Liquidity risk and optimal dividend/investment strategies
|
|
- Franklin Arnold Norton
- 5 years ago
- Views:
Transcription
1 Liquidity risk and optimal dividend/investment strategies Vathana LY VATH Laboratoire de Mathématiques et Modélisation d Evry ENSIIE and Université d Evry Joint work with E. Chevalier and M. Gaigi ICASQF, Cartagena, June 15-18, 2016
2 1 Introduction 2 The model 3 Characterization of the value functions 4 Numerical results
3 Motivations : a corporate finance problem Corporate governance problems : managerial decisions 1 Operational levels : marketing, product lines, internal organization... 2 Financial levels : capital/debt structure, type of financing... 3 Cash-flow levels/utilization : dividends and investments.
4 Motivations : a corporate finance problem Cash-flow (utilization) policy Dividend policy : payment to shareholders Share buyback (dividend or investment?) Dividend payment Singular problem : [1] Jeanblanc and Shiryaev (95), Choulli, Taksar and Zhou (03)
5 Motivations : a corporate finance problem Cash-flow (utilization) policy Investment policy : investment for future growth Organic growth : internal development of new products, technologies, factories... Merger/Acquisition : acquire a competitor for its product portfolio, geographic reach... Optimal switching problem : Brekke and Oksendal (94), Duckworth and Zervos (01), Hamadène and Jeanblanc (05), LV. and Pham (07), Pham, LV. and Zhou (09).
6 Motivations : a corporate finance problem Combining both problems : Dividend and investment Typical corporate dilemma : Total SA, ENI, BP : how to use the huge Cash-Flow generated? which investment projects? Three observed policies : 1 Trader Classified Media : return over 90% of stock value in cash. 2 Bouygues : return 25% of stock value in cash. 3 Microsoft : almost no dividend payment for years!
7 Introduction - motivations : the Jeanblanc-Shiryaev problem. A policy strategy is Z : a F-adapted càdlàg non-decreasing process We consider process X representing the dynamics of the cash reserve of a firm : dx t = µdt + σdw t dz t X 0 = x The value of the firm is defined as : [ T ˆV 0 (x) = sup E 0 Z Z 0 ˆV 0 the value function of a singular problem e ρt dz t ] characterized as the unique solution on (0, ) with growth linear condition to (V.I) 0 : [ min ρ ˆV ] 0 L 0 ˆV0, ˆV 0 1 = 0, x > 0, (1), ˆV 0 (0) = 0. (2)
8 Optimal dividend problem/singular control [1] Explicit solution : Jeanblanc and Shiryaev (95) or Radner and Shepp (96) ˆV O explicit expression dx = μ 0 dt + σdw dz X t 0 = x T0 Vˆ 0( x) = sup E e Z 0 t ρt t dzt dividend : x x ˆ0 Regime 0 0 ˆx 0 x Benchmark 0: a singular control problem
9 Introduction - motivations : assumptions In the previous problem, it is assumed : there is no investment opportunity the firm assets are infinitely liquid or illiquid Our objective : Study optimal dividend and investment control problem under constraints, i.e. relaxing the above assumptions L.V, Pham, and Villeneuve (2008) : the interaction between dividend policy and investment under uncertainty. A mixte singular and switching control problem Other related papers : Decamps and Villeneuve (05), Chevalier, L.V., Scotti (13), Guo and Tomecek (08)... In this study, we look at the dividend/investment problems but no longer assume that assets are infinitely illiquid or liquid. The firm may face some liquidity costs when buying or selling assets.
10 The model Let (Ω, F, P) be a probability space equipped with a filtration F = (F t ) t 0 and W and B be two correlated F-Brownian motions, A policy strategy is a singular/switching control α = ((τ i, q i ) i N, Z ) Z : F-adapted càdlàg non-decreasing process representing dividend policy. (τ i ) : an increasing sequence of stopping times representing investment decision times. (q i ) : q i are F τi -measurable variables representing the number of productive assets units bought/sold. We assume productive assets are risky assets whose value process S is solution of the following equation : ds t = S t (µdt + σdb t ), S 0 = s, (3)
11 The model the dynamics of the quantity of assets held by the firm Q t N is governed by : dq t = 0 for τ i t < τ i+1, Q τi = Q τ + q i, Q 0 = i q, for i N. (4) The dynamics of the cash reserve (or more precisely the firm s cash and equivalents) process of the firm is governed by : dx t = rx t dt + h(q t )(bdt + ηdw t ) dz t, for τ i t < τ i+1 X τi = X τ S τi f (q i )q i κ, X 0 = i 0, where b, r and η are positive constants h a non-negative, non-decreasing and concave function satisfying h(q) H with h(1) > 0 and H > 0. The non-negative, non-decreasing, function f represents the liquidity cost function (or impact function with the impact being temporary) with f (0) = 1 for i N. (5) Remarque. For a given fixed q, the model is closely related to the bachelier model used by Jeanblanc-Shiryaev.
12 The investment objective The bankruptcy time is defined as T := T y,α := inf{t 0, X t < 0}. We define the liquidation value as L(x, s, q) := x + (sf ( q)q κ) + We introduce the following notation S := R + (0, + ) N. The optimal firm value is defined on S := R + (0, + ) N, by [ ] T v(x, s, q) = sup E (x,s,q) e ρu dz u α A(x,s,q) 0 (6)
13 Trivial case where value function is infinite We now identify the trivial cases where the value function is infinite. Lemma If we have r > ρ or µ > ρ then v(y) = + on S. Proof : Let y := (x, s, q) S. We first assume that ρ < r. At time 0, by choosing to liquidate the firm s assets, we may get L(y) > 0 in cash. Then by waiting until a given time t > 0, we may obtain v(y) e (r ρ)t L(y). By letting t going to +, we have v(y) = +. Assume that ρ < µ. First, suppose that q 1. In this case, by doing nothing up to time t and then liquidate at time t, for any t > 0, we may obtain a lower bound of v(y) which we prove goes to when t goes to
14 Associated HJB equation From the Dynamic Programming Principle, we may obtain the following HJB equation : min{ρϕ(x, s, q) Lϕ(x, s, q); ϕ (x, s, q) 1; v(x, s, q) Hv(x, s, q)} = 0 on S, (7) x where we have set and Lϕ(x, s, q) = η2 h(q) 2 2 ϕ 2 x 2 + σ2 s 2 2 ϕ 2 s 2 + cσηsh(q) 2 ϕ s x + (rx + bh(q)) ϕ ϕ + µs x s Hϕ(x, s, q) = max ϕ (Γ(y, n)) n A(x,s,q) { with A(x, s, q) = n Z : n q and n(f (n)) x }, s Γ(y, n) = (x n(f (n))s, s, q + n). S = {(x, s, q) R + (0, + ) N}.
15 Sequence of auxiliary value functions We now introduce the following subsets of A(y) : A N (y) := {α = ((τ k, ξ k ) k N, Z ) A(y) : τ k = + a.s. for all k N + 1} We define [ ] T v N (y) = sup E (x,s,q) e ρu dz u, N N (8) 0 α A N (y) The different steps to follow We characterize recursively the value functions v N. We prove the convergence of v N to v. We compute numerically the different regions (continuation, buy and sell and dividend regions) since no explicit solution may be obtained.
16 Sequence of auxiliary value functions In the next Proposition, we recall explicit formulas for v 0 and the optimal strategy associated to this singular control problem. Proposition There exists x (q) [0, + ) such that { Vq(x) if 0 x x v 0 (x, s, q) := (q) x x (q) + V q(x (q)) if x x (q), where V q is the C 2 function, solution of the following differential equation η 2 h(q) 2 y + (rx + bh(q))y ρy = 0; 2 (9) y(0) = 0, y (x (q)) = 1 and y (x (q)) = 0. (10) Notice that x v 0 (x, s, q) is a concave and C 2 function on [0, + ) and that if h(0) = 0, it is optimal to immediately distribute dividends up to bankruptcy therefore v 0 (x, s, 0) = x.
17 Auxiliary functions : Optimal stopping We now are able to characterize our impulse control problem as an optimal stopping time problem, defined through an induction on the number of interventions N. Proposition For all (x, s, q, N) S N, we have T τ v N (x, s, q) = sup E{ e ρu dz u + e ρτ G N 1 (X x τ (τ,z ) T Z 0, Sτ s, q)1 {τ<t }}, (11) where T is the set of stopping times, Z the set of predictable and non-decreasing càdlàg processes, and G N 1 (x, s, q) := max N 1 (Γ(y, n)) and G 1 = 0, n a(x,s,q) with { a(x, s, q) := n Z : n q and nf (n) x κ }, s and Γ(y, n) := (x nf (n)s κ, s, q + n).
18 Bounds and convergence of v N We begin by stating a standard result which says that any smooth function, which is supersolution to the HJB equation, is a majorant of the value function. Proposition Let N N and φ = (φ q) q N be a family of non-negative C 2 functions on R + (0, + ) such that q N (we may use both notations φ(x, s, q) := φ q(x, s)), φ q(0, s) 0 for all s (0, ) and [ min ρφ(y) L N φ(y), φ(y) G N 1 (y), φ ] x (y) 1 for all y (0, + ) (0, + ) N, where we have set then we have v N φ. Corollary L N ϕ = η2 h(q) 2 2 [ σ 2 s 2 +1 {N>0} 2 ϕ ϕ + (rx + bh(q)) x 2 x Upper bound For all N N and (x, s, q) S, we have 2 2 ϕ s 2 + cσηsh(q) 2 ϕ ϕ + µs s x s 0 (12) ]. L(x, s, q) v N (x, s, q) x + sq + K where ρk = bh.
19 Bounds and convergence of v N Proposition(Convergence) For all y S, we have lim v N(y) = v(y). N + Proof : We obviously have v N v N+1 v for all N N. For y S and ε > 0, we may now consider a strategy α = ((τ k, ξ k ) k N, Z ) A(y) such that v(y) J α (y) + ε. Notice that, as (τ i ) i N is such that lim i + τ i = +, there exists N N such that which ends the proof. T τn J α (y) E[ e ρs dz s] + ε 0 v N (y) + ε,
20 Viscosity characterization of v N We turn to the characterization of the function v N as the unique function which satisfies the boundary condition v N (y) = G N 1 (y) on {0} (0, + ) N. (13) and is a viscosity solution of the following HJB equation : min{ρv N (y) Lv N (y); v N x (y) 1; v N (y) G N 1(y)} = 0 on (0, + ) 2 N, (14) It relies on the following Dynamic Programming Principle. DPP Let θ T, y := (x, s, q) S and set ν = T θ, we have (ν τ) v N (y) = sup E[ (τ,z ) T Z 0 e ρs dz s + e ρ(ν τ) v N ( X x (ν τ), S s ν τ, q ) 1l {τ<ν} ] (15)
21 Viscosity characterization of v N We are now able to establish the main results of this section. Theorem For all (N, q) N N, the value function v N (,, q) is continuous on (0, + ) 2. Moreover v N is the unique viscosity solution on (0, + ) 2 N of the HJB equation (14) satisfying the boundary condition (13) and the following growth condition for some positive constants C 1, C 2 and C 3. v N (x, s, q) C 1 + C 2 x + C 3 sq, (x, s, q) S,
22 Numerical results FIGURE: Description of different regions, in (x, s) for a fixed q 0.
23 Numerical results FIGURE: Description of different regions, in (x, s) for q 1 > q 0.
Optimal exit strategies for investment projects. 7th AMaMeF and Swissquote Conference
Optimal exit strategies for investment projects Simone Scotti Université Paris Diderot Laboratoire de Probabilité et Modèles Aléatories Joint work with : Etienne Chevalier, Université d Evry Vathana Ly
More informationA Model of Optimal Portfolio Selection under. Liquidity Risk and Price Impact
A Model of Optimal Portfolio Selection under Liquidity Risk and Price Impact Huyên PHAM Workshop on PDE and Mathematical Finance KTH, Stockholm, August 15, 2005 Laboratoire de Probabilités et Modèles Aléatoires
More informationA problem of portfolio/consumption choice in a. liquidity risk model with random trading times
A problem of portfolio/consumption choice in a liquidity risk model with random trading times Huyên PHAM Special Semester on Stochastics with Emphasis on Finance, Kick-off workshop, Linz, September 8-12,
More informationThe priority option: the value of being a leader in complete and incomplete markets.
s. a leader in s. Mathematics and Statistics - McMaster University Joint work with Vincent Leclère (École de Ponts) Eidgenössische Technische Hochschule Zürich, December 09, 2010 s. 1 2 3 4 s. Combining
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationOptimal portfolio strategies under partial information with expert opinions
1 / 35 Optimal portfolio strategies under partial information with expert opinions Ralf Wunderlich Brandenburg University of Technology Cottbus, Germany Joint work with Rüdiger Frey Research Seminar WU
More informationBand Control of Mutual Proportional Reinsurance
Band Control of Mutual Proportional Reinsurance arxiv:1112.4458v1 [math.oc] 19 Dec 2011 John Liu College of Business, City University of Hong Kong, Hong Kong Michael Taksar Department of Mathematics, University
More informationA Barrier Version of the Russian Option
A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr
More informationOptimization problems in financial mathematics : explicit solutions for diffusion models Boguslavskaya, E.
UvA-DARE (Digital Academic Repository) Optimization problems in financial mathematics : explicit solutions for diffusion models Boguslavskaya, E. Link to publication Citation for published version (APA):
More informationShadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs
Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs Mihai Sîrbu, The University of Texas at Austin based on joint work with Jin Hyuk Choi and Gordan
More informationPRÉPUBLICATIONS DU LABORATOIRE
Universités de Paris 6 & Paris 7 - CNRS (UMR 7599) PRÉPUBLCATONS DU LABORATORE DE PROBABLTÉS & MODÈLES ALÉATORES 4, place Jussieu - Case 188-75 252 Paris cedex 5 http://www.proba.jussieu.fr Optimal partially
More informationOptimal Impulse Control for Cash Management with Two Sources of Short-term Funds
The Eighth International Symposium on Operations Research and Its Applications (ISORA 09) Zhangjiajie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 323 331 Optimal Impulse Control for
More informationOrder book resilience, price manipulation, and the positive portfolio problem
Order book resilience, price manipulation, and the positive portfolio problem Alexander Schied Mannheim University Workshop on New Directions in Financial Mathematics Institute for Pure and Applied Mathematics,
More informationThomas Knispel Leibniz Universität Hannover
Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover knispel@stochastik.uni-hannover.de AnStAp0 Vienna, July
More informationThe Real Option Approach to Plant Valuation: from Spread Options to Optimal Switching
The Real Option Approach to Plant Valuation: from Spread Options to René 1 and Michael Ludkovski 2 1 Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University
More informationA Class of Singular Control Problems and the Smooth Fit Principle. February 25, 2008
A Class of Singular Control Problems and the Smooth Fit Principle Xin Guo UC Berkeley Pascal Tomecek Cornell University February 25, 28 Abstract This paper analyzes a class of singular control problems
More information(6, 4) Is there arbitrage in this market? If so, find all arbitrages. If not, find all pricing kernels.
Advanced Financial Models Example sheet - Michaelmas 208 Michael Tehranchi Problem. Consider a two-asset model with prices given by (P, P 2 ) (3, 9) /4 (4, 6) (6, 8) /4 /2 (6, 4) Is there arbitrage in
More informationProving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control
Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control Erhan Bayraktar University of Michigan joint work with Virginia R. Young, University of Michigan K αρλoβασi,
More informationexchange rate, optimal impulse control, quasi-variational inequalities, stopping
OPTIMAL INTERVENTION IN THE FOREIGN EXCHANGE MARKET WHEN INTERVENTIONS AFFECT MARKET DYNAMICS ALEC N. KERCHEVAL AND JUAN F. MORENO Abbreviated Title: Optimal intervention in foreign exchange Abstract.
More informationAN OPTIMAL EXTRACTION PROBLEM WITH PRICE IMPACT
AN OPTIMAL EXTRACTION PROBLEM WITH PRICE IMPACT GIORGIO FERRARI, TORBEN KOCH Abstract. A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot
More informationOptimal Stopping Problems and American Options
Optimal Stopping Problems and American Options Nadia Uys A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master
More informationOptimal Consumption, Investment and Insurance Problem in Infinite Time Horizon
Optimal Consumption, Investment and Insurance Problem in Infinite Time Horizon Bin Zou and Abel Cadenillas Department of Mathematical and Statistical Sciences University of Alberta August 213 Abstract
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationLiquidation in Limit Order Books. LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Liquidation in Limit Order Books with Controlled Intensity Erhan and Mike Ludkovski University of Michigan and UCSB
More informationDynamic Pricing for Non-Perishable Products with Demand Learning
Dynamic Pricing for Non-Perishable Products with Demand Learning Victor F. Araman Stern School of Business New York University René A. Caldentey DIMACS Workshop on Yield Management and Dynamic Pricing
More informationLiquidation of an indivisible asset with independent investment
Liquidation of an indivisible asset with independent investment Emilie Fabre Guillaume Royer Nizar Touzi January 3, 25 Abstract We provide an extension of the explicit solution of a mixed optimal stopping
More informationIn the benchmark economy, we restrict ourselves to stationary equilibria. The individual
1 1. Appendix A: Definition of Stationary Equilibrium In the benchmark economy, we restrict ourselves to stationary equilibria. The individual state variables are deposit holdings, d, mortgage balances,
More informationOptimal investment with high-watermark fee in a multi-dimensional jump diffusion model
Optimal investment with high-watermark fee in a multi-dimensional jump diffusion model Karel Janeček Zheng Li Mihai Sîrbu August 2, 218 Abstract This paper studies the problem of optimal investment and
More informationOptimal portfolio liquidation with execution cost and risk
Optimal portfolio liquidation with execution cost and risk Idris KHARROUBI Laboratoire de Probabilités et Modèles Aléatoires CNRS, UMR 7599 Université Paris 7, and CRES, e-mail: kharroubi@ensae.fr Huyên
More informationMinimization of ruin probabilities by investment under transaction costs
Minimization of ruin probabilities by investment under transaction costs Stefan Thonhauser DSA, HEC, Université de Lausanne 13 th Scientific Day, Bonn, 3.4.214 Outline Introduction Risk models and controls
More informationResearch Article Classical and Impulse Stochastic Control on the Optimization of Dividends with Residual Capital at Bankruptcy
Hindawi Discrete Dynamics in Nature and Society Volume 27, Article ID 2693568, 4 pages https://doi.org/.55/27/2693568 Research Article Classical and Impulse Stochastic Control on the Optimization of Dividends
More informationEconomics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
More informationEstimating the efficient price from the order flow : a Brownian Cox process approach
Estimating the efficient price from the order flow : a Brownian Cox process approach - Sylvain DELARE Université Paris Diderot, LPMA) - Christian ROBER Université Lyon 1, Laboratoire SAF) - Mathieu ROSENBAUM
More informationIntroduction Optimality and Asset Pricing
Introduction Optimality and Asset Pricing Andrea Buraschi Imperial College Business School October 2010 The Euler Equation Take an economy where price is given with respect to the numéraire, which is our
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationUniversity Of Calgary Department of Mathematics and Statistics
University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, 2018 1 / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National
More informationPerturbative Approaches for Robust Intertemporal Optimal Portfolio Selection
Perturbative Approaches for Robust Intertemporal Optimal Portfolio Selection F. Trojani and P. Vanini ECAS Course, Lugano, October 7-13, 2001 1 Contents Introduction Merton s Model and Perturbative Solution
More informationWorst Case Portfolio Optimization and HJB-Systems
Worst Case Portfolio Optimization and HJB-Systems Ralf Korn and Mogens Steffensen Abstract We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent,
More informationRegularity of the Optimal Exercise Boundary of American Options for Jump Diffusions
Regularity of the Optimal Exercise Boundary of American Options for Jump Diffusions Hao Xing University of Michigan joint work with Erhan Bayraktar, University of Michigan SIAM Conference on Financial
More informationPseudo-stopping times and the hypothesis (H)
Pseudo-stopping times and the hypothesis (H) Anna Aksamit Laboratoire de Mathématiques et Modélisation d'évry (LaMME) UMR CNRS 80712, Université d'évry Val d'essonne 91037 Évry Cedex, France Libo Li Department
More informationRobust pricing hedging duality for American options in discrete time financial markets
Robust pricing hedging duality for American options in discrete time financial markets Anna Aksamit The University of Sydney based on joint work with Shuoqing Deng, Jan Ob lój and Xiaolu Tan Robust Techniques
More informationOPTIMAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY
Dynamic Systems and Applications 2 2) 93-24 OPTIAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY JINGXIAO ZHANG, SHENG LIU, AND D. KANNAN 2 Center for Applied Statistics,School of Statistics, Renmin
More informationA new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009
A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance
More informationOrder book modeling and market making under uncertainty.
Order book modeling and market making under uncertainty. Sidi Mohamed ALY Lund University sidi@maths.lth.se (Joint work with K. Nyström and C. Zhang, Uppsala University) Le Mans, June 29, 2016 1 / 22 Outline
More informationOPTIMAL DIVIDEND AND FINANCING PROBLEMS FOR A DIFFUSION MODEL WITH EXCESS-OF-LOSS REINSURANCE STRATEGY AND TERMINAL VALUE
Vol. 38 ( 218 ) No. 6 J. of Math. (PRC) OPTIMAL DIVIDEND AND FINANCING PROBLEMS FOR A DIFFUSION MODEL WITH EXCESS-OF-LOSS REINSURANCE STRATEGY AND TERMINAL VALUE LI Tong, MA Shi-xia, Han Mi (School of
More informationNumerical approximation for a portfolio optimization problem under liquidity risk and costs
umerical approximation for a portfolio optimization problem under liquidity risk and costs M hamed GAIGI Vathana LY VATH Mohamed MIF Salwa TOUMI January 15, 2013 Abstract This paper concerns with numerical
More informationA constrained non-linear regular-singular stochastic control problem, with applications
Stochastic Processes and their Applications 9 (4) 67 87 www.elsevier.com/locate/spa A constrained non-linear regular-singular stochastic control problem, with applications Xin Guo a;, Jun Liu b, Xun Yu
More informationOPTIMAL COMBINED DIVIDEND AND PROPORTIONAL REINSURANCE POLICY
Communications on Stochastic Analysis Vol. 8, No. 1 (214) 17-26 Serials Publications www.serialspublications.com OPTIMAL COMBINED DIVIDEND AND POPOTIONAL EINSUANCE POLICY EIYOTI CHIKODZA AND JULIUS N.
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationOptimal Execution Tracking a Benchmark
Optimal Execution Tracking a Benchmark René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Princeton, June 20, 2013 Optimal Execution
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
More informationStochastic Areas and Applications in Risk Theory
Stochastic Areas and Applications in Risk Theory July 16th, 214 Zhenyu Cui Department of Mathematics Brooklyn College, City University of New York Zhenyu Cui 49th Actuarial Research Conference 1 Outline
More informationNecessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs
Necessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs Paolo Guasoni Boston University and University of Pisa Walter Schachermayer Vienna University of Technology
More informationTechnical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance
Technical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance In this technical appendix we provide proofs for the various results stated in the manuscript
More informationCoherent and convex monetary risk measures for bounded
Coherent and convex monetary risk measures for bounded càdlàg processes Patrick Cheridito ORFE Princeton University Princeton, NJ, USA dito@princeton.edu Freddy Delbaen Departement für Mathematik ETH Zürich
More informationIntroduction to Algorithmic Trading Strategies Lecture 3
Introduction to Algorithmic Trading Strategies Lecture 3 Trend Following Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com References Introduction to Stochastic Calculus with Applications.
More informationTechnical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance
Technical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance In this technical appendix we provide proofs for the various results stated in the manuscript
More informationOn Stopping Times and Impulse Control with Constraint
On Stopping Times and Impulse Control with Constraint Jose Luis Menaldi Based on joint papers with M. Robin (216, 217) Department of Mathematics Wayne State University Detroit, Michigan 4822, USA (e-mail:
More informationEMS SCHOOL Risk Theory and Related Topics
EMS SCHOOL Risk Theory and Related Topics Bedlewo, Poland. september 29th- october 8th, 2008 1 Credit Risk: Reduced Form Approach Tomasz R. Bielecki, IIT, Chicago Monique Jeanblanc, University of Evry
More informationOptimal portfolio liquidation with execution cost and risk
Optimal portfolio liquidation with execution cost and risk Idris KHARROUBI Laboratoire de Probabilités et Modèles Aléatoires CNRS, UMR 7599 Université Paris 7, and CRES, e-mail: kharroubi@ensae.fr Huyên
More informationInvestment/consumption problem in illiquid markets with regime-switching
Investment/consumption problem in illiquid markets with regime-switching Paul Gassiat 1), Fausto Gozzi 2), Huyên Pham 1),3) September 23, 213 1) Laboratoire de Probabilités et 2) Dipartimento di Scienze
More informationOptimal investment and contingent claim valuation in illiquid markets
and contingent claim valuation in illiquid markets Teemu Pennanen Department of Mathematics, King s College London 1 / 27 Illiquidity The cost of a market orders depends nonlinearly on the traded amount.
More informationRandom Times and Their Properties
Chapter 6 Random Times and Their Properties Section 6.1 recalls the definition of a filtration (a growing collection of σ-fields) and of stopping times (basically, measurable random times). Section 6.2
More informationA system of variational inequalities arising from finite expiry Russian option with two regimes 1
A system of variational inequalities arising from finite expiry Russian option with two regimes 1 Zhou Yang School of Math. Sci., South China Normal University, Guangzhou 510631, China Abstract: In this
More informationPoisson Approximation for Structure Floors
DIPLOMARBEIT Poisson Approximation for Structure Floors Ausgeführt am Institut für Stochastik und Wirtschaftsmathematik der Technischen Universität Wien unter der Anleitung von Privatdoz. Dipl.-Ing. Dr.techn.
More informationOn dynamic stochastic control with control-dependent information
On dynamic stochastic control with control-dependent information 12 (Warwick) Liverpool 2 July, 2015 1 {see arxiv:1503.02375} 2 Joint work with Matija Vidmar (Ljubljana) Outline of talk Introduction Some
More informationAn Introduction to Moral Hazard in Continuous Time
An Introduction to Moral Hazard in Continuous Time Columbia University, NY Chairs Days: Insurance, Actuarial Science, Data and Models, June 12th, 2018 Outline 1 2 Intuition and verification 2BSDEs 3 Control
More informationLONG-TERM OPTIMAL REAL INVESTMENT STRATEGIES IN THE PRESENCE OF ADJUSTMENT COSTS
LONG-TERM OPTIMAL REAL INVESTMENT STRATEGIES IN THE PRESENCE OF ADJUSTMENT COSTS ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider the problem of determining in a dynamical way the optimal capacity level
More informationHamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem
Hamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem Shuenn-Jyi Sheu Institute of Mathematics, Academia Sinica WSAF, CityU HK June 29-July 3, 2009 1. Introduction X c,π t is the wealth with
More informationUtility Maximization in Hidden Regime-Switching Markets with Default Risk
Utility Maximization in Hidden Regime-Switching Markets with Default Risk José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis figueroa-lopez@wustl.edu pages.wustl.edu/figueroa
More informationCoherent Risk Measures. Acceptance Sets. L = {X G : X(ω) < 0, ω Ω}.
So far in this course we have used several different mathematical expressions to quantify risk, without a deeper discussion of their properties. Coherent Risk Measures Lecture 11, Optimisation in Finance
More informationOptimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112
Optimal Control Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Review of the Theory of Optimal Control Section 1 Review of the Theory of Optimal Control Ömer
More informationSolving Sequential Decision Problems via Continuation Values 1
Solving Sequential Decision Problems via Continuation Values 1 Qingyin Ma a and John Stachurski b a, b Research School of Economics, Australian National University September 1, 016 ABSTRACT. We study a
More informationDiscussion Papers in Economics
Discussion Papers in Economics No. 10/11 A General Equilibrium Corporate Finance Theorem for Incomplete Markets: A Special Case By Pascal Stiefenhofer, University of York Department of Economics and Related
More informationarxiv: v2 [q-fin.tr] 11 Apr 2013
Estimating the efficient price from the order flow: a Brownian Cox process approach arxiv:131.3114v2 [q-fin.r] 11 Apr 213 Sylvain Delattre LPMA, Université Paris Diderot Paris 7) delattre@math.univ-paris-diderot.fr
More informationOptimal Inattention to the Stock Market with Information Costs and Transactions Costs
Optimal Inattention to the Stock Market with Information Costs and Transactions Costs Andrew B. Abel The Wharton School of the University of Pennsylvania and National Bureau of Economic Research Janice
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationWeak convergence and large deviation theory
First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More information14.461: Technological Change, Lecture 4 Competition and Innovation
14.461: Technological Change, Lecture 4 Competition and Innovation Daron Acemoglu MIT September 19, 2011. Daron Acemoglu (MIT) Competition and Innovation September 19, 2011. 1 / 51 Competition and Innovation
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationRelative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague joint work
More informationVISCOSITY SOLUTION AND IMPULSE CONTROL OF THE DIFFUSION MODEL WITH REINSURANCE AND FIXED TRANSACTION COSTS
VISCOSITY SOLUTION AND IMPULSE CONTROL OF THE DIFFUSION MODEL WITH REINSURANCE AND FIXED TRANSACTION COSTS Huiqi Guan Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. Email:
More informationHomogenization of a 1D pursuit law with delay
Homogenization of a 1D pursuit law with delay Jeremy Firozaly (Joint work with R. Monneau) Paris-Est University June 2, 2016 J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2,
More informationPartial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula
Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationAmbiguity and Information Processing in a Model of Intermediary Asset Pricing
Ambiguity and Information Processing in a Model of Intermediary Asset Pricing Leyla Jianyu Han 1 Kenneth Kasa 2 Yulei Luo 1 1 The University of Hong Kong 2 Simon Fraser University December 15, 218 1 /
More informationNonlinear Lévy Processes and their Characteristics
Nonlinear Lévy Processes and their Characteristics Ariel Neufeld Marcel Nutz January 11, 215 Abstract We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely,
More informationOn Ergodic Impulse Control with Constraint
On Ergodic Impulse Control with Constraint Maurice Robin Based on joint papers with J.L. Menaldi University Paris-Sanclay 9119 Saint-Aubin, France (e-mail: maurice.robin@polytechnique.edu) IMA, Minneapolis,
More informationSample of Ph.D. Advisory Exam For MathFinance
Sample of Ph.D. Advisory Exam For MathFinance Students who wish to enter the Ph.D. program of Mathematics of Finance are required to take the advisory exam. This exam consists of three major parts. The
More informationCapital Structure and Investment Dynamics with Fire Sales
Capital Structure and Investment Dynamics with Fire Sales Douglas Gale Piero Gottardi NYU April 23, 2013 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 23, 2013 1 / 55 Introduction Corporate
More informationIntroduction to Algorithmic Trading Strategies Lecture 4
Introduction to Algorithmic Trading Strategies Lecture 4 Optimal Pairs Trading by Stochastic Control Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Problem formulation Ito s lemma
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationStrong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term
1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes
More informationQuestion 1. The correct answers are: (a) (2) (b) (1) (c) (2) (d) (3) (e) (2) (f) (1) (g) (2) (h) (1)
Question 1 The correct answers are: a 2 b 1 c 2 d 3 e 2 f 1 g 2 h 1 Question 2 a Any probability measure Q equivalent to P on F 2 can be described by Q[{x 1, x 2 }] := q x1 q x1,x 2, 1 where q x1, q x1,x
More informationOptimal Dividend Strategies for Two Collaborating Insurance Companies
Optimal Dividend Strategies for Two Collaborating Insurance Companies Hansjörg Albrecher, Pablo Azcue and Nora Muler Abstract We consider a two-dimensional optimal dividend problem in the context of two
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationCoherent and convex monetary risk measures for unbounded
Coherent and convex monetary risk measures for unbounded càdlàg processes Patrick Cheridito ORFE Princeton University Princeton, NJ, USA dito@princeton.edu Freddy Delbaen Departement für Mathematik ETH
More information