , iΔ, } Discrete-time discounted problem is of the form: find a bounded function gαΔ : S × D × [0, Γ ] → R such that gαΔ (π, x, 0) ≡ and gαΔ (π, z, γ ) = 1S c (π)Mhr,α gαΔ (π, z, γ ) + 1S (π) max ln Eπ,z exp γ e−αΔ δ δ × ln π · e X(Δ) + gαΔ π(Δ), zΔ , γ e−αΔ , Mhr,α gαΔ (π, z, γ ) (18.48) with Mhr,α g(π, z, γ ) = sup ln Eπ,z exp γ e−αh ln π · eX(h) π ∈Sδ + γ e−αh ln e π(h), π + g π , zh , γ e−αh (18.49) An equivalent form of (18.48) is gαΔ (π, z, γ ) δ0 τ ∧TΔ Δ = sup ln Eπ,z exp γ τ ∈T Δ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 δ0 + Mhr,α gαΔ π τ ∧ TΔδ , zτ ∧T δ0 , γ e−ατ ∧TΔ (18.50) Δ with TΔδ = inf iΔ : π(iΔ) ∈ S \ Sδ0 (18.51) We have the following: Proposition 18.13 There is a unique bounded function gαΔ : S × D × [0, Γ ] → R such that gαΔ (π, z, 0) ≡ 0, which is continuous in Sδ0 , and for which (18.48) and, equivalently, (18.50) are satisfied Moreover, gαΔ (π, z, γ ) τ1 Δ = sup ln Eπ,z exp γ V i=1 e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) 18 Asymptotics of HARA Utility τj Δ ∞ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) +γ j =2 ∞ +γ i= 523 τj −1 +h +1 Δ e−α(τi +h) ln π(τi ) · eX(τi +h)−X(τi ) + ln e π − (τi + h), π i i=1 (18.52) Proof We first prove that the operator T defined by right-hand side of (18.48) as T g(π, z, γ ) = 1S c (π)Mhr,α g(π, z, γ ) + 1S (π) max ln Eπ,z exp γ e−αΔ δ δ × ln π · e X(Δ) + g π(Δ), zΔ , γ e−αΔ , Mhr,α g(π, z, γ ) transforms the class of bounded functions continuous in Sδ0 into itself Note first that by Proposition 18.5 the operator Mhr,α transforms bounded functions into bounded continuous functions By the uniform integrability of the term δ0 −α(τ ∧T δ +h) (π · eX(τ ∧T ) )γ e (notice that Γ ≤ and we have assumed (18.3)) it suffices to show the continuity of δ0 Gα (π, z, W, γ ) = sup ln Eπ,z exp γ e−ατ ∧T f (Wτ ∧T δ0 ) τ + g π τ ∧ T δ , zτ ∧T δ0 , γ e−ατ ∧T δ0 for continuous bounded functions f and g Now, similarly as in the proof of Lemma 18.6, we can use a penalty method (for time-dependent functions of π , z, and W studied in [4] and [13]) and obtain by (18.15) and (18.16) the continuity of the function Gα Consequently, for a bounded function g, the function T g is also a continuous bounded function in Sδ0 , and T g(π, z, 0) ≡ 0, whenever g(π, z, 0) ≡ Moreover by the compactness of S × D we have the uniform convergence of g(π, z, γ ) to as γ → Therefore, iterations of the operator T converge to a function gα , which is a solution to (18.48) Since the term g diminishes in the successive iterations of T , the limit does not depend on the function g, provided that g is continuous bounded in Sδ0 and g(π, z, 0) ≡ The form of (18.50) follows by iteration from (18.48) noticing that ln π · eX(τ ∧T δ0 ) + ln π(τ ) · eX(τ ∧T δ +h)−X(τ ∧T δ ) = ln π · eX(τ ∧T δ +h) In what follows wee need the following lemma Lemma 18.14 We have sup sup sup π,π ∈Sδ ,π ∈Sδ z,z ,z ∈D 0≤γ ≤Γ := L < ∞ Eπ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )}1z (zh )] Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )}1z (zh )] (18.53) 524 L Stettner Proof Suppose that L = ∞ Then there are sequences (π(n) ), (π(n) ), (π(n) ), γn → and elements z, z , z of D (since D is finite) such that Eπ(n) ,z [exp{γn e−αh ln(π(n) · eX(h) e(π(h), π(n) )}1z (zh )] Eπ (n) ,z [exp{γn e−αh ln(π(n) · eX(h) e(π(h), π(n) )}1z (zh )] → ∞ Since the numerator and denominator are bounded from above, it may happen only when Eπ (n) ,z exp γn e−αh ln(π(n) · eX(h) e π(h), π(n) 1z (zh ) → as n → ∞ Therefore, Ez [1z (zh )] = By the form of the transition density (18.12), Ez [1z (zh )] > for any z ∈ D, and we have a contradiction The following two estimations play crucial role in the vanishing discount approach in the next section Proposition 18.15 There is a constant L < ∞ such that, for π, π ∈ S, z, z ∈ D, and γ ≤ Γ , we have gαΔ (π, z, γ ) − gαΔ (π , z , γ ) ≤ L (18.54) and, for γ1 ≤ γ2 , sup gαΔ (π, z, γ1 ) − gαΔ (π, z, γ2 ) ≤ L π∈Sδ γ2 − γ1 − e−αh (18.55) Proof Notice first that gαΔ (π, z, γ ) − gαΔ (π , z , γ ) ≤ gαΔ (π, z, γ ) − Mhr,α gαΔ (π , z , γ ) δ0 τ ∧TΔ Δ ≤ sup ln Eπ,z exp γ τ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 δ0 + Mhr,α gαΔ π τ ∧ TΔδ , zτ ∧T δ0 , γ e−ατ ∧TΔ Δ + Mhr,α gαΔ π , z , γ e −ατ ∧TΔδ δ0 − Mhr,α gαΔ π , z , γ e−ατ ∧TΔ − Mhr,α gαΔ (π , z , γ ) , (18.56) and, using (18.53), we obtain Mhr,α gαΔ (π, z, γ ) − Mhr,α gαΔ (π , z , γ ) ≤ sup ln π ∈ Sδ ≤L Eπ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}] Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}] (18.57) 18 Asymptotics of HARA Utility 525 To obtain (18.55), we have to estimate δ0 Mhr,α gαΔ π , z , γ e−ατ ∧TΔ − Mhr,α gαΔ (π , z , γ ) δ0 Let γ2 = e−ατ ∧TΔ , and γ1 = γ γ2 By the conditional Hölder inequality E eγ γ2 Z Fτ ∧T δ0 ≤ E eγ Z Fτ ∧T δ0 Δ γ2 Δ for a suitably integrable random variable Z and from (18.52) we have Δ g (π, z, γ1 ) ≤ gαΔ (π, z, γ ) γ2 α (18.58) Moreover, Mhr,α gαΔ (π , z , γ1 ) − Mhr,α gαΔ (π , z , γ ) ≤ sup ln π ∈ Sδ Eπ ,z [exp{γ1 e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ1 e−αh )}] Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}] = sup a(π ), (18.59) π ∈ Sδ and by (18.58) (applied to γ := γ e−αh ), since γ2 < and ln E{Z} ≥ E{ln Z} for a positive random variable Z, we have a(π ) ≤ ln Eπ ,z [exp{γ1 e−αh ln(π · eX(h) e(π(h), π )) + γ2 gαΔ (π , zh , γ e−αh )}] Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}] ≤ ln (Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}])γ2 Eπ ,z [exp{γ e−αh ln(π · eX(h) e(π(h), π )) + gαΔ (π , zh , γ e−αh )}] = (γ2 − 1) ln Eπ ,z exp γ e−αh ln π · eX(h) e π(h), π + gαΔ π , zh , γ e−αh ≤ (γ2 − 1)Eπ ,z γ e−αh ln π · eX(h) e π(h), π + gαΔ π , zh , γ e−αh (18.60) It remains to estimate (γ2 − 1) infπ∈Sδ gαΔ (π, z, γ ) By (18.52), changing portfolio every time when it is allowed, i.e., changing portfolio after every h units of time, we have (with π(0) = π ) ∞ gαΔ (π, z, γ ) ≥ ln Eπ,z exp γ i=1 + ln e π − (ih), π i e−α(ih) ln π (i − 1)h · eX(ih)−X((i−1)h) 526 L Stettner ∞ ≥ Eπ,z e−α(ih) ln π (i − 1)h · eX(ih)−X((i−1)h) γ i=1 + ln e π − (ih), π i ≥ Kγ e−αh − e−αh (18.61) with K = infπ,π ∈Sδ ,z∈D Eπ,z {ln(π · eX(h) ) + ln e(π − (h), π ))} Therefore, by (18.60), a(π ) ≤ (γ2 − 1)γ e−αh K + ≤ ατ ∧ TΔδ γ K − e−αh ≤ (1 − γ2 )γ 2|K| − e−αh α 2|K| ≤ TΔδ 2γ |K|, −αh −αh 1−e 1−e (18.62) and summarizing (18.56)–(18.62), we obtain (18.54) We are going now to prove (18.55) For γ1 ≤ γ2 and a positive random variable Z, γ1 by the Hölder inequality we have E[Z γ1 ] ≤ (E[Z γ2 ]) γ2 Therefore, Δ Δ egα (π,z,γ1 )−gα (π,z,γ2 ) ≤ sup V τ1 Δ Eπ,z [exp{γ1 i=1 e Eπ,z [exp{γ2 i=1 e τj Δ τ1 Δ −αiΔ [ln(π((i − 1)Δ) · eX(iΔ)−X((i−1)Δ) )] −αiΔ [ln(π((i − 1)Δ) · eX(iΔ)−X((i−1)Δ) )] + γ1 ∞ j =2 + γ2 ∞ j =2 + γ1 + γ2 ∞ −α(τi +h) [ln(π(τi ) · eX(τi +h)−X(τi ) ) + ln e(π − (τi i=1 e ∞ −α(τi +h) [ln(π(τi ) · eX(τi +h)−X(τi ) ) + ln e(π − (τi i=1 e i= τj Δ i= τj −1 +h +1 Δ τj −1 +h +1 Δ τ1 Δ ≤ ln Eπ,z exp γ2 e−αiΔ [ln(π((i − 1)Δ) · eX(iΔ)−X((i−1)Δ) )] e−αiΔ [ln(π((i − 1)Δ) · eX(iΔ)−X((i−1)Δ) )] + h), π i ))]}] + h), π i ))]}] e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 ∞ τj Δ + γ2 τ +h j =2 i= j −1 +1 Δ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) 18 Asymptotics of HARA Utility ∞ + γ2 527 e−α(τi +h) ln π(τi ) · eX(τi +h)−X(τi ) i=1 γ1 γ2 −1 + ln e π − (τi + h), π i (18.63) Since ln E{Z} ≥ E{ln Z} for a positive random variable Z, and in the impulse strategies V we can restrict ourselves to the Markov strategies (depending on the current values of the processes (π(t)) and (z(t)) only), we have gαΔ (π, z, γ1 ) − gαΔ (π, z, γ2 ) ≤ − inf − V γ1 γ2 τ1 Δ × ln Eπ,z exp γ2 e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 τj Δ ∞ + γ2 j =2 ∞ + γ2 i= e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) τj −1 +h +1 Δ e−α(τi +h) ln π(τi ) · eX(τi +h)−X(τi ) + ln e π − (τi + h), π i i=1 γ1 Eπ,z γ2 ≤ − inf − V γ2 τj Δ ∞ + γ2 j =2 ∞ + γ2 i= τ1 Δ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) τj −1 +h +1 Δ e−α(τi +h) ln π(τi ) · eX(τi +h)−X(τi ) + ln e π − (τi + h), π i i=1 ∞ ≤ sup(γ2 − γ1 )Eπ,z + V e−α(τi−1 +h) (ψ¯ α ) i=2 ∞ ≤ (γ2 − γ1 ) i=1 e−αh(i−1) (ψ¯ α ) = γ2 − γ1 (ψ¯ α ) − e−αh (18.64) 528 L Stettner with ψ¯ α ≥ of the form ψ¯ α = − δ0 τ ∧TΔ Δ π ∈S ,π inf ∈Sδ0 inf Eπ ,z ,z ∈D τ e−αiΔ ln π (i − 1)Δ i=1 · eX(iΔ)−X((i−1)Δ) δ +h) + e−α(τ ∧TΔ δ +h) ln π τ ∧ TΔδ + h · eX(τ ∧TΔ e π − τ ∧ TΔδ + h , π This completes the proof of (18.55) 18.5 Long-Run PUOP In this section we first consider a discrete-time version of the long-run power utility optimal control and then by limit procedure, based on the bounds from Proposition 18.15, we obtain the continuous-time long-run Bellman equation The approach to discrete-time long-run PUOP will be based on vanishing discount Fix π¯ ∈ Sδ and z¯ ∈ D Let g¯ αΔ (π, z, γ ) = gαΔ (π, z, γ ) − gαΔ (π, ¯ z¯ , γ ) Then from (18.48) and (18.50) we have g¯ αΔ (π, z, γ ) = 1S c (π) Mhr,α g¯ αΔ (π, z, γ ) + gαΔ π, ¯ z¯ , γ ) ¯ z¯ , γ e−αh − gαΔ (π, δ + 1S (π) max ln Eπ,z exp γ e−αΔ ln π · eX(Δ) δ + g¯ αΔ π(Δ), zΔ , γ e−αΔ + gαΔ π¯ , z¯ , γ e−αΔ − gαΔ (π¯ , z¯ , γ ) , Mhr,α g¯αΔ (π, z, γ ) + gαΔ π, ¯ z¯ , γ e−αh − gαΔ (π¯ , z¯ , γ ) , (18.65) and g¯αΔ (π, z, γ ) δ0 τ ∧TΔ Δ = sup ln Eπ,z exp γ τ ∈T Δ e−αiΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 δ0 + Mhr,α g¯αΔ π τ ∧ TΔδ , zτ ∧T δ0 , γ e−ατ ∧TΔ Δ − gαΔ (π¯ , z¯ , γ ) δ +h) + gαΔ π, ¯ z¯ , γ e−α(τ ∧TΔ (18.66) 18 Asymptotics of HARA Utility 529 We have the following: Theorem 18.16 For each Δ ≤ Γ , there are a bounded function w r,Δ , which is continuous in the set Sδ0 , and a constant λΔ (γ ) such that we have w r,Δ (π, z, γ ) = 1S c (π) Mhr wr,Δ (π, z, γ ) − λΔ (γ )h δ + 1S (π) max ln Eπ,z exp γ ln π · eX(Δ) δ + w r,Δ π(Δ), zΔ , γ − λΔ (γ )Δ , Mhr w r,Δ (π, z, γ ) − λΔ (γ )h (18.67) with Mhr w(π, z, γ ) = sup ln Eπ,z exp γ ln π · eX(h) e π(h), π π ∈Sδ + w(π , zh , γ ) (18.68) and, equivalently, w r,Δ (π, z, γ ) δ0 τ ∧TΔ Δ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) := sup ln Eπ,z exp γ τ ∈TΔ i=1 0 + Mhr wr,Δ π τ ∧ TΔδ , zτ ∧T δ0 , γ − Δ Δ τ ∧ TΔδ + h Δ λ (γ ) Δ (18.69) Furthermore, |wr,Δ (π, z, γ )| ≤ L (L is the same as in (18.54)) Proof Notice first that by (18.54) we have that |g¯ αΔ (π, z, γ )| ≤ L and by (18.55), that gαΔ (π¯ , z¯ , γ e−αh ) − gαΔ (π¯ , z¯ , γ ) and gαΔ (π¯ , z¯ , γ e−αΔ ) − gαΔ (π, ¯ z¯ , γ ) are bounded from above as functions of α Therefore, by (18.59), gαΔ (π¯ , z¯ , γ e−αΔ ) − gαΔ (π, ¯ z¯ , γ ) is bounded, and there is a subsequence (αn ), limn→∞ αn = 0, and constants λΔ m (γ ) such that, for m = 1, 2, , lim g Δ n→∞ αn π, ¯ z¯ , γ e−αn mΔ − gαΔn π¯ , z¯ , γ e−αn (m−1)Δ = −ΔλΔ m (γ ) (18.70) By Proposition 18.5 one can choose a further subsequence of (αn ), for simplicity again denoted by (αn ), such that, for k = 0, 1, , Mhr,αn g¯αΔn π, z, γ e−αn kΔ → whΔ (π, z, γ , k) (18.71) for a certain function whΔ with convergence uniform on compact subsets of S00 × D Consequently, 530 L Stettner g¯ αΔn (π, z, γ ) δ0 τ ∧TΔ Δ → sup ln Eπ,z exp γ τ ∈TΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 + whΔ π τ ∧ TΔδ , zτ ∧T δ0 , γ , Δ δ +h τ ∧TΔ Δ δ0 τ ∧ TΔ Δ −Δ λΔ i (γ ) , (18.72) i=1 and, for m = 1, 2, , g¯αΔn π, z, γ e−αn mΔ δ0 τ ∧TΔ Δ → sup ln Eπ,z exp γ τ ∈TΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 + whΔ π τ ∧ TΔδ , zτ ∧T δ0 , γ , m + Δ τ ∧ TΔδ Δ m+ δ +h τ ∧TΔ Δ −Δ λΔ i (γ ) i=m+1 := q (π, z, γ , m) Δ (18.73) as n → ∞ uniformly in π ∈ Sδ Therefore, r Mhr,αn g¯αΔn π, z, γ e−αn kΔ → Mh,Δ q Δ (π, z, γ , k) (18.74) with r g(π, z, γ , k) = sup ln Eπ,z exp γ ln π · eX(h) e π(h), π Mh,Δ π ∈Sδ + q Δ π , zh , γ , k + h Δ (18.75) Hence, from (18.66) we have δ0 τ ∧TΔ Δ q Δ (π, z, γ , m) = sup ln Eπ,z exp γ τ ∈TΔ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) i=1 0 r q Δ π τ ∧ TΔδ , zτ ∧T δ0 , γ , m + + Mh,Δ Δ τ ∧ TΔδ Δ 18 Asymptotics of HARA Utility m+ 531 δ +h τ ∧TΔ Δ −Δ λΔ i (γ ) (18.76) i=m+1 The function f (γ ) := ln E[eγ Y ] with a given random variable Y , whenever is defined, is convex Consequently, gαΔ as a function of γ is also convex, and neglecting other variables, by convexity we have gαΔ (γ e−α(m−1)Δ ) − gαΔ (γ e−αmΔ ) gαΔ (γ e−αmΔ ) − gαΔ (γ e−α(m+1)Δ ) ≥ γ e−αmΔ (1 − e−αΔ ) γ e−α(m−1)Δ (1 − e−αΔ ) and e−αΔ gαΔ γ e−α(m−1)Δ − gαΔ γ e−αmΔ ≥ gαΔ γ e−αmΔ − gαΔ γ e−α(m+1)Δ Δ Δ Therefore, by (18.70), λΔ m (γ ) ≥ λm+1 (γ ) Since λm (γ )Δ is bounded, there is Δ Δ −αn mΔ ) − g¯ Δ (π, z, λΔ (γ ) such that limm→∞ λΔ m (γ ) = λ (γ ) Now, g¯ αn (π, z, γ e αn −α (m−1)Δ n γe ) is bounded and is a difference of two sequences that in the limit as αn → are monotonic, one of them is convergent to λΔ (γ ) Consequently the other sequence is also convergent, and there is a limit limm→∞ q Δ (π, z, γ , m) − q Δ (π, z, γ , m − 1) := d(π, z, γ ) Since q Δ (π, z, γ , m) = q Δ (π, z, γ , m) − q Δ (π, z, γ , m + 1) + · · · + q Δ (π, z, γ , m + k − 1) − q Δ (π, z, γ , m + k) + q Δ (π, z, γ , m + k), by the boundedness of q Δ we clearly have that d(π, z, γ ) = By the uniform continuity of q Δ (·, ·, γ , m) it follows then that there is a continuous function wr,Δ (π, z, γ ) on Sδ0 × B such that limm→∞ q Δ (π, z, γ , m) = wr,Δ (π, z, γ ), uniformly on compact subsets of Sδ0 × B Letting now m → ∞ in (18.76), we obtain wr,Δ (π, z, γ ) δ0 τ ∧TΔ Δ ln π (i − 1)Δ · eX(iΔ)−X((i−1)Δ) = sup ln Eπ,z exp γ τ ∈TΔ i=1 0 + w r,Δ π τ ∧ TΔδ , zτ ∧T δ0 , γ − Δ Δ τ ∧ TΔδ + h Δ λ (γ ) Δ which completes the proof Basing on (18.69), we can now show the main result , (18.77) 532 L Stettner Theorem 18.17 There are a constant λ and a bounded continuous function w r such that the following Bellman equation is satisfied: w r (π, z, γ ) = sup ln Eπ,z exp γ ln π · eX(τ ∧T δ0 ) τ 0 + Mhr w r π τ ∧ T δ , zτ ∧T δ0 , γ − τ ∧ T δ + h λ(γ ) (18.78) Furthermore, γ −1 λ(γ ) = sup Jγr (V ) (18.79) V Proof By the proof of the previous theorem we know that wr,Δ and λΔ are bounded uniformly in Δ Therefore, for a fixed γ , one can choose a subsequence Δn → such that λΔn (γ ) → λ and Mhr w r,Δn → v uniformly on compact subsets of S00 × D Now, letting Δn → in (18.69), we obtain that limn→∞ w r,Δn = w r and v = Mhr wr This completes the proof of (18.78) To prove the formula (18.79), by an analogy to the proof of Theorem 18.11 we have to introduce the function w¯ r , w¯ r (π, z, W, γ ) := sup ln Eπ,z exp γ ln W π · eX(τ ∧T δ0 ) τ + Mhr wr π τ ∧ T δ , zτ ∧T δ0 , γ − τ ∧ T δ + h λ(γ ) (18.80) Clearly, w¯ r (π, z, W, γ ) = γ ln W + w r (π, z, γ ) For a given impulsive strategy V = (τn , π n ), consider the following notation: for n = 1, 2, , πn (τn + h) = π n , πn (τn + h + s) = π − (τn + h + s) for s > Recall that Wt− and Wt are the wealth process before and after possible transaction at time t It is clear that, for s ≥ 0, ¯ ) ew(π,z,W,γ ≥ Eπ,z exp w¯ r π s ∧ T δ , zs∧T δ0 , W − 0 s∧T δ , γ − λ(γ ) s ∧ T δ (18.81) Therefore, Z˜ n (s) = exp w¯ r πn (τn + h + s) ∧ Tτδn +h , zτ +h+s∧T δ0 , τn +h W− τ +h+s∧Tτδn +h , γ − λ(γ ) s ∧ T δ0 ◦ θτn +h (18.82) is a Gsn = Fτn +h+s -supermartingale For any stopping time τ ≥ τn + h, since {τ − τn − h ≤ s} = {τ ≤ τn + h + s} ∈ Fτn +h+s = Gsn , we have that τ − τn − h is a (Gsn )-stopping time Therefore, if additionally τ ≤ Tτδn +h 0 (where, as before, Tτδn +h := τn + h + T δ ◦ θτn +h ), we have r E Z˜ n (τ − τn − h) Fτn +h ≤ Z˜ n (0) = ew¯ (πn (τn +h),zτn +h ,Wτn +h ,γ ) (18.83) 18 Asymptotics of HARA Utility 533 By the form of the operator Mhr , taking into account that w¯ r (π, z, W, γ ) = γ ln W + wr (π, z, γ ), we obtain ew¯ r (π − n−1 (τn ),zτn ,Wτn ,γ ) ≥ e−λ(γ )h Eπ,z exp w¯ r π n , zτn +h , Wτn +h , γ Fτn , (18.84) and therefore, for fixed T > 0, we have E exp χτn ≤T w¯ r πn τn+1 ∧ (T + h) , zτn+1 ∧(T +h) , Wτ−n+1 ∧(T +h) , γ − λ(γ ) τn+1 ∧ (T + h) − τn − h ≤E e χτn ≤T w¯ r (πn (τn +h),zτn +h ,Wτn +h ) ≤ eχτn ≤T (λ(γ )h+w¯ r (π Fτn Fτ n − n−1 (τn ),zτn ,Wτn ,γ )) Consequently, E exp χτn ≤T w¯ r πn τn+1 ∧ (T + h) , zτn+1 ∧(T +h) , Wτ−n+1 ∧(T +h) , γ − w¯ r πn−1 (τn ), zτn , Wτ−n , γ − λ(γ ) τn+1 ∧ (T + h) − τn Fτn ≤ 1, (18.85) and therefore, ∞ χτn ≤T w¯ r πn τn+1 ∧ (T + h) , zτn+1 ∧(T +h) , E exp n=1 Wτ−n+1 ∧(T +h) , γ − w¯ r πn−1 (τn ), zτn , Wτ−n , γ − λ(γ ) τn+1 ∧ (T + h) − τn ≤ Finally, using ζ (T ) = inf{n : τn ≥ T }, we rewrite the last inequality in the form E exp w¯ r πζ (T )−1 τζ (T ) ∧ (T + h) , zτζ (T ) ∧(T +h) , Wτ−ζ (T ) ∧(T +h) , γ − w¯ r (π, z, W, γ ) − λ(γ ) τζ (T ) ∧ (T + h) − τζ (T )−1 ≤1 and E exp wr πζ (T )−1 τζ (T ) ∧ (T + h) , zτζ (T ) ∧(T +h) , γ + γ ln Wτ−ζ (T ) ∧(T +h) + w¯ r (π, z, W, γ ) − λ(γ ) τζ (T ) ∧ (T + h) − τζ (T )−1 ≤ (18.86) 534 L Stettner Notice that for the strategy Vˆ defined by optimal stopping times, from (18.81) and portfolio changes accordingly to the selector of the operator Mhr we have the equalities in (18.84)–(18.86) Taking the logarithm, then dividing both sides of (18.86) by T and letting T → ∞, we obtain (18.81) and the optimality of the strategy Vˆ 18.6 General Form of the Long-Run Bellman Equations We consider in this section (for simplicity) the case with execution delay The case with decision lag can be studied in a similar way To simplify the notation, we neglect here obligatory diversification We would like to find a general, unified form of the Bellman equation, which covers the cases of logarithmic and power utility function Following [7], we can define the problem as follows: find a function w and a constant λ such that for any positive K, we have U Ke w(W,π,z) = sup Eπ,z sup U π − (τ + h) · eX(τ +h) e−λ(τ +h) τ π × Ke π − (τ + h), π ew(W (π − (τ +h)·eX(τ +h) )e(π − (τ +h),π ),π ,zτ +h ) (18.87) It can be shown (see Sect of [7]) that λ is an optimal utility growth By (18.87) we see that the mapping K → supτ U −1 Eπ,z {U (K · · · )} is positively homogeneous This is satisfied in particular, when the mapping K → U −1 E{U (KZ)} is positively homogeneous for any random variable Z By Theorem 3.1 of [7] it holds whenever U (x) = Ax γ + B with γ > 0, or U (x) = A ln x + B with A > and arbitrary B Consequently, up to normalization, U should be a power or logarithmic utility function, and in fact this radically limits the use of the general Bellman equation of the form (18.87) Notice furthermore that we can rewrite (18.87) in the form U ew(W,π,z) = sup Eπ,z U π − (τ ) · eX(τ ) e−λ(τ +h) Mw Wτ , π(τ ), zτ , τ (18.88) where M(W, π, z) = sup U −1 Eπ ,z U π · eX(h) e π(h), π ew(Wh ,π ,zh ) π ∈Sδ (18.89) As we can see in this paper, Bellman equations were independent of the initial value of the wealth process (Wt ) Consequently, we may look for a solution to the equation U ew(π,z) = sup Eπ,z U π − (τ ) · eX(τ ) e−λ(τ +h) Mw π(τ ), zτ τ with M(π, z) = sup U −1 Eπ ,z U π · eX(h) e π(h), π ew(π ,zh ) π In particular cases we obtain the following examples (18.90) 18 Asymptotics of HARA Utility 535 Example 18.18 If U (x) = ln x, then w(π, z) = sup Eπ,z ln π · eX(τ ) − λ(τ + h) + Mh w π(τ ), zτ τ with Mh w(π, z) = sup Eπ ,z ln(π · eX(h) ) + ln e π(h), π + w(π , zh ) π ∈Sδ Example 18.19 If U (x) = x γ , then eγ w(π,z) = sup Eπ,z π − (τ ) · eX(τ ) γ −λr γ (τ +h) e τ Mhr w π − (τ ), zτ with γ Mhr w(π, z) = sup Eπ ,z π · eX(h) e π(h), π π ∈Sδ γ γ w(π ,zh ) e , which modulo small transformations coincide with the Bellman equations considered in Remark 18.12 and Theorem 18.17 As we pointed out above, these two examples practically correspond to the only utility functions for which we could expect to find solutions to (18.90) References M Akian, A Sulem, M.I Taksar, Dynamic optimization of long-term growth rate for a portfolio with transaction costs and logarithmic utility Math Finance 11, 153–188 (2001) D Applebaum, Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge, 2004) T.R Bielecki, S.P Pliska, Risk sensitive dynamic asset management Appl Math Optim 39, 337–360 (1999) T Duncan, B Pasik-Duncan, L Stettner, Growth optimal portfolio under proportional transaction costs with obligatory diversification Appl Math Optim 63, 107–132 (2011) doi:101007/s00245-010-9113-x A Irle, J Sass, Good Portfolio Strategies under Transaction Costs: A Renewal Theoretic Approach Stochastic Finance (Springer, New York, 2006), pp 321–341 I Karatzas, S 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proportional transaction costs Appl Math Optim 54, 277–294 (2006) 15 T Tamura, Maximization of the long-term growth rate for a portfolio with fixed and proportional transaction costs Adv Appl Probab 40, 673–695 (2008) .. .Advanced Mathematical Methods for Finance Giulia Di Nunno Bernt Øksendal Editors Advanced Mathematical Methods for Finance Editors Giulia Di Nunno Bernt Øksendal CMA,... and the use of advanced mathematical tools for finance, from theory to practice This book was born in the same spirit of the program It presents innovations in the mathematical methods in various... Linden 6, 10099 Berlin, Germany e-mail: penner@math.hu-berlin.de G Di Nunno, B Øksendal (eds.), Advanced Mathematical Methods for Finance, DOI 10.1007/978-3-642-18412-3_1, © Springer-Verlag Berlin