March 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
|
|
- Adela Webb
- 6 years ago
- Views:
Transcription
1 ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth process never falls below a xed fraction of its maximum-to-date, and one strives to maximize the long-term growth rate of its expected utility. This problem was introduced and solved explicitly by Grossman and Zhou; we present an approach which simplies and extends their results. Introduction and Summary In a very interesting recent article, Grossman & Zhou (993) consider the classical portfolio optimization problem of Merton (97) under the \drawdown constraint" that the wealth process X () satisfy: X >max X (s); 8 t< (:) st almost surely. In other words, one admits only those portfolios () for which the corresponding wealth process X () never falls below % of its maximum-to-date, for some given constant IMA, University of Minnesota, Minneapolis, MN 55455, on leave from Columbia University, cvitanic@ima.umn.edu y Department of Statistics, Columbia University, New York, NY 7, ik@stat.columbia.edu
2 (; ). The objective is then to maximize the long-term growth rate R():= lim T T log E(X (T )) (:) of expected utility, for some power (; ), over portfolio rules () that satisfy (.). Using a mixture of analytical and probabilistic arguments, Grossman and Zhou provide an explicit solution to this problem, when investment is between a bond and one stock (modeled by geometric Brownian motion with constant coecients). They show that the optimal portfolio ^() always invests a constant proportion of the dierence X ^ max st X ^, t<, in the risky asset. Their insights are impressive, but the arguments are rather lengthy and detailed. We present in this paper an approach to the above problem, which simplies the results of Grossman & Zhou (993) { and extends them to the case of several stocks with general deterministic coecients. The model and the problem are introduced in sections and 3, respectively. The approach is based on an auxiliary nite-horizon stochastic control problem, formulated in section 4 (Problem 4., Remark 4.) in terms of the process N () in (4.). This problem admits an explicit optimal portfolio ^(), which is independent of the time-horizon T (; ) and can be found using \classical" martingale and duality arguments. It is then a relatively straightforward matter to show that this portfolio ^() is also optimal for the problem of maximizing (.); this is carried out in section 5. In section 6 we nd the optimal portfolio for the case of logarithmic utility function, and in section 7 we show that the same portfolio maximizes the long-term growth rate almost surely, not only in expectation. Moreover, this portfolio is optimal even if we allow random (adapted) market coecients. The Model Let us consider the following, by now standard, model of a nancial market M with one riskless asset (\bond", price P at time t) and a risky assets (\stocks"; prices P i at time t, id), modeled by the stochastic equations dp = P rdt; P () = (.) dp i = P i 4 bi dt + dx j= 3 ij dw j 5 ; Pi () = p i > ; (.) for i = ;:::;d: These equations are driven by the d-dimensional Brownian motion W = (W i ;:::;W d ), which generates all the randomness in the model { in the sense that the interest rate r(), the vector of appreciation rates b() =(b ();:::;b d ()) and the volatility matrix () =f ij ()g i;jd are bounded measurable processes, adapted to the augmentation F =
3 ffg t< of the ltration F W =(W(s);st), t<generated by W. We shall assume throughout that the matrix () isinvertible, and the \relative risk" process := [b ri],t<is bounded as well, where I is a d-dimensional vector with all entries equal to one. We shall also denote by := P = exp r(s)ds ; t< (:3) the \discount process" of this model. Consider now an economic agent who invests in this market according to a portfolio rule ()=( ();:::; d ()) in such away that his corresponding wealth process X ()isgoverned by the equation dx 3 dx = i X M dx 4 bi dt + ij dw j 5 " i= dx + i i= = rx dt + X X () = x; and satises the \drawdown constraint" Here (; ) is a given constant, and X M j= + M M [(b ri)dt + dw ] ; rdt (.4) P [X >M ; 8t<]=: (:5) M := max st ((s)x (s)): (:6) X M The interpretation is this: the agent does not tolerate the \drawdown of his discounted wealth, from its maximum-to-date", to be greater than or equal to the constant, at any time t ; thus, he imposes the (almost sure) constraint (.5). He invests a proportion i of the dierence X M > intheith stock, i =;:::;d, and invests P the remainder di= i X M + M of his wealth in the bond. With this interpretation in mind, we set up the formal model as follows.. Denition: For a given initial capital x>, let A (x) denote the class of measurable, F-adapted processes :[;)R d which satisfy k k dt < ; a.s. (:7) 3
4 for any given T (; ), and for which the stochastic functional/dierential equation (.4), (.6) has a unique F-adapted solution X () that obeys the constraint (.5). The elements of A (x) will be called \admissible portfolio processes". The class of Denition. is non-empty. In fact, it is shown in the Appendix that 8 >< >: for any measurable, F-adapted process :[;T]R d ; ^ =( ) is an admissible portfolio in A (x); for any x> 3 The Grossman-Zhou Problem 9 >= >; : (:8) Let U :(;)Rbe a utility function, i.e., a strictly increasing, strictly concave function of class C with U (+) = ;U () = and U(+). The convex dual of this function is given by ~U(y) := max[u(x) xy] =U(I(y)) yi(y);y > (3:) x> where I() is the inverse of U (). 3. Problem (Grossman & Zhou (993)): For some given <<, maximize the long-term rate of growth R():= lim T T log E(X (T )) (3:) of expected power-utility, over A (x). In particular, compute v() := and nd ^ A (x), for which the limit lim supremum in (3.3). T sup R() (3:3) A (x) log T E(X ^ (T )) = R(^) exists and achieves the Grossman and Zhou solved Problem 3. for d = and constant r;b ;, using rather lengthy analytical and probabilistic techniques. We present in section 5 a simple solution to this problem that allows general d and deterministic coecients r();b();() in (.), (.). 4 An Auxiliary Process and Problem For any portfolio process A (x) as in Denition (.), consider the auxiliary process N := X M (M ) ; t<: (4:) 4
5 Because the increasing process M () of (.6) is at o the set ft =X =M g, wehave from (.4), (.3), (4.): d(n ) = (N ) dw ; Consider also the processes Z :=exp (s)dw (s) W :=W+ (s)ds: (4:) k(s)k ds ; H:=Z: (4:3) From the product rule d(hn) = NdZ+Zd(N ) + dhz; Ni and (4.), (4.3) we obtain: d(hn) = HN( )dw. In other words, for any A (x) the process Z HN =( )x exp ( t )(s)dw (s) k k (s)ds (4:4) is a positive local martingale, hence supermartingale, which thus satises E [H(T )N (T )] ( )x ; 8T (; ): (4:5) We now pose an auxiliary stochastic control problem, involving the process N () of (4.). 4. An Auxiliary, Finite-Horizon, Control Problem: For a given T (; ) and utility function U :(;)R, denote by A (x; T ) the class of portfolios () that satisfy the requirements of Denition. on the nite horizon [;T], and nd ^() A (x; T ) that achieves V (; T;x):= sup EU(N(T)): (4:6) A (x;t ) There is a fairly straightforward solution to this problem along the lines of Karatzas, Lehoczky & Shreve (987), as follows: For any y>, A (x; T )wehave from (3.), (4.5): EU(N (T)) E ~ U(yH(T)) + ye(h(t)n (T)) E ~ U(yH(T)) + y( )x (4:7) The inequalities in (4.7) are equalities, if and only if y =^yand ()=^() are such that both N ^ (T ) = I(^yH(T)); (4.8) E[H(T )I(^yH(T))] = ( )x = (4.9) hold. Now ^yis uniquely determined from (4.9), and a portfolio ^ A (x) satisfying (4.8) can be found by introducing the positive martingale Q :=E[H(T)I(^yH(T))jF] = ( )x + Q(s)' (s)dw (s); t T: (4:) 5
6 The second equality follows from the representation theorem for Brownian martingales as stochastic integrals with respect to the Brownian motion W (e.g. Karatzas & Shreve (99), x3.4), and ' :[;T]R d is a measurable, F-adapted process with R T k'(s)k ds <, a.s. Comparing (4.) with (4.4) and recalling (.8), we see that In particular, (4.8) follows, and ^()= ( +' ) ()A (x; T ); H()N ^ () =Q(); a:s: (4:) V (; T;x)=E[(U I)(^yH(T))]: (4:) 4. Remark: Let us consider now Problem 4. with utility function Then, with := U(x) = x for := ( ); <<: (4:3), the formulae (4.9), (4.) become ^y ( )x = E[(H(T )) ] ; V (; T;x)= ( )x (E(H(T )) ) = : (4:4) If, in addition, the coecients r(), b(), () are deterministic, then " (H) = exp (s)dw (s) exp k(s)k ds r(s)+ + k(s)k ds and (4.), (4.), (4.4) give Q =( )x exp ( (s)dw (s) k(s)k ds ) ; ' = (4:5) ^ = ( + ) = ( ) ; independent oft; (4:6) V (; T;x)= ( )x exp ( r+ + kk dt) : (4:7) Clearly, the portfolio ^() of (4.6) is well-dened for all t<; it belongs to A (x) of Denition. for any x (; ), by (.8). 6
7 5 Solution of the Grossman-Zhou Problem We shall assume in this section that 8 >< >: the coecients r(), b(), () in the model of (.), (.) are deterministic, R and that r := lim T R T T r(s)ds; k k := lim T T T k(s)k ds exist and are nite. 9 >= >; (5:) 5. Theorem: Under the assumption (5.), the portfolio ^() of (4.6) is optimal for the Problem 3.. In fact, in the notation of (3.), (3.3), (4.7) and (5.) we have where lim T T log E(X ^ (T )) = R(^) =v()=v()+r ; (5:) V () := lim T T log V (; T;x)=r + ( + )k k = ( ) r + k k ( ) : ( ) (5:3) In order to establish this result, it will be helpful to consider the auxiliary problem v() := sup A (x) R (); " R () := lim T T log E(N (T )) ( ) : (5:4) From the fact that the portfolio ^() of (4.6) does not depend on the horizon T (; ), it is clear that lim T T log E(N (T ^ )) ( ) = R (^) =v()=v(): (5:5) It will also be helpful to note from (4.) that (N ) ( ) =() (X ) f M X ; (5:6) where the function f (x) := x ( x),x is strictly increasing on (;) and strictly decreasing on (; ). Proof of Theorem 5.: From (5.6) we obtain E(N (T )) ( ) ((T )) ( ) ( ) E(X (T )) ; (5:7) whence R () R() r v() r ; 8 A (x) (5:8) 7
8 and therefore V () v() r. In order to establish the reverse inequality, take (;) close enough to so that f () f (=), and observe from (5.6) that for an arbitrary A (x)(a (x)) we have E(N(T)) ( ) ((T )) (f (=)) E(X (T )) =((T)) ( =) E(X (T)) : (5:9) Consequently V () R () R() r ; 8 A (x); whence V () v() r ; letting " and invoking the continuity of the function V () in (5.3), we obtain V () v() r and thus the third equality of (5.): v() =V()+r = r + k k (( )) ( ) : (5:) To obtain the second equality in (5.), it suces to observe that (5.8), (5.5), (5.) imply v() R(^) R (^)+r = V ()+r = v(): Finally, the rst equality in (5.), i.e., the existence of the indicated limit, follows from the double inequality ( ) log( )+ T T T log( ( ) r(s)ds + T log E(N ^ (T )) ( )+ T ) T log E(X ^ (T )) r(s)ds + T log E(N ^ (T )) ( ) (5.) (a consequence of (5.7), (5.9)) in conjunction with (5.5), by passing to the limit as T and then letting " : 5. Remark: Formally setting = in (4.6), we recover the well-known optimal portfolio ^ = for the investment problem without the constraint (.5), with utility function U(x) = x from wealth and deterministic coecients. 6 Maximizing Long-term Rate of Expected Logarithmic Utility The methods of section 4-5 can also be used to show that the portfolio =( ) ; t< (6:) 8
9 is optimal for the problem of maximizing the long-term rate of expected logarithmic utility under the constraint (.5): lim T T E(log X (T )) lim T T E(log X (T )) = ( ) r + k k + r; 8 A (x): (6:) This problem was also considered by Grossman & Zhou (993) in their setting. It turns out that (6.) holds for general random, F-adapted coecients r(), b(), (), for which the conditions of section are satised and the limits r := lim T T Erdt; kk := lim T T exist and are nite. Indeed, consider u() = sup A(x) P(), P():= lim P () := lim T E(log T N Ekk dt (6:3) E(log T X (T )) and u() = sup A(x) P (); (T )) instead of the quantities in (3.), (3.) and (5.4), respectively. Solving the Problem 4. with U(x) =( ) log x leads to Q() ( )x =, '() in (4.), and thus the optimal portfolio of (4.) takes the form () in (6.), independent of the nite horizon T>; furthermore, (4.4) becomes V (; T;x) = log x + log( ) +( r(s)+ k(s)k )E R T ds, whence V (; T;x) u() = lim =( ) r+ k k T T Now one writes (5.6) with =, and uses exactly the same methodology as in section 5 to prove that lim T T E(log X (T )) = P( )=u()=u()+r,thus establishing (6.). 7 Maximization of Long-term Growth Rate from Investment More important than the optimality property (6.), however, is the fact that the portfolio () of (6.) maximizes the long-term growth rate from investment S() := lim T T log X (T ) lim T T log X (T )=( ) r + k k : +r ; a:s: (7:) over all A (x). Again, this comparison is valid for general random, F-adapted coecients as in section, under the proviso that the limits r := lim T T rdt; k k := lim T T 9 kk dt (7:)
10 exist and are nite, almost surely. In order to prove (7.), let us start by noticing that :=N=N, t< satises the stochastic equation d =( )dw ; () = from (4.) and It^o's rule, and is thus a positive supermartingale, for any A (x). It follows readily from this (e.g. Karatzas (989), p. 43) that lim t t log, or equivalently S ():= lim T T log(n (T )) lim T T log(n (T )) = (7.3) = ( ) r + k k =: s(); a.s. The existence of this last limit, and its value, follow from (A.6) and (7.). On the other hand, the inequality (5.6) with = leads to (as in (5.)) T log(n (T )) + r(s)ds log( ) T T T log X (T ) (7.4) Z T log(n (T )) + T r(s)ds T T log( ( =) ) almost surely, for any A (x)a (x) and any (;) suciently close to. In particular, (7.4) gives S ()+r s() := esssup S(); a.s. (7:5) A (x) in the notation of (7.)-(7.3), whence s()+r s(); similarly, S() r S () s(); whence s() r s() and in the limit as " : s() r s(), a.s. It develops that s() =s()+r = ( ) r + k k + r, and it remains to show the existence of the limit and the equality in (7.). But both of these follow by writing the double inequality (7.4) with, letting T to obtain in conjunction with (7.3) s() =s()+r lim T T log X (T ) lim T T log X (T ) s(); and then letting " to conclude lim log T T X (T )=s(), almost surely.
11 A Appendix We devote this section to the proof of the claim (.8). Clearly, it suces to show that for any x (; ) and () as in (.8), the stochastic equation d ^X =(^X ^M) dw ; ^M = max st admits a unique F-adapted solution that satises a.s. ^X(s); ^X() = x (A:) ^X >^M; 8t<: (A:) For then ^X()=() coincides with X ^ (), the wealth process corresponding to the portfolio ^ =( ) according to (.4), and (.5) is satised; and vice-versa. Suppose that ^X() isanf-adapted process that satises (A.), (A.). Following Grossman & Zhou (993), p. 69, observe that d log d Therefore, ^X ^M ^X(T ) ^M = = ^X ^M R := log( ) log dw d ^M ^M ; ^X ^M Clearly, the continuous increasing process K := log d ^M ^M ; whence := (s)dw (s) = + log ^M x ^M x k(s)k ds: : (A:3) is at away from the set n o t = ^X = ^M, i.e., away from the zero-set of the continuous nonnegative process R() of (A.3). From the theory of the Skorohod equation (e.g. Karatzas & Shreve (99), x3.6) we have then K = max (s), and from this and (A.3): st ^M M ~ :=xexp ( ) max (s) ; (A:4) st ) exp ^X X ~ :=xexp ( ) max (s) +( st Notice also (from (A.4), (A.5) and (4.)) that max (s) : (A:5) st N ^ =(^X ^M)( ^M) =( )x e : (A:6) It is straightforward to check that ~ X() satises (A.), (A.).
12 References [] S.J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Math. Finance 3 (3), (993), pp. 4{76. [] I. Karatzas, Optimization problems in the theory of continuous trading, SIAM J. Control & Optimization 7, (989) pp [3] I. Karatzas, J.P. Lehoczky and S.E. Shreve, Optimal portfolio and consumption decisions for a \small investor" on a nite horizon, SIAM J. Control & Optimization 5, (987), pp. 57{586. [4] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus ( nd edition), Springer-Verlag, New York (99). [5] R.C. Merton, Optimum consumption and portfolio rules in a continuous time model, J. Econom. Theory 3, (97), pp. 373{43, Erratum: Ibid 6 (973), pp. 3{4.
Ulam Quaterly { Volume 2, Number 4, with Convergence Rate for Bilinear. Omar Zane. University of Kansas. Department of Mathematics
Ulam Quaterly { Volume 2, Number 4, 1994 Identication of Parameters with Convergence Rate for Bilinear Stochastic Dierential Equations Omar Zane University of Kansas Department of Mathematics Lawrence,
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 informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
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 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 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 informationSensitivity analysis of the expected utility maximization problem with respect to model perturbations
Sensitivity analysis of the expected utility maximization problem with respect to model perturbations Mihai Sîrbu, The University of Texas at Austin based on joint work with Oleksii Mostovyi University
More informationPortfolio Optimization in discrete time
Portfolio Optimization in discrete time Wolfgang J. Runggaldier Dipartimento di Matematica Pura ed Applicata Universitá di Padova, Padova http://www.math.unipd.it/runggaldier/index.html Abstract he paper
More informationPathwise Construction of Stochastic Integrals
Pathwise Construction of Stochastic Integrals Marcel Nutz First version: August 14, 211. This version: June 12, 212. Abstract We propose a method to construct the stochastic integral simultaneously under
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
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 informationA note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing
A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing Claudio Fontana Laboratoire Analyse et Probabilité Université d'évry Val d'essonne, 23 bd de France, 91037, Évry
More informationThe main purpose of this chapter is to prove the rst and second fundamental theorem of asset pricing in a so called nite market model.
1 2. Option pricing in a nite market model (February 14, 2012) 1 Introduction The main purpose of this chapter is to prove the rst and second fundamental theorem of asset pricing in a so called nite market
More informationStochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.
Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of
More informationA MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT
A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields
More informationWe suppose that for each "small market" there exists a probability measure Q n on F n that is equivalent to the original measure P n, suchthats n is a
Asymptotic Arbitrage in Non-Complete Large Financial Markets Irene Klein Walter Schachermayer Institut fur Statistik, Universitat Wien Abstract. Kabanov and Kramkov introduced the notion of "large nancial
More informationof space-time diffusions
Optimal investment for all time horizons and Martin boundary of space-time diffusions Sergey Nadtochiy and Michael Tehranchi October 5, 2012 Abstract This paper is concerned with the axiomatic foundation
More informationarxiv: v1 [math.pr] 24 Sep 2018
A short note on Anticipative portfolio optimization B. D Auria a,b,1,, J.-A. Salmerón a,1 a Dpto. Estadística, Universidad Carlos III de Madrid. Avda. de la Universidad 3, 8911, Leganés (Madrid Spain b
More informationPortfolio Optimization with unobservable Markov-modulated drift process
Portfolio Optimization with unobservable Markov-modulated drift process Ulrich Rieder Department of Optimization and Operations Research University of Ulm, Germany D-89069 Ulm, Germany e-mail: rieder@mathematik.uni-ulm.de
More informationTHE ASYMPTOTIC ELASTICITY OF UTILITY FUNCTIONS AND OPTIMAL INVESTMENT IN INCOMPLETE MARKETS 1
The Annals of Applied Probability 1999, Vol. 9, No. 3, 94 95 THE ASYMPTOTIC ELASTICITY OF UTILITY FUNCTIONS AND OPTIMAL INVESTMENT IN INCOMPLETE MARKETS 1 By D. Kramkov 2 and W. Schachermayer Steklov Mathematical
More informationPORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS
International Journal of Theoretical and Applied Finance c World Scientific Publishing Company PORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS HONGCAN LIN Department of Statistics and Actuarial Science,
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 informationIOANNIS KARATZAS Mathematics and Statistics Departments Columbia University
STOCHASTIC PORTFOLIO THEORY IOANNIS KARATZAS Mathematics and Statistics Departments Columbia University ik@math.columbia.edu Joint work with Dr. E. Robert FERNHOLZ, C.I.O. of INTECH Enhanced Investment
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 informationOn the dual problem of utility maximization
On the dual problem of utility maximization Yiqing LIN Joint work with L. GU and J. YANG University of Vienna Sept. 2nd 2015 Workshop Advanced methods in financial mathematics Angers 1 Introduction Basic
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 informationCorrections to Theory of Asset Pricing (2008), Pearson, Boston, MA
Theory of Asset Pricing George Pennacchi Corrections to Theory of Asset Pricing (8), Pearson, Boston, MA. Page 7. Revise the Independence Axiom to read: For any two lotteries P and P, P P if and only if
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 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 informationOPTIONAL DECOMPOSITION FOR CONTINUOUS SEMIMARTINGALES UNDER ARBITRARY FILTRATIONS. Introduction
OPTIONAL DECOMPOSITION FOR CONTINUOUS SEMIMARTINGALES UNDER ARBITRARY FILTRATIONS IOANNIS KARATZAS AND CONSTANTINOS KARDARAS Abstract. We present an elementary treatment of the Optional Decomposition Theorem
More informationConjugate duality in stochastic optimization
Ari-Pekka Perkkiö, Institute of Mathematics, Aalto University Ph.D. instructor/joint work with Teemu Pennanen, Institute of Mathematics, Aalto University March 15th 2010 1 / 13 We study convex problems.
More informationMinimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization
Finance and Stochastics manuscript No. (will be inserted by the editor) Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization Nicholas Westray Harry Zheng. Received: date
More informationTrading Volume in Dynamically Efficient Markets
Trading Volume in Dynamically Efficient Markets Tony BERRADA HEC Montréal, CIRANO and CREF Julien HUGONNIER University of Lausanne, CIRANO and FAME Marcel RINDISBACHER Rotman School of Management, University
More informationA Representation of Excessive Functions as Expected Suprema
A Representation of Excessive Functions as Expected Suprema Hans Föllmer & Thomas Knispel Humboldt-Universität zu Berlin Institut für Mathematik Unter den Linden 6 10099 Berlin, Germany E-mail: foellmer@math.hu-berlin.de,
More informationValuation and Pricing of Electricity Delivery Contracts The Producer s View
SWM ORCOS Valuation and Pricing of Electricity Delivery Contracts The Producer s View Raimund M. Kovacevic Research Report 2015-04 February, 2015 Operations Research and Control Systems Institute of Statistics
More informationALEKSANDAR MIJATOVIĆ AND MIKHAIL URUSOV
DETERMINISTIC CRITERIA FOR THE ABSENCE OF ARBITRAGE IN DIFFUSION MODELS ALEKSANDAR MIJATOVIĆ AND MIKHAIL URUSOV Abstract. We obtain a deterministic characterisation of the no free lunch with vanishing
More informationQUANTITATIVE FINANCE RESEARCH CENTRE
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 241 December 2008 Viability of Markets with an Infinite Number of Assets Constantinos Kardaras ISSN 1441-8010 VIABILITY
More informationConvex Stochastic Control and Conjugate Duality in a Problem of Unconstrained Utility Maximization Under a Regime Switching Model
Convex Stochastic Control and Conjugate Duality in a Problem of Unconstrained Utility Maximization Under a Regime Switching Model by Aaron Xin Situ A thesis presented to the University of Waterloo in fulfilment
More informationSTOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER. Department of Mathematics. University of Wisconsin.
STOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER Department of Mathematics University of Wisconsin Madison WI 5376 keisler@math.wisc.edu 1. Introduction The Loeb measure construction
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 informationThe Asymptotic Theory of Transaction Costs
The Asymptotic Theory of Transaction Costs Lecture Notes by Walter Schachermayer Nachdiplom-Vorlesung, ETH Zürich, WS 15/16 1 Models on Finite Probability Spaces In this section we consider a stock price
More informationMathematical Finance
ETH Zürich, HS 2017 Prof. Josef Teichmann Matti Kiiski Mathematical Finance Solution sheet 14 Solution 14.1 Denote by Z pz t q tpr0,t s the density process process of Q with respect to P. (a) The second
More informationNonlinear representation, backward SDEs, and application to the Principal-Agent problem
Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem
More informationSEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM. 1. Introduction This paper discusses arbitrage-free separable term structure (STS) models
SEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM DAMIR FILIPOVIĆ Abstract. This paper discusses separable term structure diffusion models in an arbitrage-free environment. Using general consistency
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 informationPoint Process Control
Point Process Control The following note is based on Chapters I, II and VII in Brémaud s book Point Processes and Queues (1981). 1 Basic Definitions Consider some probability space (Ω, F, P). A real-valued
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 informationExample I: Capital Accumulation
1 Example I: Capital Accumulation Time t = 0, 1,..., T < Output y, initial output y 0 Fraction of output invested a, capital k = ay Transition (production function) y = g(k) = g(ay) Reward (utility of
More information1 Markov decision processes
2.997 Decision-Making in Large-Scale Systems February 4 MI, Spring 2004 Handout #1 Lecture Note 1 1 Markov decision processes In this class we will study discrete-time stochastic systems. We can describe
More informationSuper-replication and utility maximization in large financial markets
Super-replication and utility maximization in large financial markets M. De Donno P. Guasoni, M. Pratelli, Abstract We study the problems of super-replication and utility maximization from terminal wealth
More informationConstrained Dynamic Optimality and Binomial Terminal Wealth
To appear in SIAM J. Control Optim. Constrained Dynamic Optimality and Binomial Terminal Wealth J. L. Pedersen & G. Peskir This version: 13 March 018 First version: 31 December 015 Research Report No.
More informationOn the relation between the Smith-Wilson method and integrated Ornstein-Uhlenbeck processes
Mathematical Statistics Stockholm University On the relation between the Smith-Wilson method and integrated Ornstein-Uhlenbeck processes Håkan Andersson Mathias Lindholm Research Report 213:1 ISSN 165-377
More informationKevin X.D. Huang and Jan Werner. Department of Economics, University of Minnesota
Implementing Arrow-Debreu Equilibria by Trading Innitely-Lived Securities. Kevin.D. Huang and Jan Werner Department of Economics, University of Minnesota February 2, 2000 1 1. Introduction Equilibrium
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 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 informationRobust Mean-Variance Hedging via G-Expectation
Robust Mean-Variance Hedging via G-Expectation Francesca Biagini, Jacopo Mancin Thilo Meyer Brandis January 3, 18 Abstract In this paper we study mean-variance hedging under the G-expectation framework.
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 informationWhite noise generalization of the Clark-Ocone formula under change of measure
White noise generalization of the Clark-Ocone formula under change of measure Yeliz Yolcu Okur Supervisor: Prof. Bernt Øksendal Co-Advisor: Ass. Prof. Giulia Di Nunno Centre of Mathematics for Applications
More informationBackward martingale representation and endogenous completeness in finance
Backward martingale representation and endogenous completeness in finance Dmitry Kramkov (with Silviu Predoiu) Carnegie Mellon University 1 / 19 Bibliography Robert M. Anderson and Roberto C. Raimondo.
More information[A + 1 ] + (1 ) v: : (b) Show: the derivative of T at v = v 0 < 0 is: = (v 0 ) (1 ) ; [A + 1 ]
Homework #2 Economics 4- Due Wednesday, October 5 Christiano. This question is designed to illustrate Blackwell's Theorem, Theorem 3.3 on page 54 of S-L. That theorem represents a set of conditions that
More informationSolving the Poisson Disorder Problem
Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer-Verlag, 22, (295-32) Research Report No. 49, 2, Dept. Theoret. Statist. Aarhus Solving the Poisson Disorder Problem
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 informationA Proof of the EOQ Formula Using Quasi-Variational. Inequalities. March 19, Abstract
A Proof of the EOQ Formula Using Quasi-Variational Inequalities Dir Beyer y and Suresh P. Sethi z March, 8 Abstract In this paper, we use quasi-variational inequalities to provide a rigorous proof of the
More informationPortfolio optimisation with small transaction costs
1 / 26 Portfolio optimisation with small transaction costs an engineer s approach Jan Kallsen based on joint work with Johannes Muhle-Karbe and Paul Krühner New York, June 5, 2012 dedicated to Ioannis
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 informationRandom G -Expectations
Random G -Expectations Marcel Nutz ETH Zurich New advances in Backward SDEs for nancial engineering applications Tamerza, Tunisia, 28.10.2010 Marcel Nutz (ETH) Random G-Expectations 1 / 17 Outline 1 Random
More informationSnell envelope and the pricing of American-style contingent claims
Snell envelope and the pricing of -style contingent claims 80-646-08 Stochastic calculus I Geneviève Gauthier HEC Montréal Notation The The riskless security Market model The following text draws heavily
More informationHandout 4: Some Applications of Linear Programming
ENGG 5501: Foundations of Optimization 2018 19 First Term Handout 4: Some Applications of Linear Programming Instructor: Anthony Man Cho So October 15, 2018 1 Introduction The theory of LP has found many
More informationOPTIMAL STOPPING OF A BROWNIAN BRIDGE
OPTIMAL STOPPING OF A BROWNIAN BRIDGE ERIK EKSTRÖM AND HENRIK WANNTORP Abstract. We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian
More informationSolution of Stochastic Optimal Control Problems and Financial Applications
Journal of Mathematical Extension Vol. 11, No. 4, (2017), 27-44 ISSN: 1735-8299 URL: http://www.ijmex.com Solution of Stochastic Optimal Control Problems and Financial Applications 2 Mat B. Kafash 1 Faculty
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
More informationAn Analytic Method for Solving Uncertain Differential Equations
Journal of Uncertain Systems Vol.6, No.4, pp.244-249, 212 Online at: www.jus.org.uk An Analytic Method for Solving Uncertain Differential Equations Yuhan Liu Department of Industrial Engineering, Tsinghua
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 informationarxiv: v1 [q-fin.mf] 26 Sep 2018
Trading Strategies Generated by Path-dependent Functionals of Market Weights IOANNIS KARATZAS DONGHAN KIM arxiv:189.1123v1 [q-fin.mf] 26 Sep 218 September 27, 218 Abstract Almost twenty years ago, E.R.
More informationECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko
ECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko Indirect Utility Recall: static consumer theory; J goods, p j is the price of good j (j = 1; : : : ; J), c j is consumption
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationPortfolio optimisation under non-linear drawdown constraints in a semimartingale financial model
Noname manuscript No. (will be inserted by the editor) Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model Vladimir Cherny Jan Ob lój the date of receipt and
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 informationDYNAMIC ASSET ALLOCATION AND CONSUMPTION CHOICE IN INCOMPLETE MARKETS
DYNAMIC ASSET ALLOCATION AND CONSUMPTION CHOICE IN INCOMPLETE MARKETS SASHA F. STOIKOV AND THALEIA ZARIPHOPOULOU The University of Texas at Austin Abstract. We study the optimal investment consumption
More informationEQUITY MARKET STABILITY
EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference,
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationTHE ARROW PRATT INDEXES OF RISK AVERSION AND CONVEX RISK MEASURES THEY IMPLY
THE ARROW PRATT INDEXES OF RISK AVERSION AND CONVEX RISK MEASURES THEY IMPLY PAUL C. KETTLER ABSTRACT. The Arrow Pratt index of relative risk aversion combines the important economic concepts of elasticity
More informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationMaximum Process Problems in Optimal Control Theory
J. Appl. Math. Stochastic Anal. Vol. 25, No., 25, (77-88) Research Report No. 423, 2, Dept. Theoret. Statist. Aarhus (2 pp) Maximum Process Problems in Optimal Control Theory GORAN PESKIR 3 Given a standard
More informationSuperreplication under Volatility Uncertainty for Measurable Claims
Superreplication under Volatility Uncertainty for Measurable Claims Ariel Neufeld Marcel Nutz April 14, 213 Abstract We establish the duality formula for the superreplication price in a setting of volatility
More informationA monetary value for initial information in portfolio optimization
A monetary value for initial information in portfolio optimization this version: March 15, 2002; Jürgen Amendinger 1, Dirk Becherer 2, Martin Schweizer 3 1 HypoVereinsbank AG, International Markets, Equity
More informationThe Bellman Equation for Power Utility Maximization with Semimartingales
The Bellman Equation for Power Utility Maximization with Semimartingales Marcel Nutz ETH Zurich, Department of Mathematics, 8092 Zurich, Switzerland marcel.nutz@math.ethz.ch First Version: December 9,
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationConvex polytopes, interacting particles, spin glasses, and finance
Convex polytopes, interacting particles, spin glasses, and finance Sourav Chatterjee (UCB) joint work with Soumik Pal (Cornell) Interacting particles Systems of particles governed by joint stochastic differential
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 informationJan Kallsen. Stochastic Optimal Control in Mathematical Finance
Jan Kallsen Stochastic Optimal Control in Mathematical Finance CAU zu Kiel, WS 15/16, as of April 21, 2016 Contents 0 Motivation 4 I Discrete time 7 1 Recap of stochastic processes 8 1.1 Processes, stopping
More informationLOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES
LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES ADRIAN D. BANNER INTECH One Palmer Square Princeton, NJ 8542, USA adrian@enhanced.com RAOUF GHOMRASNI Fakultät II, Institut für Mathematik Sekr. MA 7-5,
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationRobust linear optimization under general norms
Operations Research Letters 3 (004) 50 56 Operations Research Letters www.elsevier.com/locate/dsw Robust linear optimization under general norms Dimitris Bertsimas a; ;, Dessislava Pachamanova b, Melvyn
More informationWinter Lecture 10. Convexity and Concavity
Andrew McLennan February 9, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 10 Convexity and Concavity I. Introduction A. We now consider convexity, concavity, and the general
More informationarxiv:submit/ [q-fin.pm] 25 Sep 2011
arxiv:submit/0324648 [q-fin.pm] 25 Sep 2011 Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung Qingshuo Song Jie Yang September 25, 2011 Abstract We study
More information1 Introduction The problem considered here is to solve for the optimal investment decision of an investor who must withdraw funds (e.g., to pay for so
Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time Sid Browne Columbia University Original: January 9, 1995 Final Version: October 3, 1997 Appeared in: Mathematics of
More informationCCFEA Technical Report WP Symmetries and the Merton portfolio selection model
CCFEA Technical Report WP041-10 Symmetries the Merton portfolio selection model V Naicker, JG O Hara P. G. L. Leach # Quantum Information Computation Group, School of Physics, University of KwaZulu-Natal,
More informationApproximation Algorithms for Maximum. Coverage and Max Cut with Given Sizes of. Parts? A. A. Ageev and M. I. Sviridenko
Approximation Algorithms for Maximum Coverage and Max Cut with Given Sizes of Parts? A. A. Ageev and M. I. Sviridenko Sobolev Institute of Mathematics pr. Koptyuga 4, 630090, Novosibirsk, Russia fageev,svirg@math.nsc.ru
More information