Portfolio optimization in the case of an exponential utility function and in the presence of an illiquid asset
aa r X i v : . [ q -f i n . P M ] M a y arXiv manuscript No. (will be inserted by the editor) Portfolio optimization in the case of an exponentialutility function and in the presence of an illiquidasset
Ljudmila A. Bordag
Received: May 11, 2020/ Accepted: date
Abstract
We study an optimization problem for a portfolio with a risk-free, aliquid, and an illiquid risky asset. The illiquid risky asset is sold in an exogenousrandom moment with a prescribed liquidation time distribution. The investorprefers a negative or a positive exponential utility function. We prove that bothcases are connected by a one-to-one analytical substitution and are identicalfrom the economic, analytical, or Lie algebraic points of view.It is well known that the exponential utility function is connected withthe HARA utility function through a limiting procedure if the parameter ofthe HARA utility function is going to infinity. We show that the optimizationproblem with the exponential utility function is not connected to the HARAcase by the limiting procedure and we obtain essentially different results.For the main three dimensional PDE with the exponential utility functionwe obtain the complete set of the nonequivalent Lie group invariant reductionsto two dimensional PDEs according to an optimal system of subalgebras of theadmitted Lie algebra. We prove that in just one case the invariant reduction isconsistent with the boundary condition. This reduction represents a significantsimplification of the original problem.
Keywords portfolio optimization · illiquidity · Lie group analysis · invariantreductions Mathematics Subject Classification (2010) · · · Ljudmila A. BordagFaculty of Natural and Environmental Sciences, University of Applied Sciences Zit-tau/G¨orlitz, Theodor-K¨orner-Allee 16, D-02763 Zittau, GermanyE-mail: [email protected] Ljudmila A. Bordag
We study an optimization problem for a portfolio with an illiquid, a liquidrisky, and a risk-free asset in the framework of continuous time. We supposethat the illiquid asset is sold in an exogenous random moment T with a pre-scribed liquidation time distribution. When we started investigating an opti-mization problem where the time-horizon is an exogenous random variable wehave had in mind previous crises and some typical regulations in EU coun-tries. But now the corona crises make this problem setting highly actual formost countries in Europe. It seems that in a few weeks the problem settingdescribed in this paper will be very actual for many households, small andmiddle entrepreneurs as in summertime a lot of them will fear default. Theprevious crises showed that the situation will force them to sell their factoryor house because they cannot pay obligations in time. It was a rather typicalsituation in previous crises. As it is set up now one cannot sell one piece ata time from own house or own small shop or factory; one cannot go with theprice down, because it is just on the smallest possible level, and one needs a bitliquidity to stay alive and close most of own debts. During crisis time manyinvestors change the risk tolerance dramatically and their behavior changestowards a strong conservative one. One can find a lot of references in the shortreview [7] devoted to assessing risk tolerance in dependence on economic cyclesas well as a discussion about different parameter choices and forms of utilityfunctions and building financial models. We suppose that during an economiccrisis some investors will prefer to use instead of a HARA utility function aCARA utility function, for instance an exponential utility function.There is a huge amount of papers devoted to the classical optimal invest-ment problem with a random endowment as further generalizations of thefamous Mertons model like in [6], [8]. The main difference in our problem set-ting is that the portfolio includes an illiquid asset with a prescribed liquidationtime distribution. For the first time the optimization problem in this form withthe HARA utility function was introduced in the paper [4] and later studied inpapers [5], [3], [2]. If the illiquid asset in the portfolio is a real estate, a factory,a plant, or store then you can sell it as a whole only. It is a typical situationwith the selling of houses, small factories or shops that you know the marketsituation and a typical liquidation time distribution. If it is the seller marketthen it will be similar to an exponential distribution. If it is the buyer marketthen the liquidation time distribution will be similar to a Weibull distribution.We study in this paper both liquidation time distributions as special cases.The influence of the risk tolerance preferred by the investor on the so-lution of an optimization problem was studied before for different portfoliosettings with and without an illiquid asset. We primary consider the results inconnection with the selection of the utility function.In the paper [14] the authors started with the classical Merton’s optimiza-tion problem used in [12]. The portfolio contains one liquid risky asset anda risk-free money market account. The trading takes place within the fixedfinite time horizon. The authors explore the question of risk management un- ortfolio optimization in the case of an exponential utility function 3 der different risk preferences of the investor. They study the optimal wealthprocess and the portfolio process across different utilities and provide transfor-mations between two such processes corresponding to two arbitrary utilities.It is possible to find a deterministic transformation using the local absoluterisk tolerance function associated with the corresponding utility function. Thistransformation is defined by a solution of a linear heat equation with the risktolerance function as a coefficient by the second spatial derivative. Because ofthe classical problem setting it is possible to study the influence of the chosenutility and the risk tolerance on the wealth process and the different charac-teristics of the optimal portfolio in detail. The authors prove that the mainrole plays the curvature of the risk tolerance function of the preferred utilityfunction. Certainly we cannot expect such tractability from an optimizationproblem with an illiquid asset, but we can use this model as a benchmark forthe case if the volume of the illiquid position of the studied portfolio vanishes.The dependence of optimal liquidation strategies from the risk aversion ofinvestors was studied in the work of Schied and Sch¨oneborn [19]. The authorsconsider the infinite time horizon in the optimal portfolio liquidation problemand use a stochastic control approach. In this model a large investor trades onerisky and one risk-free asset. Thereby due to insufficient liquidity of the riskyasset the investor‘s trading rate moves the market price for the risky asset.The authors obtain for the value function and the optimal strategy nonlinearparabolic PDEs. They have to determine the adaptive trading strategy thatmaximizes the expected utility of the proceeds of a large asset sale. Withalauthors studied the financial influence of different types of investor’s utilityfunctions. They found that the optimal strategy is aggressive or passive in-the-money in dependence on the investor‘s risk tolerance, i.e. if the utility functiondisplays increasing or decreasing risk aversion. The authors proved that suchstrategies are rational for investors with different absolute risk aversion profiles.Another approach to a liquidation problem of an illiquid asset is providedin paper [13]. It is devoted to the problem of how efficiently liquidate largeassets positions up to an exogenous fixed terminal time. The author supposesthat the investor prefers the exponential utility function and seeks to maximizethe expected utility of the terminal value of his wealth. The portfolio containsan illiquid asset called a primary risky asset, a liquid asset that is imperfectlycorrelated with the primary asset and is called a proxy risky asset as well asa riskless money market account that pays zero interest rate. In practice theinvestor tries to reduce the price impact by trading a large number of assetsand to hedge market risk of the liquidated portfolio. As a common strategyone chooses splitting of the given order into smaller pieces and to trade thesepieces sequentially over time. The author can find optimal strategies explicitlyand study their properties. The strategies depend on time and parametersof the model only and are solutions of a linear ordinary differential equationof the second order. The author proves that this case is a generalization ofthe original Merton’s model studied in [12]. He also noticed that the explicitand simple results for optimal strategies were possible to obtain just by usingfinite terminal time and because of the investor used the exponential utility Ljudmila A. Bordag function. A more realistic setting, for instance where the investor receivesmultiply orders at random times or the liquidation time is not fixed in thebeginning leads to an essential more complicated model. In comparison to ourcase the illiquid asset in [13] does not pay any dividends and the investor canalso split the illiquid asset and sell them piece by piece as well as the investorhas no consumption during the lifetime of the portfolio.Study of optimization problems with three assets including an illiquid assetleads to three dimensional nonlinear Hamilton–Jacobi–Bellman (HJB) equa-tions. The corresponding nonlinear three dimensional partial differential equa-tions (PDEs) include a lot of parameters describing the behavior of assets andare challenging for analytical and numerical methods. To simplify the inves-tigated problem one tries to find an inner symmetry of such equation andreduce the number of independent variables at least to two or if possible toone. Usually, low dimensional problems better studied and are, therefore, eas-ier to handle. We use in this paper the powerful method of Lie group analysis.This method is very well known for more than 100 years. It is very often usedin the area of mathematical physics and over the past 20 years also in financialmathematics [1]. Nearly all known explicit solutions for ODEs or PDEs werefound or can be found algorithmically by this method. This method is up tonow the most appropriate method to find algorithmically substitutions to re-duce a high dimensional PDE to a lower dimensional one or even to an ODE.The Lie algebraic structure of the corresponding HJB equation is one of thetwo main results of this paper. The Lie algebraic structure of the HJB equa-tion reveals important structural properties of the considered equation. Forinstance, for the linear Black-Scholes equation the corresponding Lie algebraicstructure gives rise to famous substitutions which reduce it to the heat equa-tion or it allowed to obtain the fundamental solution and the explicit formulasfor European Call- or Put. For a nonlinear equation like studied here we cannotget any fundamental solution but we can obtain reductions of the high dimen-sional HJB equation to simpler ones. We study the complete set of all possiblereductions and describe the unique reduction to two dimensional PDE whichsatisfies the boundary conditions. We present also the explicit form of thecorresponding investment-consumption strategies in invariant variables.Thisreduction also means that for all further investigation it is sufficient to use thetwo dimensional PDE instead of the three dimensional main HJB equation.Our paper is organized as follows. In Section 2 we introduce the economicproblem setting in detail. There is also provided a theorem that the HJBequation with a HARA utility function possesses a unique viscosity solutionwhich was earlier proved in [4]. It will be now adapted to the case exponentialutility function. In Section 3 we provide the Lie group analyses of the opti-mization problems with a general liquidation time distribution and differentutility functions. We prove that the cases HARA utility and exponential util-ity are completely different cases. The usual limiting procedure between theHARA and exponential utility functions gives us the wrong results for the cor-responding Lie algebraic structures. In Section 4 for the optimization problemwith the exponential utility function we chose an optimal system of subalge- ortfolio optimization in the case of an exponential utility function 5 bras of the admitted Lie algebra and provide the complete set of all invariantreductions of the corresponding three dimensional PDE. Because this is anessential step, the complete prove with all details is given and the meaningof each reduction is explained. In each case we prove if the invariant substi-tutions are compatible with the boundary condition. In Section 5 we discussthe connection between different results and see that the radical change of theinvestment consumption strategies is connected with the chosen exponentialutility function.
We study an optimization problem for a portfolio in the framework of contin-uous time. An investor has a portfolio with three assets: an illiquid, a liquidrisky, and a risk-free asset. The investor has an illiquid asset that has somepaper value and can not be sold until some moment T that is random witha prescribed liquidation time distribution. The investor tries to maximize heraverage consumption investing into a liquid risky asset that is partly corre-lated with the illiquid one. The investor is free to choose a utility function incorrespondence with her risk tolerance. In our previous papers [4], [5], [3] weassumed that the investor chooses a hyperbolic absolute risk aversion (HARA)utility function or a logarithmic (LOG) utility function as a special case of theHARA utility. Now we suppose that the investor has a quite different risk tol-erance as before and chooses an exponential utility function. We notice that arisk tolerance R ( c ) of an investor is defined as R ( c ) = − U ′ ( c ) U ′′ ( c ) for any utilityfunction U ( c ). For the HARA utility function the risk tolerance R ( c ) is a linearfunction of c and for the exponential utility function it is a constant.Because the form of utility functions define the form of HJB equations andthe limiting procedures connecting the HARA utility with the logarithmicaland exponential utility functions play an important role in this paper we de-scribe these relations in more detail. In the previous papers [4], [5], [3] and [2]we used the HARA and LOG utility functions and studied connection betweenboth optimization problems. We used the HARA utility function in the form U HARA ( c ) = 1 − γγ (cid:18)(cid:18) c − γ (cid:19) γ − (cid:19) , < γ < . (1)It is easily to see that as γ → U HARA ( c ) γ → −→ U LOG ( c ) = ln c. (2)In common literature is often noticed that we obtain an exponential utilityfunction as a limit case of a HARA utility function by γ → ∞ . This assertionis correct just if the HARA utility function takes a special form, for instance, Ljudmila A. Bordag for the HARA utility in the form (1) it is not the case. It is easy to prove thatif we take the HARA utility in the form U HARA ( c ) = 1 − γγ (cid:18) ac − γ + 1 (cid:19) γ , < γ < , a > , (3)then we obtain by the limiting procedure an exponential utility function U HARA ( c ) γ →∞ −→ U EXP n ( c ) = − e − ac . (4)It is so called negative utility function (denoted as EXPn). The most commonform of the the exponential utility function is U EXP p ( c ) = 1 a (1 − e − ac ) , a > . (5)We call it a positive exponential utility function (and denote it as EXPp).Both negative and positive exponential utility functions differ just by an ad-ditive and a multiplicative constant a . We will study later both optimizationproblems, with EXPn and EXPp utility functions.The forms U HARA and U HARA of the HARA utility function are oftenused and from an economical point of view both of them have properties ofa HARA type utility function. From the analytical point of view the HARAutility functions U HARA and U HARA are different.In the case of U HARA we obtain by limiting transition γ → R ( c ) = − U ′ ( c ) U ′′ ( c ) = c − γ , (6)and by γ → R LOG ( c ) = c as it to expect. But we get neither afinite limit by γ → ∞ of the function U HARA ( c ) nor a relevant value for therisk aversion R ( c ) in this case.In the second case for the utility function U HARA (3) we obtain for therisk tolerance the expression R ( c ) = ac + 1 − γa (1 − γ ) = c − γ + 1 a . (7)Here U HARA tends by the limiting transition γ → ∞ to the EXPn utilityfunction (4) and the risk tolerance takes a constant value R ( c ) = a − as it is toexpect in the case of an exponential utility function. But here in contradictionto the first case of U HARA we cannot obtain any meaningful expression by thelimiting procedure γ →
0, it means we do not obtain a transition to the LOGutility function.In other words to study the connection between two optimization prob-lems with a HARA utility function and with a logarithmic utility functionwe should use the HARA utility for instance in the form U HARA ( c ) to beable to provide the limiting procedure γ → ortfolio optimization in the case of an exponential utility function 7 with an exponential utility we should use another form of the HARA utility,for instance, of type U HARA ( c ) to be able to make the limiting transition for γ → ∞ in corresponding formulas.Because of the relation (4) to correct comparison of the results for the opti-mization problem with the HARA utility function U HARA ( c ) with the resultsfor an optimization problem with an exponential utility function we need tostudy first the optimization problem with the negative exponential utility func-tion U EXP n ( c ). From the other side the positive form of the utility function(5) is often used and it may be interesting to provide results also for this formof the utility function. The optimization problems with both utility functionsdescribe economically equivalent situations. We decided to study both casesbecause of both types exponential utility functions are widely used. We willprove that the optimization problems with the EXPn and EXPp utility func-tions possesses also equivalent analytical and Lie algebraic structures. Thereexist one-to-one analytical substitution which provides the equivalence rela-tion between two of these optimization problems which we provide explicitlylater in Section 3.2.We obtain for the HARA utility function R ( c ) = c − γ , 0 < γ < γ is a parameter of the HARA utility function (given in the form introducedlater in (1)), and R ( c ) = c for the LOG utility function. It means also that theabsolute risk tolerance R ( c ) is increasing or decreasing with the consumption c in these two cases.For an exponential utility function (independently for the positive expo-nential (EXPp) utility function or for the negative exponential (EXPn) utilityfunction) the risk tolerance R ( c ) is a constant. In other words, all the timethe absolute risk tolerance stays unaltered in the framework of these opti-mization problems. We see that even though both the LOG and EXPn utilityfunctions can be regarded as a limit case of the HARA utility function they de-scribe quite different economical situations: in the first case, the risk tolerancechanges with the consumption c and in the case of the EXPn or EXPp utilityfunctions the risk tolerance do not depend on the level of the consumption atall. It means that now the investor has a constant risk tolerance. Maybe itexplains that both optimization problems studied before and presented nowhave quite different analytic and Lie algebraic structures as we show it later.2.1 Formulation of the optimization problemThe investor’s portfolio includes a risk-free bond B t , a risky asset S t and anon-traded asset H t that generates stochastic income, i.e. dividends or costs ofmaintaining the asset. The liquidation time of the portfolio T is a randomly-distributed continuous variable. The risk free bank account B t , with the in-terest rate r , follows dB t = rB t dt, t ≤ T, (8) Ljudmila A. Bordag where r is constant. The lower case index t denotes the spot value of the assetat the moment t .The stock price S t follows the geometrical Brownian motion dS t = S t ( α dt + σ dW t ) , t ≤ T, (9)with the continuously compounded rate of return α > r and the standarddeviation σ . The illiquid asset H t , that can not be traded up to the time T and its paper value is correlated with the stock price and is governed by dH t H t = ( µ − δ ) dt + η (cid:16) ρ dW t + p − ρ dW t (cid:17) , t ≤ T, (10)where µ is the expected rate of return of the risky illiquid asset, ( W t , W t )are two independent standard Brownian motions, δ is the rate of dividendpaid by the illiquid asset, η is the standard deviation of the rate of return,and ρ ∈ ( − ,
1) is the correlation coefficient between the stock index and theilliquid risky asset. The parameters µ , δ , η , ρ are all assumed to be constant.The randomly distributed time T is an exogenous time and it does not de-pend on the Brownian motions ( W t , W t ). The probability density function ofthe liquidation time distribution is denoted by φ ( t ), whereas Φ ( t ) denotes thecumulative distribution function, and Φ ( t ), the survival function, also knownas a reliability function, Φ ( t ) = 1 − Φ ( t ). We skip here the explicit notionof the possible parameters of the distribution in order to make the formulasshorter. In dependence on the rate of illiquidity the liquidation time distribu-tion can take different forms. Typically one use the simplest one parameterexponential distribution with the reliability function Φ ( t ) = e − κt , where κ is the parameter of the distribution or a more advanced Weibull distributionwith Φ ( t ) = e − ( t/λ ) k with two parameters λ and k . We will take these twodistributions as examples in our investigation. We notice that the exponentialdistribution is a special case of the Weibull distribution by k = 1 and κ = 1 /λ .We assume that the investor consumes at rate c ( t ) from the liquid wealthand the allocation-consumption plan ( π, c ) consists of the allocation of theportfolio with the cash amount π = π ( t ) invested in stocks, the consumptionstream c = c ( t ) and the rest of the capital kept in bonds. The consumptionstream c ( t ) is admissible if and only if it is positive and there exists a strategythat finances it. Further on we sometimes omit the dependence on t in someof the equations for the sake of clarity of the formulas. All the income isderived from the capital gains and the investor must be solvent. In otherwords, the liquid wealth process L t must cover the consumption stream. Thewealth process L t is the sum of cash holdings in bonds, stocks and random dividends from the non-traded asset minus the consumption stream, i.e. itmust satisfy the balance equation dL t = (cid:0) rL t + δH t + π t ( α − r ) − c t (cid:1) dt + π t σ dW t . ortfolio optimization in the case of an exponential utility function 9 The investor wants to maximize the overall utility consumed up to the randomtime of liquidation T , given by U ( c ) := E (cid:20)Z ∞ Φ ( t ) U ( c ) dt (cid:21) , (11)as it was shown in [4]. It means we work with the problem (11) that correspondsto the value function V ( l, h, t ), which is defined as V ( l, h, t ) = max ( π,c ) E (cid:20)Z ∞ t Φ ( t ) U ( c ) dt | L ( t ) = l, H ( t ) = h (cid:21) , (12)where l could be regarded as an initial capital and h as a paper value of theilliquid asset. The value function V ( l, h, t ) satisfies the HJB equation V t ( l, h, t ) + 12 η h V hh ( l, h, t ) + ( rl + δh ) V l ( l, h, t ) +( µ − δ ) h V h ( l, h, t ) + max π G [ π ] + max c ≥ H [ c ] = 0 , (13) G [ π ] = 12 V ll ( l, h, t ) π σ + V hl ( l, h, t ) ηρπσh + π ( α − r ) V l ( l, h, t ) , (14) H [ c ] = − c V l ( l, h, t ) + Φ ( t ) U ( c ) , (15)with the boundary condition V ( l, h, t ) → , as t → ∞ . (16)In [5] and [4] the authors have already demonstrated that the formulatedproblem has a unique solution under certain conditions. Namely, the followingtheorem was proved Theorem 1 [4]. There exists a unique viscosity solution of the correspondingHJB equation (12)-(16) if1. U ( c ) is strictly increasing, concave and twice differentiable in c ,2. lim t →∞ Φ ( t ) E [ U ( c ( t ))] = 0 , Φ ( t ) ∼ e − κt or faster as t → ∞ ,3. U ( c ) ≤ M (1 + c ) γ with < γ < and M > ,4. lim c → U ′ ( c ) = + ∞ , lim c → + ∞ U ′ ( c ) = 0 . In this paper we restrict ourselves to the case of exponential utility func-tions that satisfy three first conditions of Theorem 1 by definition. We checkedthe proof of the theorem in [4] and see that the condition lim c → U ′ ( c ) → ∞ can be replaced by the condition lim c → U ′ ( c ) > U ′ ( c ) | c =0 > , lim c → + ∞ U ′ ( c ) = 0 (17)which will be satisfied by negative or positive exponential utility functions.It means that there exist a unique viscosity solution to the HJB equation(12)-(16) with an exponential utility function.Now we can adjust and reformulate Lemma proved in [4] about the prop-erties of the value function as follows Lemma 1
Under the conditions (1) − (3) from Theorem 1 and the condition(17)the value function V ( t, l, h ) (12) has the following properties: (i) V ( l, h, t ) is concave and non-decreasing in l and in h , (ii) V ( l, h, t ) is strictly increasing in l , (iii) V ( l, h, t ) is strictly decreasing in t starting from some point, (iv) 0 ≤ V ( l, h, t ) ≤ O ( | l | γ + | h | γ ) uniformly in t . In the next sections, we will first study three dimensional PDE which weobtain from the HJB equation after formal maximization, then we will try tosimplify this three dimensional PDE as far as possible using its internal alge-braic structure. The properties of the value function listed in Lemma 1 we willuse to define the reduction which keeps all properties of the original optimiza-tion problem. It follows that if one can find a solution to the reduced equationit will be also the unique viscosity solution of the optimization problem.
First we study the case of an optimization problem with the EXPn utility func-tion (4). As usual we provide a formal maximization of (14) and (15) for thechosen utility function in the HJB equation (13) and get a three dimensionalnonlinear PDE.The HJB equation (13) after the formal maximization procedure will takethe form V t ( l, h, t ) + 12 η h V hh ( l, h, t ) + ( rl + δh ) V l ( l, h, t ) + ( µ − δ ) hV h ( l, h, t ) − ( α − r ) V l ( l, h, t ) + 2( α − r ) ηρhV l ( l, h, t ) V lh ( l, h, t ) + η ρ σ h V lh ( l, h, t )2 σ V ll ( l, h, t )+ 1 a V l ( l, h, t ) ln V l ( l, h, t ) − a (cid:0) Φ ( t ) (cid:1) V l ( l, h, t ) − ln aa V l ( l, h, t ) = 0 , (18) V → , t −→ ∞ . Here the investment π ( l, h, t ) and consumption c ( l, h, t ) strategies look as fol-lows in terms of the value function V(l,h, t) π ( l, h, t ) = − ηρσhV lh ( l, h, t ) + ( α − r ) V l ( l, h, t ) σ V ll ( l, h, t ) , (19) c ( l, h, t ) = 1 a ln (cid:18) Φ ( t ) aV l ( l, h, t ) (cid:19) . (20)Equation (18) is a nonlinear three dimensional PDE with the three indepen-dent variables l, h, t such equations are demanding by study with analyticalor numerical methods. The Lie group analysis of a nonlinear PDE is a propertool to obtain the Lie algebra admitted by this equation. Using the generators ortfolio optimization in the case of an exponential utility function 11 of this symmetry algebra one can reduce the dimension of the equation (18)and make a problem better tractable.Roughly speaking to obtain internal Lie algebraic structure of a differentialequation on any function V ( l, h, t ) we present this equation as an algebraicequation, for instance like ∆ ( l, h, t, V, V l , V h , V t , V ll , V ll , V lh , V hh ) = 0 in somespecial space called jet bundle. This space is denoted by j ( n ) , where n isthe order of the highest derivative in the differential equation. All derivativeswill be now considered as new dependent variables. Thereafter we study theproperties of the solution manifold of this equation, which is now a surface inthe jet bundle j ( n ) . We take a generator U of a point transformation in thecorresponding jet bundle and act on the solution manifold to define invariantsubspaces. One obtains a large system of partial differential equations on thecoefficients of the generator U . Usually this system does not has any nontrivialsolution at all, and correspondingly the studied differential equation does notadmit any symmetry. In seldom cases, one gets nontrivial generators of thepoint transformations admitted by the equation. The symmetry propertieswill then used to simplify the studied equation and one obtains a so calledreduced equation. In detail one can find the description of this method in [15],[9] or in [1] where a short and comprehensive introduction in this method isgiven as well as applications to other PDEs arising in financial mathematics.Here we formulate the main theorem of Lie group analysis for the opti-mization problem with the EXPn utility function. Theorem 2
The HJB equation (18) with the EXPn utility function (4) andwith a general liquidation time distribution Φ ( T ) admits the four dimensionalLie algebra L EXP n spanned by generators U , U , U , U , i.e. L EXP n = < U , U , U , U > , where U = 1 ar ∂∂l − V ∂∂V , U = ∂∂V , (21) U = − ar (cid:18) e rt Z e − rt d ln Φ ( t ) (cid:19) ∂∂l + 1 r ∂∂t , U = e rt ∂∂l . with following nontrivial commutation relations [ U , U ] = U , [ U , U ] = U . (22) Except finite dimensional Lie algebra L EXP n (21) the equation (18) admitsalso an infinite dimensional algebra L ∞ = < ψ ( h, t ) ∂∂V > where the function ψ ( h, t ) is any solution of the linear parabolic PDE ψ t ( h, t ) + 12 η h ψ hh ( h, t ) + ( µ − δ ) hψ h ( h, t ) = 0 . (23) Proof
As in [15], [9] or [1] we introduce the second jet bundle j (2) and presentthe equation (18) in the form ∆ ( l, h, t, V, V l , V h , V t , V ll , V ll , V lh , V hh ) = 0 as afunction of these variables in the jet bundle j (2) . We look for generators of theadmitted Lie algebra in the form U = ξ ( l, h, t, V ) ∂∂l + ξ ( l, h, t, V ) ∂∂h + ξ ( l, h, t, V ) ∂∂t + η ( l, h, t, V ) ∂∂V , (24) where the functions ξ , ξ , ξ , η can be found using the over determined systemof determining equations U (2) ∆ ( l, h, t, V, V l , V h , V t , V ll , V ll , V lh , V hh ) | ∆ =0 = 0 , (25)where U (2) is the second prolongation of U in j (2) . We look at the actionof U (2) on ∆ ( l, h, t, V, V l , V h , V t , V ll , V ll , V lh , V hh ) located on its solution sub-variety ∆ = 0 and obtain an overdetermined system of PDEs on the func-tions ξ , ξ , ξ and η from (24). This system has 130 PDEs on the functions ξ , ξ , ξ , η . The most of them are trivial and lead to following conditions onthe functions ( ξ ) h = 0 , ( ξ ) V = 0 , ( ξ ) ll = 0 , ( ξ ) l = ξ ( t ) , ( ξ ) l = 0 , ( ξ ) V = 0 , ( ξ ) l = 0 , ( ξ ) h = 0 , ( ξ ) V = 0 , ( η ) l = 0 , ( η ) V V = 0 , ( η ) V = η ( h, t ) . Consequently, the unknown functions in (24) have the following structure ξ ( l, h, t, V ) = ξ ( t ) l + ξ ( t ) , ξ ( l, h, t, V ) = ξ ( h, t ) , ξ ( l, h, t, V ) = ξ ( t ) ,η ( l, h, t, V ) = η ( h, t ) V + η ( h, t ) . (26)Here ξ ( t ) , ξ ( t ) , ξ ( t, h ) , ξ ( t ) , η ( t, h ) , and η ( t, h ) are some functions whichwill be defined later. To find these unknown functions we should have a closerlook at the nontrivial equations of the obtained system, that are left. After allsimplifications, we get the system of seven PDEs η t + η h η hh + ( µ − δ ) hη h = 0 , (27) ξ t − ξ l = 0 , ( µ − δ )( ξ − hξ h + hξ t ) − η h ξ hh + η h η h = 0 ,hξ t + 2( ξ − hξ h ) = 0 , a η + rξ − a ξ t + δξ − a Φ t Φ ξ = 0( α − r ) ξ t + 2 ηρhη h = 0 , ( α − r )( ξ − hξ h + hξ t ) + ηρσ h η h = 0 . We introduce the differential operator L = ∂∂t + η h ∂ ∂h + ( µ − δ ) h ∂∂h usingthis operator we can rewrite the first equation in the above system as condi-tions on the functions η ( h, t ) and η ( h, t ) which appears in the last equationof (26) correspondingly as L η ( h, t ) = 0 and L η ( h, t ) = 0 . Other equationsin the above system do not contained the function η ( h, t ) at all. If we nowdenote η ( h, t ) = ψ ( h, t ) then we see that we proved the last statement of thetheorem, see (23). ortfolio optimization in the case of an exponential utility function 13 Solving the system (27) for an arbitrary function Φ ( t ) we obtain ξ = c e rt + η ar − ξ a e rt Z e − rt Φ t Φ dt, ξ = 0 ,ξ − const., η = η V + η + ψ ( h, t ) , c , η , η − const. (28)The equations (28) contain four arbitrary constants ξ , c , η , η and afunction η ( h, t ) = ψ ( h, t ) which is an arbitrary solution of L ψ ( h, t ) = 0. For-mulas (28) define four generators of the finite dimensional Lie algebra L EXP n (21) and the infinitely dimensional algebra L ∞ (23) as it was described inTheorem 2. ⊓⊔ Remark 1
The found four dimensional Lie algebra describes the symmetryproperty of the equation (18) for any function Φ ( t ). In [4], [5] we have provedthe theorem for existence and uniqueness of the solution of HJB equation fora liquidation time distribution for which Φ ( t ) ∼ e − κt or faster as t → ∞ ,therefore we will regard this type of the distribution studying the analyticalproperties of the equation further on.First we explain the meaning of some generators of the Lie algebra listedin Theorem 2. We start with the second generator U = ∂∂V . It means that theoriginal value function V ( l, h, t ) which is a solution of the equation (18), canbe shifted on any constant and still be a solution of the same equation. Neitherallocation π or consumption function c will change their values because theyalso depend only on the derivatives of the value function. In some sense itis a trivial symmetry, since the equation (18) contains just the derivatives of V ( l, h, t ) so we certainly can add a constant to this function and it still willbe a solution of the equation. Following this symmetry does not give a rise toany reductions of the studied three dimensional PDE and this symmetry donot satisfy the boundary condition V ( l, h, t ) → , t −→ ∞ because of that itis not interesting by solving of the possed optimization problem.The fourth generator U = e rt ∂∂l means that the value of the independentvariable l can be shifted on the arbitrary value de rt , i.e. the shift l → l + de rt , d − const. leaves the solution unaltered. From economical point of view it meansthat the absolut value of the initial capital is not important for this problem.We can arbitrary shift the initial liquidity l on a bank account d, d > d, d < l + de rt should be positiv in the initial time moment. Thevalue function V ( l, h, t ) as a solution of the equation (18) and the allocation-consumption strategy ( π, c ) will be unaltered. This symmetry is trivial and itdoes not provide any reductions of the original three dimensional PDE.We also get the infinitely dimensional algebra L ∞ = < ψ ( h, t ) ∂∂V > wherethe function ψ ( h, t ) is any solution of the linear PDE ψ t + η h ψ hh + ( µ − δ ) hψ h = 0, see Theorem 2. It has a special meaning - we can add any solution ψ ( h, t ) of this equation to the value function V ( l, h, t ) without any changesof the allocation-consumption strategy ( π, c ). From economical point of viewit means that the additional use of some financial instrument which is the solution of ψ t + η h ψ hh + ( µ − δ ) hψ h = 0 do not change the investment-allocation strategies in this optimization problem. The boundary condition V ( l, h, t ) → , t −→ ∞ leads to the following boundary condition on thesolution of this equation ψ ( h, t ) → , t −→ ∞ . It means it is a financialinstrument which value is defined just by the paper value of the illiquid as-set and time only, can not change the allocation-consumption strategy ( π, c ).We notice also that after the substitution h = e x , t = − τ /η , ψ ( t, h ) = h ( µ − δ − η ) /η e − ( µ − δ − η ) τ/η v ( τ, x ) we obtain on the function v ( τ, x ) theparabolic equation of the type v t = v xx , which is well studied. The solutionmethods as well as the fundamental solution of this equation are well known.3.1 Relation between two optimization problems: one with the HARA andone with the EXPn utility functionIn our previous papers we studied the optimization problem with an illiquidasset in the case if the investor used the HARA utility function (1) or thelogarithmic utility function (2). It is well known that both problems are con-nected by limiting procedure if γ →
0. In the previous paper [3] we provedthat also analytic and Lie algebraic structures of both optimization problemsalso connected with the same limiting procedure.We noticed before that the EXPn utility function is connected to theHARA utility function U HARA ( c ) with the limiting procedure by γ → ∞ ,see (4).It means also that we cannot use directly the results of the Lie groupanalysis obtained in previous works [3] and [2] to compare the admitted Liealgebras for the optimization problem with the HARA utility function in theform U HARA ( c ) with the results in this work for an optimization problemwith an exponential utility function. Because of that we should recalculatethe results of the Lie group analysis for the new form of the HARA utilityfunction. We remember that we first provide the formal maximization in theHJB equation (13) and correspondingly to the chosen utility function we obtaina three dimensional PDE. In our previous works [3] and [2] we used the utilityfunction U HARA ( c ) in the form (1) and got following PDE V t ( t, l, h ) + 12 η h V hh ( t, l, h ) + ( rl + δh ) V l ( t, l, h ) + ( µ − δ ) hV h ( t, l, h ) − ( α − r ) V l ( t, l, h ) + 2( α − r ) ηρhV l ( t, l, h ) V lh ( t, l, h ) + η ρ σ h V lh ( t, l, h )2 σ V ll ( t, l, h )+ (1 − γ ) γ Φ ( t ) − γ V l ( t, l, h ) − γ − γ − − γγ Φ ( t ) = 0 , V t →∞ −→ . (29)Now if we insert in the HJB equation (13) the HARA utility function U HARA (3) then we obtain the PDE in the form V t ( t, l, h ) + 12 η h V hh ( t, l, h ) + ( rl + δh ) V l ( t, l, h ) + ( µ − δ ) hV h ( t, l, h ) ortfolio optimization in the case of an exponential utility function 15 − ( α − r ) V l ( t, l, h ) + 2( α − r ) ηρhV l ( t, l, h ) V lh ( t, l, h ) + η ρ σ h V lh ( t, l, h )2 σ V ll ( t, l, h )+ (1 − γ ) γ Φ ( t ) − γ V l ( t, l, h ) − γ − γ − − γa V l ( t, l, h ) = 0 , V t →∞ −→ . (30)The equations (29) and (30) differ analytically in the last terms, from an eco-nomical point of view they describe equivalent optimization problems. The Liegroup analysis of the first equation was provided in [3]. Now we use the samemethod and find the admitted Lie group for the equation (30). We formulatethe results of in the following theorem Theorem 3
The equation (30) admits the three dimensional Lie algebra L HARA spanned by generators L HARA = < U , U , U > , where U = ∂∂V , U = e rt ∂∂l , U = (cid:18) l + 1 − γar (cid:19) ∂∂l + h ∂∂h + γV ∂∂V , (31) for any liquidation time distribution. Moreover, if and only if the liquidationtime distribution has the exponential form, i.e. Φ ( t ) = de − κt , where d, κ areconstants the studied equation admits a four dimensional Lie algebra L HARA with an additional generator U = ∂∂t − κV ∂∂V , (32) i.e. L HARA = < U , U , U , U > .Except finite dimensional Lie algebras (36) and (32) correspondingly equation(30) admits also an infinite dimensional algebra L ∞ = < ψ ( h, t ) ∂∂V > wherethe function ψ ( h, t ) is any solution of the linear PDE ψ t ( h, t ) + 12 η h ψ hh ( h, t ) + ( µ − δ ) hψ h ( h, t ) = 0 . (33) The Lie algebra L HARA has the following non-zero commutator relations [ U , U ] = γ U , [ U , U ] = U (34) The Lie algebra L HARA has the following non-zero commutator relations [ U , U ] = γ U , [ U , U ] = − κ U , [ U , U ] = U , [ U , U ] = − r U . (35)We will not provide the proof of the Theorem 3 because it is quite similar tothe proof of the previous Theorem 2 for the equation (18). In the paper [3]we obtained for the equation (29) with a general liquidation time distributionanother three dimensional Lie algebra L HARA with generators of the followingform U = ∂∂V , U = e rt ∂∂l , U = l ∂∂l + h ∂∂h + (cid:18) γV − (1 − γ ) Z Φ ( t ) dt (cid:19) ∂∂V , (36) as well as an infinite dimensional algebra L ∞ = < ψ ( h, t ) ∂∂V > where thefunction ψ ( h, t ) is any solution of the linear PDE ψ t ( h, t ) + η h ψ hh ( h, t ) +( µ − δ ) hψ h ( h, t ) = 0 . It is easy to see that both algebras L HARA and L HARA have the same commutation relations and are isomorph. We prove that theadmitted Lie algebras are also similar. Indeed if we take the substitutions˜ l = l − − γar , ˜ h = h, ˜ t = t, ˜ V = V + 1 − γγ Z Φ ( t ) dt (37)then the equation (29) on the function ˜ V (˜ l, ˜ h, ˜ t ) will be replaced by the equa-tion (30) on the value function V ( l, h, t ). Correspondingly the generators of thealgebra (36) will take the form of the generators of (31). It means that the Liealgebras L HARA and L HARA are isomorph and similar and the optimizationproblems with the utility functions U HARA and with U HARA are equivalentnot only from an economical and an analytical but also from the Lie algebraicpoint of view. Now we have a correct form of generators of the Lie algebrato study a limiting procedure by γ → ∞ . Indeed using the properties of thegenerators of the Lie algebra we obtain from (31) U ∞ = ∂∂V , U ∞ = e rt ∂∂l , U ∞ = 1 ar ∂∂l − V ∂∂V , (38)We apply now the limiting transition to the main equation (30) and see thatwe do not get the corresponding PDE in the form (18), despite the fact thatthe U HARA and EXPn utility are connected with this limiting procedure. Weget different analytical structures on this step.Now we compare this Lie algebraic structure with the described in Theo-rem 2. First we see that both three dimensional PDEs have the same infinitedimensional algebra L ∞ = < ψ ( h, t ) ∂∂V > . Then we compare the finite di-mensional algebra (38) with (21) and see that the finite dimensional algebrasin these cases are essentially different. In the case of the U HARA ( c ) utilityfunction and a general liquidation time distribution we have after limitingtransition γ → ∞ the three dimensional algebra (38) and in the case of theEXPn utility function we got the four dimensional algebra (21). These alge-bras do not connect with the limiting procedure by γ → ∞ as well as the boththree dimensional PDEs (30) and (18) are not connected with this limitingprocedure. Other sides it is easy to see that all three generators (38) coincidewith the three of the four generators of (21). Lie algebra (21) is in some wayextension of the Lie algebra (38). It means that using the exponential utilityfunctions like EXPn or EXPp makes the corresponding optimization problemsmoother from the Lie algebraic point of view.We see that by limiting procedure γ → ∞ neither the analytic nor theLie algebraic structure of the optimization problem will be preserved. If in theprevious cases of the HARA and LOG utility functions it was sufficient to studythe case of the HARA utility and then just take a limit by γ → ortfolio optimization in the case of an exponential utility function 17 function in own rights step-by-step independently from the case of the HARAutility function.In the next section we study the optimization problem with the EXPputility function. Because the utility functions are defined up to additive andmultiplicative constants the results should be in some sense equivalent. Wewill prove this equivalence explicitly for the optimization problems with theEXPn and EXPp utility functions in the next section.3.2 Relation between two optimization problems correspondingly one withthe EXPn and one with the EXPp utility functionThe EXPp utility function (5) is very close to the EXPn function (4), yet thisparticular case rather popular therefore we analyze it separately.The whole approach is very similar to the method described at the begin-ning of this section therefore we omit some details here. In the case of theEXPp utility function (5) the HJB equation after the formal maximizationprocedure will take the following form V t + 12 η h V hh + ( rl + δh ) V l + ( µ − δ ) hV h (39) − ( α − r ) V l + 2( α − r ) ηρhV l V lh + η ρ σ h V lh σ V ll + 1 a V l ln V l − a (cid:0) Φ ( t ) (cid:1) V l + 1 a Φ ( t ) = 0 , V → , t → ∞ . The main PDE (18) for the EXPn utility function differs from this one forthe EXPp utility function by one term only. If in (18) the last term was − (cid:0) a ln a (cid:1) V l now it is a Φ ( t ), i.e. now equation (39) has a free term a Φ ( t )without the dependent variable V ( l, h, t ) as it was before.Analogously to the previous chapter we can formulate and prove the maintheorem of Lie group analysis for the HJB optimization problem with thepositive exponential utility function. Theorem 4
The equation (39) admits the four dimensional Lie algebra L EXP p spanned by generators U , U , U , U , i.e. L EXP p = < U , U , U , U > ,where U = 1 ar ∂∂l − (cid:18) V + 1 a Z Φ ( t ) dt (cid:19) ∂∂V , U = ∂∂V , (40) U = − ar e rt (cid:18)Z e − rt d ln Φ ( t ) (cid:19) ∂∂l + 1 r ∂∂t − ar Φ ( t ) ∂∂V , U = e rt ∂∂l . for any liquidation time distribution Φ ( t ) . Except finite dimensional Lie algebra L EXP p the equation (39) admits also an infinite dimensional algebra L ∞ = <ψ ( h, t ) ∂∂V > where the function ψ ( h, t ) is any solution of the linear PDE ψ t ( h, t ) + 12 η h ψ hh ( h, t ) + ( µ − δ ) hψ h ( h, t ) = 0 . (41) The Lie algebra L EXP p has following two non-zero commutator relations [ U , U ] = U , [ U , U ] = U . (42)If we compare nontrivial commutation relations (22) for L EXP n and (42)for L EXP p we see that they coincide. It means that both algebras L EXP p and L EXP n are isomorph because they have the same set of the structureconstants. It means also that both Lie symmetry algebras L EXP n and L EXP p correspond to the type A ⊕ A after the classification provided in [18].If we can prove that the both Lie algebras are also similar, then both op-timization problems with the EXPp and with the EXPn utility functions areequivalent and we provide at the same time the equivalence substitution. Thesimilarity of two algebras means that they are not just isomorphic but alsothat there exists an analytical substitution that provided analytical identitybetween corresponding generators. We can then use this substitution to trans-form the equation (39) to (18) or both equations to one the same equation.It is easy to see that if we make following transformations of the variables l, h, t, V in equation (39) l = ˜ l − ln aar , h = ˜ h, t = ˜ t, (43) V ( l, h, t ) = ˜ V (˜ l, ˜ h, ˜ t ) − a Z Φ ( t ) dt. then the final equation in variables ˜ l, ˜ h, ˜ t, ˜ V coincide with (18). Because thesubstitution (43) is an invertible analytical one-to-one substitution we haveto do with two identical optimization problems. The analytical and the Liealgebraic structures of the optimization problems with the EXPn and EXPputility functions are equivalent and it is enough to study one of these problemsin detail. Remark 2
The EXPp and the EXPn utility functions are connected by anaffine transformation. Because of that, it seems that we do not need the Liegroup analysis to define transformations like in (43). If we look at the for-mula (12) we can guess that at liest the transformation of the value functionshould look like in (43). But the maximization procedure used in formula(12) is non-trivial and not all transformations applied to the utility functionwill be preserved after this procedure. For instance it is rather difficult justby looking at the formula (12) to see that the liquidity variable should bealso shifted. Other sides we have seen that the utility functions U HARA and U HARA are connected with an algebraic relation but the corresponding PDEsare connected with the linear transformation (37) which is very similar to (43).In both cases, the Lie group analysis gives us the correct answer to how thecorresponding optimization problems are connected. ortfolio optimization in the case of an exponential utility function 19 L EXP n and related invariantreductions of the corresponding three dimensional PDE To find all reductions and in this way to find all classes of the nonequivalentgroup invariant solutions of a differential equation Ovsiannikov [16] has in-troduced the idea of an optimal system of subalgebras for a given symmetryalgebra of the differential equation. This idea is now widely used for PDEsand systems of ODEs arising in different areas of sciences [17], [10], [11].Now we will study a complete set of possible reductions of the three di-mensional PDE (18) to two dimensional PDEs. For this purpose we need anoptimal system of subalgebras of L EXP n . As before in [3] we use an optimalsystem developed in [18] for the real four dimensional Lie algebras of this type.To make the comparison of the results transparent we introduce in this Sectionthe same notations for the generators of L EXP n as in [18] and in [3].In the basis (21) of L EXP n there are only two non-zero commutation re-lations (22). If we introduce notations like in the paper [18], i.e. we denote U i = e i where i = 1 , . . . , e , e ] = e , [ e , e ] = e . (44)Now we can see that L EXP n corresponds to the algebras of the type A ⊕ A in the classification of [18] where also optimal systems of subalgebras forall real three and four dimensional solvable Lie algebras are provided. Thecorresponding system of optimal subalgebras of L EXP n is listed in Table 1. Table 1
The optimal system of one-, two- and three- dimensional subalgebras of L EXPn ,where ω is a parameter such that −∞ < ω < ∞ .Dimension of System of optimal subalgebras of algebra L EXPn the subalgebra1 h = h e i , h = h e i , h = h e i , h = h e + ωe i , h = h e ± e i ,h = h e ± e i , h = h e ± e i h = h e , e i , h = h e , e i , h = h e , e i , h = h e , e i ,h = h e + ωe , e i , h = h e + ωe , e i , h = h e ± e , e i ,h = h e ± e , e i , h = h e + e , e ± e i h = h e , e , e i , h = h e , e , e i , h = h e , e , e i , h = h e , e , e i ,h = h e ± e , e , e i , h = h e + ωe , e , e i Now we are going to study all possible invariant reductions of the mainequation (18).Let us first note that the subgroups H , H and H generated by subalgebras h = (cid:10) ∂∂V (cid:11) , h = (cid:10) e rt ∂∂l (cid:11) and h = (cid:10) ∂∂V ± e rt ∂∂l (cid:11) correspondingly, do not giveus any interesting reductions so we omit the detailed study of these cases here.We start with a first interesting and nontrivial case. Case H ( h ). The subalgebra h is spanned by the generator e h = < e > = (cid:28) − ar (cid:18) e rt Z e − rt d ln Φ ( t ) (cid:19) ∂∂l + 1 r ∂∂t (cid:29) . (45) To find all invariants of the subgroup H we solve the related characteristicsystem of equations dl − ar (cid:0) e rt R e − rt d ln Φ ( t ) (cid:1) = dt r = dV dh , (46)where the last two equations of the system present a formal notation thatshows that the independent variable h and the dependent variable V are ac-tually invariants under the action of the subgroup H . We can obtain otherindependent invariants solving the system above. So we obtain a set of inde-pendent invariants inv = z = l + 1 ar e rt Z e − rt d ln Φ ( t ) − ar ln Φ ( t ) − ar (1 + ln a ) , inv = h, (47) inv = W ( z, h ) = V ( l, h, t ) . (48)The invariants (47) can be used as the new independent variables z, h andthe invariant (48) as the new dependent variable W ( t, z ) to reduce the threedimensional PDE (18) to a two dimensional one12 η h W hh + ( µ − δ ) hW h + ( rz + δh ) W z + 1 a w z ln W z (49) − ( α − r ) W z + 2( α − r ) ηρhW z W zh + η ρ σ h W zh σ W zz = 0 . In (16) we formulate the boundary condition and in Lemma 1 we formulatethe main properties of the value function. Now we have to reformulate theboundary condition on the function W ( z, h ) after the substitution (47). Tomake further remarks transparent we take first as an example the simplesform of the liquidation time distribution and suppose that Φ ( t ) = e − κt , i.e.we have to do with exponential liquidation time distribution. Then the newvariable z will take the form z = l + κar t + 1 ar (cid:16) κr − − ln a (cid:17) . (50)It means that z is increasing if l or t are growing up. But it leads to contradic-tion between the properties of the function W ( z, h ) = V ( l, h, t ). One sides theboundary condition demands that the value function tends to zero for t → ∞ ,other sides that the same function is strictly increasing by l → ∞ . Becauseafter the invariant substitution the new variable z is the sum of these two oldvariables l and t we are not able to solve this contradiction. The similar incon-sistency problem arising if we use another form of the function Φ ( t ). Followingthis reduction cannot be used to solve the optimization problem. Case H ( h ). Now we look for invariants of the subgroup H . The corre-sponding subalgebra h is spanned by the generator e + ωe , i.e. h = (cid:28) ar (cid:18) − ωe rt Z e − rt d ln Φ ( t ) (cid:19) ∂∂l + ωr ∂∂t − V ∂∂V (cid:29) . (51) ortfolio optimization in the case of an exponential utility function 21 We need to regard two special cases ω = 0 and ω = 0 here. If ω = 0 then h = < e > = (cid:28) ar ∂∂l − V ∂∂V (cid:29) . (52)The invariants of the group H are inv = h, inv = t, inv = W ( h, t ) = V ( l, h, t ) e arl From the last relation follows that V ( l, h, t ) = W ( h, t ) e − arl . We see that inthis case the value function has the form V ( l, h, t ) = e − arl W ( h, t ) and thecomplete dependence on l is described just by the factor e − arl . It means thatwe obtain a decreasing function V ( l, h, t ) in the variable l in contradiction tothe properties of a value function (see Lemma 1). It means that this reductiondo not provide any meaningful solutions for our problem.Now we can move according to a standard procedure to find the invariantsof H when ω = 0. We obtain three independent invariants using a correspond-ing characteristic system inv = z = l + 1 ar e rt Z e − rt d ln Φ ( t ) − ar ln Φ ( t ) − taω , inv = h, (53) inv = W ( z, h ) = V ( l, h, t ) e rω t . (54)Analogously substituting expressions for the invariants z as the new in-dependent variable and W ( z, h ) as the new dependent variables into (18) weget 12 η h W hh + ( µ − δ ) hW h + ( rz + δh ) W z + 1 a w z ln W z − ( α − r ) W z + 2( α − r ) ηρhW z W zh + η ρ σ h W zh σ W zz (55) − a (cid:18) ω + (1 + ln a ) (cid:19) W z − rω W = 0 . We prove now a compatibility of the invariant substitutions (53) and theboundary condition (16). As before we look for the new invariant variables(53)-(54) in the case of exponential liquidation time with Φ ( t ) = e − κt thenthese formulas take the form z = l + κω − rarω t + κar , inv = h, (56) V ( t, l, h ) = W ( z, h ) e − rω t , ω = 0 . (57)From (56) follows that if we chose an arbitrary parameter ω = r/κ then thevariable z up to a constant shift coincides with the old variable l . The relation(57) shows that the boundary condition (16) will be satisfied for any solutionof (55). We see also that for other positive values of the parameter ω theinvariant variables (56)-(57) are compatible with the boundary condition (16). Similar to the case of the exponential time distribution we can study othertypes of liquidation time distributions. For instance, we look at the frequentlyused Weibull distribution with Φ ( t ) = e − ( t/λ ) k where the invariant variableswill take the form z = l + kar k +1 λ k e rt Γ ( k, rt ) + 1 arλ k t k − aω t, inv = h, (58) V ( t, l, h ) = W ( z, h ) e − rω t , ω = 0 , (59)here Γ ( k, rt ) is the upper incomplete gamma function.For the studied optimization problem the most interesting case appears ifthe liquidation time distribution has a lokal maximum like we expect it in thereal world. The Weibull distribution has a local maximum for the parameter k >
1. Because of the asymptotic behavior of the expression e rt Γ ( k, rt ) → r k − t k − as t → ∞ we obtain that for k > z → l + arλ k t k as t → ∞ . It means that also for a Weibull distribution we have compatibility ofthe invariant substitutions (58)-(59) with the boundary condition (16).We notice that the investment π ( z, h ) and consumption c ( z, t, h ) in thecase H look as π ( z, h ) = (cid:18) − ηρσhW zh + ( α − r ) W z σ W zz (cid:19) , c ( z, t, h ) = 1 a ln (cid:18) Φ ( t ) aW z (cid:19) + raω t, ω > , where W ( z, h ) is a solution of the equation (55). Case H ( h ). According to the first line of Table 1 the subalgebra h corresponding to the subgroup H algebra is spanned by h = < e ± e > = (cid:28)(cid:18) ar ± e rt (cid:19) ∂∂l − V ∂∂V (cid:29) . (60)Using a standard procedure to determine the invariants of the subgroup H weobtain three independent invariants as a solution of the characteristic system,they have a form inv = h, inv = t, inv = v ( h, t ) = e ar ± arert l V ( l, h, t ) . (61)It means also that the complete dependence of the value function V ( l, h, t ) onthe variable l is described just by the factor e − ar ± arert l . If t > r ln( ar ) thenthe value function will be decreasing function in l by choosing the plus sign inthe denominator of the fraction − ar ± are rt l and it will be increasing functionin l if we choose the minus sign in the denominator of the fraction.Because the value function for the optimization problem should be increas-ing function in l so we need to study just this one case. Therefore we chooseas a new dependent variable the function v ( h, t ) = e ar − arert l V ( l, h, t ). Substi-tuting the new dependent variable v ( h, t ) into (18) we get a two dimensionalPDE v t + η h v hh + ( µ − δ ) hv h + rare rt − (cid:16)(cid:16) aδh − (cid:16) rΦ ( t ) are rt − (cid:17)(cid:17) v + v ln v (cid:17) − ( α − r ) σ v − ( α − r ) ηρhσ v h − ( are rt − η ρ h a r v h v = 0 , v ( h, t ) t →∞ −→ . ortfolio optimization in the case of an exponential utility function 23 After Lemma 1 the value function V ( l, h, t ) cannot have an exponential growthin l like we obtain it now. It means that the invariant substitution (61) isinconsistent with the possed optimization problem. Case H ( h ). The last one dimensional subalgebra in the list of the optimalsystem of subalgebras in Table 1 is spanned by e ± e h = < e ± e > = (cid:28) ± (cid:18) − ar (cid:18) e rt Z e − rt d ln Φ ( t ) (cid:19) ∂∂l + 1 r ∂∂t (cid:19) + ∂∂V (cid:29) . (62)According to a standard procedure, we obtain following invariants of thesubgroup H inv = z = l + 1 ar e rt Z e − rt d ln Φ ( t ) − ar ln Φ ( t ) , inv = h, (63) inv = W ( z, h ) = V ( t, l, h ) ∓ rt. (64)Using these invariants (63), (64) as the new variables z, h, W ( z, h ) and substi-tuting them into (18) we obtain a two dimensional PDE on W ( z, h )12 η h W hh + ( µ − δ ) hW h + ( rz + δh ) W z + 1 a w z ln W z (65) − ( α − r ) W z + 2( α − r ) ηρhW z W zh + η ρ σ h W zh σ W zz + 1 a (1 + ln a ) W z ± r = 0In this case we see the inconsistence between the boundary condition (16)which demands that V ( t, l, h ) → t → ∞ and the invariant substitutions(63), (64) which say that the expression V ( t, l, h ) ∓ rt depends just on z, h andnot from the variable t .Totally there are four meaningful reductions of the three dimensional PDE(18) for the case of the EXPn utility function and the general liquidation timedistribution Φ ( t ) by using one dimensional subalgebras of the algebra L EXP n .Just one of these reductions which corresponds to the case H with ω = 0, i.e.the substitutions (53), (54) are consistent with the boundary condition (16)and the two dimensional PDE (55) is a corresponding reduction. This equationcan be studied further with numerical methods.In Table 1 are listed also two and three dimensional subalgebras of L EXP n .Using these subalgebras maybe we can find the deeper reductions of the PDE(18) for instance to ordinary differential equations. Case H ( h ). We take the first two dimensional subalgebra listed in Ta-ble 1, i.e. the subalgebra h = < e , e > . We rewrite the characteristic systemsto the first generator e in terms of the invariants of e (47), (48) then e takesthe form e = ar ∂∂z − W ∂∂W . Solving a corresponding characteristic system weobtain a new invariant inv e = v ( h ) = W ( z, h ) e arz , (66) which we use now as a new dependent variable to reduce the equation (49) toan ODE η h v ′′ − η ρ h (cid:0) v ′ (cid:1) v + (cid:16) ( µ − δ ) σ − ( α − r ) ηρσ (cid:17) hv ′ − (cid:16) arδh + ( α − r ) σ (cid:17) v − rv ln ( − arv ) = 0 . (67)In terms of original variables l, h, t and V ( l, h, t ) the substitution looks asfollows V ( l, h, t ) = v ( h ) e − arz , (68) z = l + 1 ar e rt Z e − rt d ln Φ ( t ) − ar ln Φ ( t ) − ar (1 + ln a ) . Now we obtain a reduction of the three dimensional PDE (18) to an ODE. Butwe cannot use this reduction, because it is inconsistent with the properties ofthe value function V ( l, h, t ) listed in the Lemma 1. The value function is anincreasing function in variable l and V ( l, h, t ) >
0, it means also v ( h ) shouldbe a positive function. From the first expression in (68) follows that V ( l, h, t )is decreasing in z and following in the variable l and from the equation (67)follows that the expression ln ( − arv ) is well defined just for negative functions v ( h ). Case H ( h ). Similar to the previous case we study now the case of h = < ( e + ωe ) , e > and after the substitution V ( t, l, h ) = v ( h ) exp (cid:18) − arl − e rt Z e − rt d ln Φ ( t ) + ln Φ ( t ) (cid:19) (69)we obtain an ODE on the function v ( h ) η h v ′′ − η ρ h (cid:0) v ′ (cid:1) v + (cid:16) ( µ − δ ) σ − ( α − r ) ηρσ (cid:17) hv ′ − (cid:16) arδh + ( α − r ) σ (cid:17) v + r (1 + ln a ) v − rv ln ( − arv ) = 0 . (70)If we can find a positive solution to this equation then we get the solution tothe original optimization problem. It is easy to see that the last term in theequation (70) will be complex-valued if the function v ( h ) >
0. It means thatit is not possible to find a positive solution of this equation. This reduction isnot compatible with the conditions possed on the optimization problem. Likein the previous case we see that also other properties of the value functionlisted in the Lemma 1 cannot be satisfied if the value function takes the form(69).All other two - and three - dimensional subalgebras listed in Table 1 donot give any meaningful reductions of the original equation (18), so we willnot regard them in detail.We studied the complete set of all possible reductions of the original threedimensional PDE (18) to simpler differential equations. We see that not all of ortfolio optimization in the case of an exponential utility function 25 the reductions are reasonable for the optimization problem. Just one of themrepresented by the two dimensional PDE satisfies all conditions. It is the mainresult of this Section and we formulate this result as a theorem
Theorem 5
The main three dimensional HJB equation (18) admits the uniquesymmetry reduction to the two dimensional PDE (55) after the substitutions(56) - (57) which satisfies all conditions of the posed optimization problem.The corresponding investment - consumption strategies are given in (60).
In this paper we study a portfolio optimization problem for a basket consistingof a risk free liquid, risky liquid and risky illiquid assets where the investorprefer to use an exponential utility function. The illiquid asset is sold in arandom moment T with a known distribution of the liquidation time. It is adistribution with a survival function Φ ( t ), satisfying very general conditionslim t →∞ Φ ( t ) E [ U ( c ( t ))] = 0 and Φ ( t ) ∼ e − κt or faster as t → ∞ . Typicallyone suppose that the liquidation time distribution is an exponential one, i.e. Φ ( t ) = e − κt , t ≥ , κ > , or of the Weibull type with Φ ( t ) = e − ( t/λ ) k ,with t ≥ , k > , λ >
0. The Weibull distribution turns to the exponentialdistribution by k = 1 and it can be understand as a generalization of theexponential distribution. Based on the economical motivation we choose k > γ → U HARA γ → −→ U LOG as well as a three dimensional HJB equation(13) corresponding to the HARA utility function formally transforms into theHJB equation with the LOG utility function. Then we proved independentlyfrom the form of the survival function Φ ( t ) that the Lie algebraic structure ofthe PDE with Lie logarithmic utility can be seen as a limit of the algebraicstructure of the PDE with the HARA utility function as γ → Φ ( t ). The results areformulated in Theorem 2 and Theorem 4. We obtained that each of these PDEs admitted the four dimensional Lie algebras L EXP n and L EXP p corre-spondingly. These algebras are isomorph and similar, it means that the studiedPDEs (18) and (39) are equivalent up to the one-to-one analytical substitu-tion. In other words the optimization problems are identical from any point ofview: an economical, analytical or Lie algebraic one.We also investigated a connection between the optimization problem withthe HARA utility function (3) and with the EXPn utility function in Sec-tion 3.1. Even though the HARA utility function is connected to the negativeexponential utility function by γ → ∞ as we mentioned in (4) we do not getthe expected connection between the corresponding optimization problems.Instead of that we obtain quite different structures of the invariant variablesby the study of the symmetry reductions of the main equation (18). In the caseof the HARA utility function a typical invariant variables were the fraction lh and time t . It means that in the case of the HARA or LOG utility function thevalue function depends in the first place from the relation between the valuesof the liquid and illiquid assets. It is completely independent of the absolutevalue of his liquid part or from the absolute value of his illiquid part of wealth.For instance, the investor in the HARA case, as we proved it before, shouldincrease his consumption rapidly if the relation l/h falls, independently howmany millions of dollars the investor has as liquid part at the moment.Here in the case of the exponential utility function the situation is quitedifferent. As follows from equation (53), the behavior of the investor dependsnow on two variables, on the value of the illiquid asset h and on the com-bined variable z which contains the liquid part of wealth and an economicallymodulated time. As a consequence, the absolute value of the illiquid part ofwealth plays a large role. The variable z tells us that the influence of a largeamount of a liquid asset plays the same role as the possibility to wait a longtime. By the way, this difference in the behavior of the invariant variables andthe radical change of the investment consumption strategies is to explain bythe fact that the risk tolerance in the case of the HARA utility function is alinear function of the consumption c and in the case of the exponential utilitythe risk tolerance is just a constant R ( c ) = a .A further difference between the optimization problems with the HARAand EXPn utility functions is related to the structure of the admitted Lie alge-bras. In the cases of the HARA and LOG utility functions the correspondingthree dimensional PDEs have admitted three dimensional main Lie algebras.Just by the special choice of a liquidation time distribution, i.e. only for theexponential function Φ ( t ) = e − κt we got an extension of these Lie algebras tothe four dimensional ones. Here in the cases of the exponential utility func-tions, independently EXPn or EXPp, we obtain from the beginning the fourdimensional Lie algebras as the symmetry algebras of the corresponding PDEs.It is remarkable that in these cases the four dimensional Lie algebras do notallow any extension independently from the form of Φ ( t ). It can be seen bythe solving of the system of equation (27) in the proof of Theorem 2.In the previous paper [3] we proved that the algebra L LOG can be obtainedas a limit case of L HARA by γ →
0. Here we see that L HARA (or correspond- ortfolio optimization in the case of an exponential utility function 27 ingly L HARA ) and L EXP n are quite different and they do not connect by γ → ∞ as well as they do not have any connections between analytical struc-tures of their generators independently on the form of the liquidation timedistribution.In our paper we pay attention to the internal structure of the admittedalgebra L EXP n to obtain convenient and useful reductions of the main equa-tion (18). Further on we use the system of optimal subalgebras provided in[18] and get corresponding nonequivalent invariant reductions of the three di-mensional PDEs (18) to two dimensional PDEs. They describe the completeset of solutions that can not be transformed into each other with the help ofthe transformations of the admitted symmetry group. We show that the threedimensional PDE can be reduced to a corresponding two dimensional ones inSection 4. The low dimensional PDEs are much more convenient for furtheranalytical or numerical studies. We also provide the formulas for the optimalinvestment-consumption policies in invariant variables using solutions of thereduced equation. We demonstrate that between meaningful reductions thereexists one (55) which is consistent with the boundary condition (16) and withthe expected properties of the value function.We remark also a different level of influence of the parameters on theHJB equation and on the admitted Lie algebraic structure. The HJB equationcontains seven parameters r, α, σ , µ, δ, η, ρ which define the behavior of theliquid and illiquid asset, and one parameter a which is fixed by the exponentialutility function. There are also some parameters which define the liquidationtime distribution, for instance, it is the parameter κ if we take the exponentialdistribution with Φ ( t ) = e − κt or two parameters λ and k if we take the Weibulldistribution with Φ ( t ) = e − ( t/λ ) k . If we look at the structure of the Lie algebrasprovided in Theorem 2 and Theorem 4 we see that the generators of thealgebras are defined by the parameters r, a and parameters of the liquidationtime distribution only. The algebras change their structure if one or some ofthese parameters vanishing. Roughly said the most influence on the form ofthe solution of this optimization problem has interest rate r , the type of theinvestor’s utility function and a marked defined liquidation time distributionfor the illiquid asset. These parameters define the invariant variables and theanalytical structure of the solutions.Summing up, we carry a complete Lie group analysis for the optimizationproblems with negative and positive exponential utility functions and for ageneral liquidation time distribution. We determine the reduced equation andcorresponding optimal policies as it formulates in the Theorem 5. Acknowledgements
The author is thankful to prof. L. Vostrikova-Jacod for interestingdiscussions and for organizing a very successful conference
Advanced Methods in MathematicalFinance, Angers, 2018, France , where the author got the idea to write this paper.8 Ljudmila A. Bordag
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