Lifetime Ruin under High-watermark Fees and Drift Uncertainty
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Lifetime Ruin under HWM Fees and Drift Uncertainty
Junbeom Lee · Xiang Yu · Chao ZhouAbstract
This paper aims to make a new contribution to the study of lifetime ruin problem byconsidering investment in two hedge funds with high-watermark fees and drift uncertainty. Due tomulti-dimensional performance fees that are charged whenever each fund profit exceeds its historicalmaximum, the value function is expected to be multi-dimensional. New mathematical challenges ariseas the standard dimension reduction cannot be applied, and the convexity of the value function andIsaacs condition may not hold in our ruin probability minimization problem with drift uncertainty.We propose to employ the stochastic Perron’s method to characterize the value function as the uniqueviscosity solution to the associated
Hamilton–Jacobi–Bellman (HJB) equation without resorting tothe proof of dynamic programming principle. The required comparison principle is also established inour setting to close the loop of stochastic Perron’s method.
Keywords
Lifetime ruin · multiple hedge funds · high-watermark fees · drift uncertainty · stochasticPerron’s method · comparison principle Hedge funds have existed for many decades in financial markets and have become increasingly popularin recent times. As opposed to the individual investment, hedge funds pool capital and invest in avariety of assets and it is administered by professionals. Hedge fund managers charge performancefees for their service to individual investors as some regular fees proportional to fund’s componentassets plus a fraction of the fund’s profits. The most common scheme entails annual fees of ofassets and of fund profit whenever the profit exceeds its historical maximum—the so-called high-watermark . In the present paper, we are interested in investment opportunities among several hedgefunds and we intend to study a stochastic control problem given the path-dependent trading frictionsas multi-dimensional high-watermark fees. Junbeom LeeDepartment of Sales and Trading, Yuanta Securities Korea, 04538 Seoul, KoreaTel.: +822-3770-5993Fax: +822-3770-5998E-mail: [email protected] YuDepartment of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong KongTel.: +852-2766-6930Fax: +852-2362-9045E-mail: [email protected] ZhouDepartment of Mathematics, Institute of Operations Research and Analytics and Suzhou Research Institute, NationalUniversity of Singapore, Singapore 119076, SingaporeTel.: +65-6516-2938Fax: +65-6779-5452E-mail: [email protected] Junbeom Lee et al.
The existing research on high-watermark fees mainly has focused on the asset management prob-lem from the point of view of the fund manager, see some examples by [21], [29], [1], [23] and [24].Meanwhile, the high-watermark process is also mathematically related to wealth drawdown constraintsstudied in [22], [17], [19] and also discussed in [15] after the transformation into expectation constraint.Recently, the high-watermark fees have been incorporated also into Merton problem for individualinvestor together with consumption choice in [26] and [27]. In the presence with consumption control,analytical solutions can no longer be promised as in some of the previous work for fund managers. Afteridentifying the state processes, the path-dependent feature from high-watermark fees can be hiddenso that the dynamic programming argument can be recalled to derive the HJB equation heuristically.The homogeneity of power utility function in [26] and [27] enables the key dimension reduction ofthe value function and the associated HJB equations can be reduced into ODE problems. Althoughthe regularity can hardly be expected, classical Perron’s method can be applied and the nice upgradeof regularity of the viscosity solution can be exercised afterwards using the convexity property ofthe transformed one-dimensional value function. As the last step, the verification theorem can beconcluded with the aid of the smoothness of value function and standard Itô calculus.In the present paper, we focus on the standpoint of the individual investor who confronts multiplehedge fund accounts in the market. However, we aim to minimize the probability that the investoroutlives her wealth, also known as the probability of lifetime ruin, instead of the Merton problemon portfolio or consumption. We determine the optimal investment strategy of an individual amongsome hedge funds who targets a given rate of consumption by minimizing the probability that the ruinoccurs before the death time. For the studies of lifetime ruin probability problem, readers can refer to[34, 10, 12, 11, 35]. In contrast to Merton problem, the dimension reduction of the value function willfail for our probability minimization problem. The auxiliary controlled state process, the so-calledprocess of distance to pay performance fees defined in (2.7), can no longer be absorbed to simplify thePDE problem. Furthermore, comparing with [26] and [27] or the lifetime ruin problem with ambiguityaversion in [11], we need to handle a genuine multi-dimensional control problem with reflections asthere exist multiple hedge funds in the market. In other words, the distance process itself is alreadymulti-dimensional, which spurs many new mathematical challenges. To wit, one can still exploit theclassical Perron’s method as in [26], [27] and [11], and obtain the existence of viscosity solution to theassociated HJB equation. Nevertheless, the upgrade of regularity of the viscosity solution can hardlybe attained for our multi-dimensional problem. Consequently, the proof of verification theorem, whichrequires certain regularity of the solution, cannot be completed. To relate the value function to theviscosity solution in our setting using classical Perron’s method, we have to provide the technical proofof dynamic programming principle at the beginning.In addition, the individual investor usually cannot keep a real-time track of the performance ofhedge funds from fund managers. Moreover, a reliable estimation of the return from hedge fundthat consists of a bunch of various assets is almost impossible in practice. Even in the hedge fundperformance report, the predicted future return in short term from fund manager is provided as acertain range instead of a fixed number. It is more realistic to assume that the investor allows driftmisspecification and starts with a family of plausible probability measures of the underlying model.This leads to a robust investment strategy with Knightian model uncertainty. In particular, we assumethat the investor would like to use the available data as a reference model and work on a robust controlproblem with the penalty on other plausible models based on the deviation from the reference one. Onenew mathematical challenge from this formulation is that the value function may lose convexity forsome parameters and the Issacs condition may fail. Adding our previous difficulties coming from multi-dimensional performance fees, the feedback optimal investment strategy and the saddle point choiceof probability measure cannot be obtained. The combination of market imperfections such as tradingfrictions together with model ambiguity renders many problems mathematically intractable. Someworkable examples in this direction can only be found in robust Merton problem with proportionaltransaction costs, see [28], [14] and [18]. The methodology introduced in these paper may not workfor our purpose with path-dependent high-watermark fees. ifetime Ruin under HWM Fees and Drift Uncertainty 3
To tackle our stochastic control problem, we choose to employ the stochastic Perron’s method(SPM) and characterize the value function as the unique viscosity solution to the associated HJBequation. This stochastic version of Perron’s method, introduced by [7], can avoid the technical andlengthy proof of dynamic programming principle (DPP) and can obtain it as a by-product. We chooseSPM over the weak DPP introduced in [13] because SPM can better handle the path-dependentstructure of our control problem with additional model uncertainty. Let us note that the comparisonprinciple is needed anyway in both methods. SPM requires the comparison principle to complete thesqueeze argument and establish the equivalence between value function and the viscosity solution,while weak DPP needs the comparison principle to guarantee the uniqueness of the viscosity solutionto the associated HJB equation. We actually find that the proof of comparison principle for SPM isrelatively easier as the applicable class of state processes can be larger than that of weak DPP. Werefer a short list of previous work on stochastic control using SPM such as [7], [9], [5], [6], [8], [30],[31], [32], [4] and [33].To establish the viscosity semisolution property of stochastic envelopes, it is usually crucial tocheck the boundary viscosity semisolution property. In our framework, we can take advantage of theproblem structure from lifetime ruin probability minimization and explicitly construct a stochasticsuper-solution and a stochastic sub-solution which satisfy the desired boundary conditions. We notethat our arguments using stochastic Perron’s method differ from [12] that solves the lifetime ruinproblem with transaction costs and [4] that examines the robust optimal switching problem. Somenontrivial issues need to be carefully addressed, which are caused by the uncertainty of drift term andthe structure of the auxiliary state process defined as the distance to pay fees. The path-dependentrunning maximum part coming from high-watermark fees do not appear in [12] nor [4], which deservessome novel and tailor-made treatment in the present paper.The rest of the paper is organized as follows. Section 2 introduces the market model with multiplehedge funds and related high-watermark fees, the default time as well as the set up with drift uncer-tainty. The robust lifetime ruin problem is defined afterwards. In Section 3, we derive the associatedHJB equation for the control problem heuristically and define the viscosity solution accordingly. Themain theorem to characterize the value function as the unique viscosity solution is presented. Section4 provides the proof of all main results using stochastic Perron’s method. The proof of the comparisonprinciple of the HJB equation is also reported therein. ( Ω, G , G , P ) be a filtered probability space such that G satisfies the usual conditions and E denotethe expectation operator under P . Let ( W t ) t ≥ denote an independent -dimensional Brownian motionand F := ( F t ) t ≥ be the natural filtration generated by ( W t ) t ≥ and it is assumed that F t ⊂ G t . Later,we will characterize G = ( G t ) t ≥ more precisely.We consider the financial market consisting of one risk-less bond with interest rate r > and twohedge fund accounts ( F it ) t ≥ , i ∈ { , } , described by d F it = µ i F it d t + σ i F it d W t , for some constant µ i ≥ and constant vector σ i ∈ R . To simplify the presentation, we only focus ontwo hedge funds henceforth. The mathematical arguments and main results can be easily extended tothe multi-dimensional case of N ≥ hedge funds without any technical difficulty. We shall denote F := (cid:20) F F (cid:21) , µ := (cid:20) µ µ (cid:21) , σ := (cid:20) σ σ (cid:21) , and assume that σ is invertible. Junbeom Lee et al.
Contrary to some standard investment problems in liquid risky assets such as stocks, we areconsidering the model when the investor is facing the wealth allocation among some hedge fundaccounts that charge proportional fees on the profit as trading frictions. In particular, the investorneeds to pay some high-watermark fees to the fund manager whenever the accumulative profit reachesthe highest value. The 2/20-rule is common for hedge funds in the sense that per year of thetotal investment and of the additional profits are paid to the fund manager whenever the high-watermark exceeds the previously attained profit maximum. To explain this in a more explicit manner,let π = ( π , π ) ∈ R denote the investment strategy in two hedge funds F . The accumulativeprofit P π = [ P ,π , P ,π ] ⊤ from the hedge fund before the deduction of the high-watermark fee, ischaracterized by the stochastic integral P i,πt := Z t π is d F is F is . (2.1)In practice, there is also a benchmark profit from which the manager’s performance is measured, see[27], and the high-watermark fee is deducted when the high-watermark of the fund is higher thanthe benchmark level. The initial high-watermark fee is denoted by some non-negative constant vector y = [ y , y ] ⊤ . Let F B ∈ R be the benchmark process given by d F Bt = diag( F Bt )[ µ B d t + σ B d W t ] , for some µ B ∈ R , σ B ∈ R × . We denote by B π = [ B ,π , B ,π ] ⊤ the accumulated benchmark profitprocess if the same strategy π is adopted, i.e., B i,πt := Z t π is d F Bs F Bs . Let q = [ q , q ] ⊤ represent the proportional rates of high-watermark fee of each hedge fund and P y,π = [ P ,y,π , P ,y,π ] be the realized profit after charging the high-watermark fee. Moreover, wedefine M i,y,π as the historical high-watermark of the i -th hedge fund. The realized profit process P i,y,π , i ∈ { , } , is given by ( d P i,y,πt := d P i,πt − q i d M i,y,πt , P i,y,π = 0 ,M i,y,πt := sup ≤ s ≤ t (cid:8) ( P i,y,πs − B i,πs ) ∨ y i (cid:9) , y i ≥ . (2.2)To represent (2.2) in a more convenient form, let us define M i,y,πt := sup ≤ s ≤ t (cid:8) ( P i,πs − B i,πs ) ∨ y i (cid:9) , i ∈ { , } . (2.3)Then by (2.2), M i,y,πt − y i = sup ≤ s ≤ t (cid:8) [ P i,πs − B i,πs ] − y i (cid:9) + = sup ≤ s 00 1 + q (cid:21) . Sometimes, we omit the superscripts x, y, π for simplicity and we also denote Z := ( X, Y , Y ) , z := ( x, y , y ) . ( W t ) t ≥ , which is defined as a random variable τ D : ( Ω, G ) → ( R + , B ( R + )) satisfying P ( τ D = 0) = 0 and P ( τ D > t ) > , for any t ≥ . From this point onward, the full marketfiltration G is precisely defined by G = ( G t ) t ≥ := ( F t ∨ σ ( { τ D ≤ u } : u ≤ t )) t ≥ . It is worth notingthat τ D is a G -stopping time but may fail to be an F -stopping time. In what follows, we assume thatthere exists a constant λ D > such that G Dt := P ( τ D > t |F t ) = e − λ D t . We call λ D the intensity of default time τ D with respect to F . Under this assumption, ( M Dt ) t ≥ := (cid:0) τ D ≤ t − λ D ( t ∧ τ D ) (cid:1) t ≥ (2.9)is a ( G ) -martingale. Moreover, for any F -martingale ( ξ t ) t ≥ , ( ξ t ∧ τ D ) t ≥ is a G -martingale. Therefore, ( W t ) t ≥ is a G -Brownian motion; see [20]. Junbeom Lee et al. Remark 2.1 1. In view of the existence of the intensity, τ D is totally inaccessible . In other words,the default of the investor comes with total surprise. On the other hand, a ruin time , whichwill be introduced later, is defined as a hitting time that the controlled wealth process crossesa given level and it is therefore predictable . In the present paper, we envision an individualinvestor who chooses her portfolio to minimize the probability involving the ruin time beforethe default time occurs.2. Although investment strategies are defined as G -adapted processes, the full filtration G isnot fully observable for the investor. However, in this filtration setup, for any G -adaptedprocess, we can find an F -reduction, where F is the observable information. Therefore, thestrictly G -adapted strategies only describe an immediate action taken by the investor at thedefault time. Note that an F -adapted process is not necessarily determined independently ofthe default time τ D , because the (constant) default intensity λ D is trivially F -adapted.2.3 Life Time Ruin Problem with Drift UncertaintyBased on previous building blocks, we are ready to introduce the primary stochastic control problemthat the investor confronts. In particular, the investor concerns the viability of her investment beforethe default time and she wishes to maintain the amount of her wealth above a certain level, say R ≥ ,before the default time happens. To this end, it is natural to introduce the so-called ruin time τ x,y,πR := inf { t ≥ X x,y,πt ≤ R } . Mathematically speaking, the investor chooses π from an admissible set A so that τ R occurs as lateas possible. As the investor cannot control the totally inaccessible time τ D , she aims to minimize theprobability that the ruin occurs before the default time.However, we consider a more practical scenario in the present paper that the return of hedge fundsmay not be revealed by fund manager to the investor very frequently. The investor usually can onlyget access to the performance of the fund from some reports on regular dates. Moreover, as the hedgefund consists of components from various assets, the estimation of return can hardly be provided ona timely basis. Based on these observations, it is reasonable to assume that the investor may not havea precise knowledge of the dynamics of hedge funds. This naturally leads to the so-called Knightianmodel uncertainty.In this paper, we will only focus on the case with drift uncertainty, i.e. the investor conceivesa family of plausible return terms from the hedge fund dynamics and proceeds to solve the controlproblem in a robust sense. Indeed, the precise estimation of the drift term is much more challengingthan the estimation of volatility term, which motivates our research. In particular, we aim to minimizethe probability of lifetime ruin by choosing wealth allocation among multiple hedge funds with high-watermark fees and drift uncertainty, which is new to the existing literature. To this end, let us firstintroduce a class of probability measures equivalent to the reference probability P and denote thisclass by L . Definition 2.1 Q ∈ L if for any ≤ t , d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G t = exp (cid:18) − Z t k θ s k d s + Z t θ ⊤ s d W s (cid:19) , (2.10)for some G -predictable process θ valued in a closed set L ⊆ R containing such that E Q h R ∞ e − λ D s k θ s k d s i < ∞ , E h exp (cid:16) R t k θ s k d s (cid:17)i < ∞ , for any t ≥ . ifetime Ruin under HWM Fees and Drift Uncertainty 7 In what follows, an equivalent measure Q is generated by θ by the representation in (2.10), andwe call Q the θ -measure. The investor intends to minimize the ruin probability under some Q ∈ L ,but the deviation of the measure from P is penalized by a relative entropy process up to the defaulttime τ D : H t ( Q | P ) := E Q h log (cid:16) d Q d P (cid:12)(cid:12)(cid:12) G t (cid:17)i , for t ≥ . (2.11)The investor’s robust stochastic control problem is then defined by V ( x, y ; ε ) := inf π ∈ A sup Q ∈ L n Q ( τ x,y,πR < τ D ) − ε H τ D ( Q | P ) o . (2.12)Here A denotes the set of all admissible controls defined in the following sense. Definition 2.2 π ∈ A if π is G -predictable and valued in a compact set K ⊆ R such that (0 , ∈ K . Remark 2.2 The coefficient ε in the penalty term of (2.12) corresponds to the investor’s level of modelambiguity about the reference probability P . For instance, the case ε → implies that sup Q ∈ L n Q ( τ x,y,πR < τ D ) − ε H τ D ( Q | P ) o → P ( τ x,y,πR < τ D ) , which indicates that the investor is completely confident about the probability measure P . On theother hand, if the agent is extremely uncertain as ε → ∞ , we get that sup Q ∈ L n Q ( τ x,y,πR < τ D ) − ε H τ D ( Q | P ) o → sup Q ∈ L Q ( τ x,y,πR < τ D ) , which reduces to the best case scenario. It is worth noting that the formulation involving the penaltyterm only works for drift uncertainty. If some plausible probabilities are mutually singular due tovolatility uncertainty, i.e. there is no dominating reference probability P , the entropy cannot be definedas in (2.11). Another interesting issue we can consider in the robust framework is to incorporate theinvestor’s ambiguity attitude towards a given set of plausible priors. Similar to [25], one can employthe alpha-maxmin preference and formulate the ruin probability problem under model uncertainty as inf π ∈ A " α sup Q ∈ L Q ( τ x,y,πR < τ D ) + (1 − α ) inf Q ∈ L Q ( τ x,y,πR < τ D ) . This formulation allows for both drift and volatility uncertainty and the constant coefficient α ∈ [0 , can represent how much ambiguity averse the investor is. Nevertheless, this problem becomes timeinconsistent and we need to look for some equilibrium portfolio strategies instead of the optimal one,which is beyond the scope of this paper and will be left as future research. Remark 2.3 The compactness of K in the definition of admissible set A can be understood that theinvestor does not take an extreme strategy and the immediate liquidation is also admissible. Moreover,as π is G -predictable, it is also F -predictable before τ D . Therefore, there is a unique continuous P π satisfying (2.1). Thanks to (2.3) and (2.5), P y,π is well-defined. More importantly, the compactnessof K is necessary for the associated HJB equation to be continuous. Otherwise, it becomes difficult toprove the comparison principle for its viscosity solutions because the typical doubling argument relieson Crandall-Ishii’s lemma and the closure of super/sub-jets , which require the compactness of K . Inother words, if the comparison principle is already guaranteed, we can relax the conditions on A onlywith care for P y,π to be well-defined. Junbeom Lee et al. In this section, we first heuristically derive the HJB equation associated with the value functionusing dynamic programming argument or martingale optimality principle. For technical reason, whendefault occurs, we assign a coffin state ∆ to the underlying process Z . Moreover, for any domain inwhat follows, we consider its one point compactification and any function u is extended by assigning u ( ∆ ) = 0 . Denote the ( Q , G ) -Brownian motion by W Q , where Q is generated by θ . For t < τ D , (2.8)can be written as ( d X x,y,πt = [ rX x,y,πt − c + π ⊤ t ( µ r∆ + σθ )] d t + π ⊤ t σ d W Q t − q ⊤ d M y,πt , X = x, d Y y,πt = − diag( π t )[( µ B∆ + σ B∆ θ ) d t + σ B∆ d W Q t ] + diag( + q ) d M y,πt , Y = y. (3.1)To obtain the associated HJB equation, we apply Itô’s formula to a smooth function ϕ that d ϕ ( Z t ) − ε k θ t k d t = (cid:2) − λ D [ ϕ ( Z t ) − ϕ ( ∆ )] + ( rX t − c ) ϕ x + A π t ,θ t [ ϕ ]( Z t ) (cid:3) d t + X i =1 , (cid:2) q i ϕ x ( Z t ) − (1 + q i ) ϕ y i ( Z t ) (cid:3) Y it =0 d M it + (cid:2) ϕ x ( Z t ) π ⊤ t σ − ∇ y ϕ ( Z t ) ⊤ diag( π t ) σ B∆ (cid:3) d W Q t + [ ϕ ( ∆ ) − ϕ ( Z t − )] d M Dt , (3.2)where ∇ y ϕ := [ ∂ y ϕ, ∂ y ϕ ] ⊤ and A π,θ [ ϕ ]( x, y , y ) := − ε k θ k + b [ π, θ ] ⊤ ∇ ϕ + 12 Tr( Σ [ π ] ∇ ϕ ) ,b [ π, θ ] := (cid:20) π ⊤ ( µ r∆ + σθ ) − diag( π )( µ B∆ + σ B∆ θ ) (cid:21) ,Σ [ π ] := (cid:20) π ⊤ σ − diag( π ) σ B∆ (cid:21) (cid:20) π ⊤ σ − diag( π ) σ B∆ (cid:21) ⊤ . Recall that ϕ ( ∆ ) = 0 in (3.2). Now, let us deduce related boundary conditions. Recalling (2.12), wecan set V ( R, y , y ) = 1 for any y i ≥ . In addition, if X t = c/r at t ≥ , the optimal strategy isliquidating the risky position so that X s = c/r for any s ≥ t . Therefore, V ( c/r, y , y ) = 0 for any y i ≥ . Thus, motivated by these boundary conditions, we need to consider the following regions andboundaries O := { ( x, y , y ) : R < x < c/r, y ≥ , y ≥ } , O + := { ( x, y , y ) : R < x < c/r, y > , y > } ,∂ O i := { ( x, y , y ) ∈ O : R < x < c/r, y i = 0 } ,∂ O := { ( x, y , y ) ∈ O : R < x < c/r, y = 0 or y = 0 } ,∂ O R := { ( R, y , y ) : y > , y > } ,∂ O c/r := { ( c/r, y , y ) : y > , y > } . Note that O = O + ∪ ∂ O , ∂ O = ∂ O R ∪ ∂ O c/r ∪ ∂ O ∪ ∂ O , and ∂ O = ∂ O ∪ ∂ O . Moreover, for anyset A , we let cl ( A ) denote the closure of A in what follows. We then consider the following operators F [ ϕ ]( z ) := λ D ϕ ( z ) − ( rx − c ) ϕ x ( z ) − inf π ∈K sup θ ∈L A π,θ [ ϕ ]( z ) , B i [ ϕ ]( z ) := q i ϕ x ( z ) − (1 + q i ) ϕ y i ( z ) , i ∈ { , } , ifetime Ruin under HWM Fees and Drift Uncertainty 9 and the associated HJB equation can be (formally) written as F [ ϕ ]( z ) = 0 , on z ∈ O + , B [ ϕ ]( z ) = 0 , on z ∈ ∂ O , B [ ϕ ]( z ) = 0 , on z ∈ ∂ O ,ϕ ( z ) = 1 , on z ∈ ∂ O R ,ϕ ( z ) = 0 on z ∈ ∂ O c/r . (3.3)Our ultimate goal is to show that the value function V defined in (2.12) is the unique viscosity solutionof the HJB equation (3.3).To this end, we first need to be careful for the boundary conditions on ∂ O , which should bedefined using semi-continuous envelope of viscosity solutions. To be precise, we denote the lower (resp.upper) semi-continuous envelope of B i , i ∈ { , } , by B ∗ (resp. B ∗ ) . On ∂ O , we will consider B ∗ [ ϕ ] := B [ ϕ ] , on ∂ O \ ∂ O , B [ ϕ ] , on ∂ O \ ∂ O , min {B [ ϕ ] , B [ ϕ ] } , on ∂ O ∩ ∂ O , and B ∗ is defined in the same way by replacing B ∗ = min {B , B } using B ∗ = max {B , B } on theboundary ∂ O ∩ ∂ O . Furthermore, we denoteUSC b ( A ) := { bounded u.s.c functions on A } , LSC b ( A ) := { bounded l.s.c functions on A } . The precise definition of viscosity sub/super solutions is given as below. Definition 3.1 (Viscosity solution) (i) v ∈ USC b (cl( O )) is a viscosity sub-solution of (3.3) if for any test function ϕ such that z ∈ O is a maximum point of v − ϕ at zero, we have F [ ϕ ]( z ) ≤ , on z ∈ O + , min (cid:8) F [ ϕ ]( z ) , B ∗ [ ϕ ]( z ) (cid:9) ≤ , on z ∈ ∂ O ,v ( z ) ≤ , on z ∈ ∂ O R ,v ( z ) ≤ , on z ∈ ∂ O c/r . (3.4)(ii) v ∈ LSC b (cl( O )) is a viscosity super-solution of (3.3) if for any test function ϕ such that z ∈ O is a minimum point of v − ϕ at zero, we have F [ ϕ ]( z ) ≥ , on z ∈ O + , max (cid:8) F [ ϕ ]( z ) , B ∗ [ ϕ ]( z ) (cid:9) ≥ , on z ∈ ∂ O ,v ( z ) ≥ , on z ∈ ∂ O R ,v ( z ) ≥ , on z ∈ ∂ O c/r . (3.5)(iii) v is a viscosity solution of (3.3) if v is both viscosity sub-solution and super-solution. Remark 3.1 The definition of viscosity solutions is inextricably involved with min/max when theboundary conditions are given on derivatives. Consider ( p, X ) ∈ J , ±O ϕ ( z ) for some z ∈ ∂ O , where J , ±O denote the closure of the second order superjet/subjet. Then there exists ( z n , p n , X n ) ∈ J , ±O such that ( z n , p n , X n ) → ( z , p, X ) . However, in this case, we cannot guarantee that z n ∈ ∂ O for any n ∈ N . For more detailed discussion, readers can refer to Section 7 in [16]. Now, we are ready to state the main result of this paper. Theorem 3.1 (The Main Theorem) The value function V , defined at (2.12), is a unique viscositysolution of the HJB equation (3.3). The proof of the theorem is split into several steps, which will be provided in the next sections.In summary, the first step is to define stochastic sub/super-solutions . We continue to show that supre-mum (resp. infimum) of stochastic sub-solutions (resp. stochastic super-solutions ) is a viscosity super-solution (resp. sub-solution). Then the main theorem can be concluded with the help of the followingcomparison principle of the HJB equation, whose proof is reported in the next section. Proposition 3.1 (Comparison Principle) Assume u and v be a sub-solution and super-solutionof (3.3), respectively. Then u ≤ v in cl( O ) . This section contributes to the proof of Theorem 3.1 using stochastic Perron’s method, which helps usto avoid the lengthy and technical proof of dynamic programming principle. To begin, we first needthe concept of random initial conditions and exit times . Definition 4.1 We call ( τ, ξ ) a random initial condition if τ is a G -stopping time valued in J , τ D K , ξ = ( ξ X , ξ Y , ξ Y ) is a G τ -measurable random variable valued in cl( O ) ∪ { ∆ } , and ξ = ∆ if and onlyif τ = τ D . We denote R as the set of all random initial conditions . Definition 4.2 The exit time of X τ,ξ,π from O , denoted by τ τ,ξ,πE , is defined by τ τ,ξ,πE := inf { t ≥ τ : X τ,ξ,πt / ∈ O} . stochastic sub-solutions of (3.3) and establishes theresult that the stochastic envelope of stochastic sub-solutions is a viscosity super-solution of (3.3).In a nutshell, stochastic sub-solutions are functions that become G -submartingales by operating on Z = ( X, Y ) . The purpose of defining the stochastic sub-solutions is to provide one direction of dynamicprogramming principle to some extent that inf π ∈ A sup Q ∈ L E Q (cid:20) V ( Z τ,ξ,πρ ) − ε Z ρτ k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≥ V ( ξ ) , (4.1)for any random initial condition ( τ, ξ ) and G -stopping time ρ such that τ ≤ ρ . Definition 4.3 (Stochastic sub-solutions) If v ∈ LSC b (cl( O )) satisfies(SB1) v ≤ on ∂ O R and v ≤ on ∂ O c/r ,(SB2) for any ( τ, ξ ) ∈ R , π ∈ A , and G -stopping time ρ ∈ [ τ, τ τ,ξ,πE ] , there exists a θ -measure Q ∈ L such that E Q (cid:20) v ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≥ v ( ξ ) , (4.2)where v in (4.2) is understood as its extension to cl( O ) ∪ { ∆ } by allocating v ( ∆ ) = 0 ,then v is called a stochastic sub-solution of (3.3). In addition, we denote by V − the class of allstochastic sub-solutions of (3.3). ifetime Ruin under HWM Fees and Drift Uncertainty 11 For the remaining of the paper, stochastic sub-solutions means stochastic sub-solutions of (3.3).In addition, to understand the meaning of the extension up to ∆ , we can simply consider τ = τ D andderive that τ = τ E = τ D = ρ and ξ = ∆ . Then both sides in (4.2) equal to zero and the equation istrivially satisfied. Remark 4.1 Note that we do not impose the oblique-type boundary condition B arising from thehigh-watermark fees in the definition of stochastic sub-solutions. The Dirichlet boundary conditionsare from the associated financial problems, namely the ruin probability minimization problem. Suchboundary conditions are invariant given the underlying processes, i.e., the same Dirichlet boundaryconditions are imposed regardless of the SDE for Z = ( X, Y ) . However, the oblique-type boundarycondition B comes from the structure of the process, the running maximum of the process, as F does.Therefore, we can deal with B and F together in the same manner in applying SPM. This in turnshows another advantage of stochastic Perron’s method that is effective to handle control problemwith high-watermark fee, especially with multiple hedge funds. Therefore, it is redundant to includethe oblique-type boundary condition in Definition 4.3, which actually will make the argument morecomplicated because it is difficult to verify that V − is closed under the maximum operation withcondition B .Our first task is to find one stochastic sub-solution so that V − is not empty. One can think of(4.2) as an upper-bound, in other words, stochastic sub-solution can be found by considering a “bettersituation”. If there is no fee in reaching the high-watermark, the case is clearly better for the investor.The minimal ruin probability in this frictionless market was already studied by [34, 10], which willturn out to be a stochastic sub-solution in our case. Put U ( x ) := ((cid:0) c − rxc − rR (cid:1) κ , R ≤ x ≤ c/r, , c/r < x,κ := 12 r (cid:2) ( r + λ D + R ) + q ( r + λ D + R ) − rλ D (cid:3) ,Σ := 12 µ ⊤ ∆ ( σσ ⊤ ) − µ ∆ . Before proceeding, note that U is a solution of the following differential equation: ( λ D U ( x ) + Σ [ U ′ ( x )] / U ′′ ( x ) + ( c − rx ) U ′ ( x ) = 0 , R < x < c/r, U ( R ) = 1 , U ( c/r ) = 0 . (4.3) Lemma 4.1 Let ψ − ( x, y ) := U ( x ) . Then ψ − ∈ V − .Proof. It is obvious that ψ − is continuous and satisfies (SB1) in definition 4.3. To prove that ψ − isa stochastic sub-solution, let us consider an arbitrary random initial condition ( τ, ξ ) , π ∈ A , and a G -stopping time ρ ∈ [ τ, τ τ,ξ,πE ] . Then we will show that (SB2) is satisfied with the reference measure P . In other words, we choose θ = 0 (4.4)in the representation of (2.10). For the rest of this proof, we omit the super-scripts τ, ξ, π for simplicity.Define a process ( X t ) t ≥ given by X τ = X τ , X τ D := ∆ , and d X t = [ rX t − c + π ⊤ t µ ∆ ] d t + π ⊤ t σ d W t , for t < τ D . In other words, X is a process without high-watermark fees, thus X ≤ X on J τ, τ E K . As U is non-increasing in [ R, ∞ ) , E [ ψ − ( X ρ , Y ρ ) |G τ ] = E [ ρ<τ D U ( X ρ ) |G τ ] ≥ E [ ρ<τ D U ( X ρ ) |G τ ] = E [ U ( X ρ ) |G τ ] (4.5) Then, it suffices to show E [ U ( X ρ ) |G τ ] ≥ U ( X τ )(= ψ − ( ξ )) . We first consider the event U := { X τ ∈ [ R, c/r ) } ∈ G τ and let ν := inf { t ≥ τ : X t ≥ c/r } . On the event U , E [ U ( X ρ ) |G τ ] ≥ E [ U ( X ρ ∧ ν ) |G τ ] . In addition, applying Itô’s formula on the event U yields U ( X ρ ∧ ν ) = U ( X τ ) + Z ρ ∧ ντ n U ′ ( X t )[( rX t − c ) + π ⊤ t µ ∆ ] + U ′′ ( X t ) 12 k σ ⊤ π t k − λ D U ( X t ) o d t + Z ρ ∧ ντ U ′ ( X t ) π ⊤ t σ d W t − Z ρ ∧ ντ U ( X t − ) d M Dt The d t -integral term is non-negative. Moreover, U , U ′ , and π are bounded, so the local martingalesterms are martingales. Therefore, we have U E [ U ( X ρ ∧ ν ) | G τ ] ≥ U U ( X τ ) . (4.6)On the other hand, on the event U c = { X τ ∈ [ c/r, ∞ ) ∪ { ∆ }} , it clearly follows that U ( X τ ) = 0 ≤ E [ U ( X ρ ) |G τ ] . Therefore, thanks to (4.5)-(4.6), we obtain E [ ψ − ( X ρ , Y ρ ) |G τ ] ≥ E [ U ( X ρ ) |G τ ] = E [ U U ( X ρ ) + U c U ( X ρ ) |G τ ] ≥ U U ( X τ ) = U ( X τ )= ψ − ( ξ ) . (4.7)Thus by (4.4), ψ − satisfies (SB2).To show the stochastic envelope of stochastic sub-solutions is a viscosity super-solution, we firstshow V − is closed under maximum operation. Lemma 4.2 If v , v ∈ V − , then v ∨ v ∈ V − .Proof. It is easy to check that v ∨ v ∈ LSC b (cl( O )) and v ∨ v satisfies (SB1) in definition 4.3. Let ( τ, ξ ) ∈ R , π ∈ A , ρ be a G -stopping time valued in interval [ τ, τ τ,ξ,πE ] . Because v and v are stochasticsub-solutions, there exist Q and Q satisfying (SB2). We denote by θ i , i ∈ { , } , the processes thatgenerate Q i . To find the measure satisfying (SB2) for v ∨ v , we define A := { v ( ξ ) > v ( ξ ) } ∈ G τ and θ := J τ, ∞ K [ A θ + A c θ ] , i.e., on the stochastic interval J τ, ∞ K , d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G · = A d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G · + 1 A c d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G · . Moreover, let Q denote the measure generated by θ . Then as v is a stochastic sub-solution and A ∈ G τ , we have A v ( ξ ) ≤ A E Q (cid:20) v ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = (cid:18) d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:19) − E (cid:20) A d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G ρ (cid:26) v ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E (cid:20) A d Q d P (cid:12)(cid:12)(cid:12)(cid:12) G ρ (cid:26) v ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ A E Q (cid:20) ( v ∨ v )( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.8) ifetime Ruin under HWM Fees and Drift Uncertainty 13 The second equality above is obtained by definition 2.1 and boundness of v . Similarly, we obtain A c v ( ξ ) ≤ A c E Q (cid:20) ( v ∨ v )( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.9)Combining (4.8) and (4.9), we have ( v ∨ v )( ξ ) ≤ E Q (cid:20) ( v ∨ v )( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . Thus, ( v ∨ v ) satisfies (SB2) with Q .In the next theorem, we will use lemma 4.2 to construct a “bump” function to argue by contradic-tion. Theorem 4.1 The lower stochastic envelope of V − , v − := sup v ∈V − v, (4.10) is a viscosity super-solution of (3.3).Proof. Lemma 4.1 already asserts that v − ≥ ψ − . Therefore, we have v − ≥ on O R and v − ≥ on O c/r . It remains to show that for this v − and any test function ϕ such that z ∈ O is a minimum pointof v − − ϕ at zero, we have ( F [ ϕ ]( z ) ≥ , on z ∈ O + , max (cid:8) F [ ϕ ]( z ) , B ∗ [ ϕ ]( z ) (cid:9) ≥ , on z ∈ ∂ O . We first show the claim above holds on the boundary part ∂ O ∩ ∂ O .Let us consider the region B a ( z ) of a ball with center z ∈ cl( O ) and the radius a intersectingwith O that B a ( z ) := { z ∈ cl( O ) : k z − z k < a } . To argue by contradiction, we suppose that there exist z = ( x , , ∈ ∂ O ∩ ∂ O and some ϕ ∈ C ( O ) such that v − − ϕ attains its strict minimum of zero at z and max (cid:8) F [ ϕ ]( z ) , B [ ϕ ]( z ) , B [ ϕ ]( z ) (cid:9) < . (4.11)Therefore it follows that there exists a constant θ ϕ ∈ L such that λ D ϕ ( z ) − ( rx − c ) ϕ x ( z ) − inf π ∈K A π,θ ϕ [ ϕ ]( z ) < . (4.12)Using ϕ , we will construct a bump function that still is in V − , in which it contradicts to (4.10). Bycontinuity of F and B i , i ∈ { , } , we can choose a small ball B a ( z ) , a > , such that for any z ∈ cl( B a ( z )) , max n λ D ϕ ( z ) − ( rx − c ) ϕ x ( z ) − inf π ∈K A π,θ ϕ [ ϕ ]( z ) < , B [ ϕ ]( z ) , B [ ϕ ]( z ) o < . (4.13)As v − − ϕ is l.s.c and cl( B a ( z )) \ B a ( z ) is compact, there exists δ > satisfying v − − ϕ ≥ δ, on cl( B a ( z )) \ B a ( z ) . As a result of Proposition 4.1 in [7] and Lemma 4.2, we can choose a non-decreasing sequence { v n } ⊆V − such that v n ր v − . By Lemma 2.4 in [9], we can pick v := v N such that v − ϕ ≥ δ/ on cl( B a ( z )) \ B a ( z ) . Then we further choose < η < δ/ small enough such that ϕ η := ϕ + η satisfies max n λ D ϕ η ( z ) − ( rx − c ) ϕ ηx ( z ) − inf π ∈K A π,θ ϕ [ ϕ η ]( z ) , B [ ϕ η ]( z ) , B [ ϕ η ]( z ) o < , (4.14)on cl( B a ( z )) . By this construction, we have ϕ η ≤ ϕ − δ/ ≤ v on cl( B a ( z )) \ B a ( z ) , (4.15) ϕ η ( z ) = ϕ η ( z ) + η = v − ( z ) + η > v − ( z ) . (4.16)Let us define v η := ( v ∨ ϕ η , cl( B a ( z )) ,v, otherwise . Then we will show that v η ∈ V − and this is a contradiction by (4.10) and (4.16).To this end, we consider an arbitrary ( τ, ξ ) ∈ R , π ∈ A , and a G -stopping time ρ ∈ J τ, τ τ,ξ,πE K .Our goal is to find a probability measure satisfying (SB2) for v η . As v is a stochastic sub-solution, forany strategy π we can find ( θ v,πt ) t ≥ producing a probability measure Q v,π ∈ L that satisfies (SB2)for v . Define Γ := (cid:8) ξ ∈ B a ( z ) and v ( ξ ) < ϕ η ( ξ ) (cid:9) ∈ G τ , and let τ a (resp. ξ a ) denote the exit time (resp. exit position) of the ball B a ( z ) , i.e., τ a := inf { t ∈ [ τ, τ τ,ξ,πE ] : Z τ,ξ,πt / ∈ B a ( z ) } ,ξ a := Z τ,ξ,πa . By ( θ v,πt ) t ≥ and θ ϕ in (4.14), define ( e θ t ) t ≥ as e θ πt := t ≥ τ ( θ ϕ Γ + θ v,πt Γ c ) Note that ξ a ∈ ∂B a ( z ) ∪ { ∆ } and ( τ a , ξ a ) ∈ R . Therefore, for ( τ a , ξ a ) and π ∈ A , there exists θ v,a,π producing Q v,a,π given by (2.10) that satisfies (SB2) for v . Then define θ π := J ,τ a K e θ π + K τ a , ∞ K θ v,a,π , (4.17)and Q π be the measure by θ π . Then for any π ∈ R , we show that Q π is the measure for v η to satisfy(SB2) from which we obtain the contradiction.In particular, we can obtain a contradiction from the place where the measure by θ ϕ,π is taken.Itô’s formula on the event Γ yields ϕ η ( Z τ,ξ,πρ ∧ τ a ) − ϕ η ( Z τ,ξ,πτ ) = Z ρ ∧ τ a τ (cid:2) A π,θ π [ ϕ η ] + 12 ε k θ πt k − λ D ϕ η + ( rX τ,ξ,πt − c ) ϕ ηx (cid:3) ( Z τ,ξ,πt ) d t − X i =1 , Z ρ ∧ τ a τ B i [ ϕ η ]( Z τ,ξ,πt ) d M it − Z ρ ∧ τ a τ ϕ η ( Z τ,ξ,πt − ) d M Dt + Z ρ ∧ τ a τ (cid:2) ϕ ηx ( Z τ,ξ,πt ) π ⊤ t σ − ∇ y ϕ η ( Z τ,ξ,πt ) ⊤ diag( π t ) σ B∆ (cid:3) d W Q π t . (4.18) ifetime Ruin under HWM Fees and Drift Uncertainty 15 On the compact set cl( B a ( z )) , ϕ η and ∇ ϕ η are bounded. Therefore, ϕ η ( Z τ,ξ,π ) and ∇ ϕ η ( Z τ,ξ,π ) arebounded on J τ, ρ ∧ τ a K . Moreover, π is valued in the compact set K . Therefore, the last two terms in(4.18) are G -martingales. Then by (4.11), we have E Q π (cid:2) Γ v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1)(cid:12)(cid:12) G τ (cid:3) ≥ E Q π (cid:20) Γ n ϕ η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≥ Γ ϕ η (cid:0) Z τ,ξ,πτ (cid:1) = Γ ϕ η ( ξ )= Γ v η ( ξ ) . Note that at the last equality, we do not exclude the case that τ = τ D , i.e., ξ = ∆ . Recall that on Γ c , we have v ( ξ ) = v η ( ξ ) and θ π = θ v,π which is the ( τ, ξ ) -optimal control of v . Let Q v,π denote the θ v,π -measure. By (SB2), it follows that Γ c v η ( ξ ) = Γ c v ( ξ ) ≤ E Q v,π (cid:20) Γ c n v (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ v,πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ E Q π (cid:20) Γ c n v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . Hence, we obtain that v η ( ξ ) ≤ E Q π (cid:20) v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ πt k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.19)Now, to replace ρ ∧ τ a with ρ in (4.19), we first consider the event Λ := { ρ > τ a } ∈ G τ a ∧ ρ . Since v = v η at ∂B a ( z ) and on K τ a , ρ K ∩ ( Λ × R + ) , we have θ π = θ v,a,π . Then denoting by Q v,a,π the θ v,a,π -measure, Λ v η ( ξ a ) = Λ v ( ξ a ) ≤ E Q v,a,π (cid:20) Λ n v ( Z τ,ξ,πρ ) − ε Z ρτ a k θ v,a,πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21) ≤ E Q π (cid:20) Λ (cid:8) v η ( Z τ,ξ,πρ ) − ε Z ρτ a k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21) . (4.20)Moreover, by (4.19) together with (4.20), we can get v η ( ξ ) ≤ E Q π (cid:20) v η ( Z τ,ξ,πρ ∧ τ a ) − ε Z τ a ∧ ρτ k θ πt k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E Q π (cid:20) Λ c n v η ( Z τ,ξ,πρ ) − ε Z ρτ k θ πt k d t o + Λ n v η ( ξ a ) − ε Z τ a τ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.21)By (4.20), we have E Q π (cid:20) Λ n v η ( ξ a ) − ε Z τ a τ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E Q π (cid:20) E Q π (cid:20) Λ n v η ( ξ a ) − ε Z τ a τ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ E Q π (cid:20) Λ n v η ( Z τ,ξ,πρ ) − ε Z ρτ k θ πt k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.22)Therefore, in view of (4.21) and (4.22), we deduce that v η ∈ V − , which clearly contradicts (4.10).Hence, it follows that v − is a viscosity super-solution of (3.3) at z ∈ ∂ O ∩ ∂ O .We can deal with points in other regions z / ∈ ∂ O ∩ ∂ O in similar ways. To be more precise, for z ∈ O + (resp. z ∈ ∂ O i , i ∈ { , } ), we suppose that there exist a function ϕ ∈ C ( O ) such that v − − ϕ attains its strict minimum of zero at z and F [ ϕ ]( z ) < (cid:0) resp . max {F [ ϕ ]( z ) , B i [ ϕ ]( z ) } < (cid:1) . Then, by employing similar contradiction arguments, we can conclude that v − is indeed a viscositysuper-solution of (3.3). stochastic super-solutions can be defined to facilitate the derivation of the otherdirection of DPP as inf π ∈ A sup Q ∈ L E Q (cid:20) V ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ V ( ξ ) . Note that the item (SP2) in the next definition is precisely motivated by the inequality above. Definition 4.4 (Stochastic super-solutions) If v ∈ USC b (cl( O )) satisfies(SP1) v ≥ on ∂ O R and v ≥ on ∂ O c/r ,(SP2) for any random initial condition ( τ, ξ ) , there exists π ∈ A such that for any G -stopping time ρ ∈ [ τ, τ τ,ξ,πE ] and Q ∈ L , E Q (cid:20) v ( Z τ,ξ,πρ ) − ε Z ρτ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ v ( ξ ) , (4.23)where v in (4.2) is understood as its extension to cl( O ) ∪ { ∆ } by allocating v ( ∆ ) = 0 ,then v is called a stochastic super-solution of (3.3). In addition, we let V + denote the class of allstochastic super-solutions of (3.3).We can find a stochastic super-solution by considering a “worse scenario”. Consider a situationthat the investor does not invest in the hedge funds, i.e., π = 0 . Then, the investor’s wealth follows d X t = [ rX t − c ] d t, X = x . We thus, can obtain that p ( x ) := P ( τ x,y, R < τ D ) = (cid:16) c − rxc − rR (cid:17) λDr . Lemma 4.3 Let ψ + ( x, y ) := p ( x ) . Then ψ + ∈ V + .Proof. It is obvious that ψ + ∈ USC b (cl( O )) and satisfies (SP1). Let ( τ, ξ ) be a random initial condition and we choose π = 0 for the strategy. Thus, for τ < τ D , d X τ,ξ,πt = [ rX τ,ξ,πt − c ] d t. Consider ρ ∈ [ τ, τ τ,ξ,πE ] as a G -stopping time. In the rest of the proof, we suppress the superscripts τ, ξ, π . By Itô’s formula, we have p ( X ρ ) − p ( X τ ) = Z ρτ n p ′ ( X t )[ rX t − c ] − λ D p ( X t ) o − Z ρτ p ( X s − ) d M Ds = − Z ρτ p ( X s − ) d M Ds As for any equivalent probability measure Q given by (2.10), M D is ( Q , G ) -martingale, it follows that E Q [ p ( X ρ ) |G τ ] = p ( X τ ) for any Q ∈ L . Therefore, for any θ -measure Q ∈ L , E Q (cid:20) ψ + ( Z ρ ) − ε Z ρτ k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E Q (cid:20) p ( Z ρ ) − ε Z ρτ k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ E Q [ p ( X ρ ) |G τ ] = p ( X τ ) = ψ + ( ξ ) . Therefore, ψ + satisfies (SP2), and we can deduce that ψ + ∈ V + . ifetime Ruin under HWM Fees and Drift Uncertainty 17 As in the previous section, we need to show V + is stable under minimum operation. The prooffollows closely the argument to prove Lemma 4.2, so we omit it. Lemma 4.4 If v , v ∈ V + , then v ∧ v ∈ V + . Then lemma 4.4 will be used to construct a bump function in the following theorem. Theorem 4.2 The lower stochastic envelope of V + , v + := inf v ∈V + v, (4.24) is a viscosity sub-solution of (3.3)Proof. By Lemma 4.3, v + ≤ ψ + . Therefore, we have v + ≤ on O R and v + ≤ on O c/r . As in the proofof theorem 4.1, it is sufficient to verify the sub-solution property of v + only on the boundary part ∂ O ∩ ∂ O . Using the same notation of balls that intersect O , we again will prove by contradiction.Suppose that there exist z = ( x , , ∈ ∂ O ∩ ∂ O and ϕ ∈ C ( O ) such that v − − ϕ attains itsstrict maximum of zero at z and min (cid:8) F [ ϕ ]( z ) , B [ ϕ ]( z ) , B [ ϕ ]( z ) (cid:9) > . (4.25)Again, as in the construction of a bump function in theorem 4.1, we can choose constants π ϕ ∈ K , η > , a > , and a stochastic super-solution v ∈ V + such that ϕ η = ϕ + η ≥ v, on cl( B a ( z )) \ B a ( z ) ,λ D ϕ η − ( rx − c ) ϕ ηx − sup θ ∈L A π ϕ ,θ [ ϕ η ] > , on cl( B a ( z )) , min (cid:8) B [ ϕ η ] , B [ ϕ η ] (cid:9) > , on cl( B a ( z )) ,ϕ η ( z ) < v − ( z ) , (4.26)and we define v η := ( v ∧ ϕ η , cl( B a ( z )) ,v, otherwise . (4.27)Then we will show that v η ∈ V + . To show that v η satisfies (SP2), let ( τ, ξ ) ∈ R . Since v ∈ V + , wecan choose ( π vt ) t ≥ for v to satisfy (SP2). Then with π ϕ in (4.26), we define ˜ π t ≥ as ˜ π t := t ≥ τ ( π ϕ Γ + π vt Γ c ) . Let us denote Γ := (cid:8) ξ ∈ B a ( z ) and v ( ξ ) < ϕ η ( ξ ) (cid:9) , and let τ a (resp. ξ a ) denote the exit time (resp. exit position) of the ball B a ( z ) . Since ( τ a , ξ a ) ∈ R and v ∈ V + , we can choose π v,a ∈ A such that for any Q ∈ L and G -stopping time valued in J τ a , τ τ,ξ,π v,a E K , v satisfies (SP2). Finally, we let π := J ,τ a K ˜ π + K τ a , ∞ K π v,a . (4.28) We will show that v η , with π , satisfies (SP2). Consider an arbitrary G -stopping time ρ ∈ [ τ, τ τ,ξ,πE ] and θ -measure Q ∈ L . Applying Itô’s formula on the event Γ yields, for any θ -measure Q , ϕ η ( Z τ,ξ,πρ ∧ τ a ) − ϕ η ( Z τ,ξ,πτ ) = Z ρ ∧ τ a τ (cid:2) A π,θ [ ϕ η ] + 12 ε k θ t k − λ D ϕ η + ( rX τ,ξ,πt − c ) ϕ ηx (cid:3) ( Z τ,ξ,πt ) d t − X i =1 , Z ρ ∧ τ a τ B i [ ϕ η ]( Z τ,ξ,πt ) d M it − Z ρ ∧ τ a τ ϕ η ( Z τ,ξ,πt − ) d M Dt + Z ρ ∧ τ a τ (cid:2) ϕ ηx ( Z τ,ξ,πt ) π ⊤ t σ − ∇ y ϕ η ( Z τ,ξ,πt ) ⊤ diag( π t ) σ B∆ (cid:3) d W Q t . (4.29)Therefore, by (4.26) and (4.27), we have E Q (cid:20) Γ n v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ E Q (cid:20) Γ n ϕ η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≤ Γ ϕ η (cid:0) Z τ,ξ,πτ (cid:1) = Γ ϕ η ( ξ )= Γ v η ( ξ ) . Recall that on Γ c , we have v ( ξ ) = v η ( ξ ) and π = π v . Since v is a stochastic super-solution by itsconstruction, we have Γ c v η ( ξ ) = Γ c v ( ξ ) ≥ E Q (cid:20) Γ c n v (cid:0) Z τ,ξ,π v ρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≥ E Q (cid:20) Γ c n v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . Thus, we deduce that v η ( ξ ) ≥ E Q (cid:20) v η (cid:0) Z τ,ξ,πρ ∧ τ a (cid:1) − ε Z τ a ∧ ρτ k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.30)To replace ρ ∧ τ a with ρ , consider Λ := { ρ > τ a } ∈ G τ a ∧ ρ . Recall that v = v η at ∂B a ( z ) and on K τ a , ρ K ∩ ( Λ × R + ) , we have π = π v,a . It then follows that Λ v η ( ξ a ) = Λ v ( ξ a ) ≥ E Q (cid:20) Λ n v ( Z τ,ξ,π v,a ρ ) − ε Z ρτ a k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21) ≥ E Q (cid:20) Λ (cid:8) v η ( Z τ,ξ,πρ ) − ε Z ρτ a k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21) . (4.31)By (4.31), one can derive that E Q (cid:20) Λ n v η ( ξ a ) − ε Z τ a τ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E Q (cid:20) E Q (cid:20) Λ n v η ( ξ a ) − ε Z τ a τ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ a (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) ≥ E Q (cid:20) Λ n v η ( Z τ,ξ,πρ ) − ε Z ρτ k θ t k d t o(cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.32)Therefore, thanks to (4.31) and (4.32), the inequality holds that Λ v η ( ξ ) ≥ Λ E Q (cid:20) v η (cid:0) Z τ,ξ,πρ (cid:1) − ε Z ρτ k θ t k d t (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.33)We can obtain the inequality on Λ c in the similar fashion as in the proof of Theorem 4.1. Hence, itcan be shown that v η ∈ V + , which contradicts (4.26) and our claim holds. ifetime Ruin under HWM Fees and Drift Uncertainty 19 B 6 = 0 in a viscosity sense. Then by the definition of viscosity solution, thetest function should satisfy F = 0 and this in turn will provide a contradiction. In what follows, wedenote q := [ q , − − q , ⊤ and q := [ q , , − − q ] ⊤ .To explain the idea to choose a test function, let z , z ′ ∈ R . As always, to push the variablesinto a diagonal entry, we need k z − z ′ k /α for some α > , in the test function. Moreover, since thedomain O is not bounded, for the test function to have a maximum in a compact set, one may wantto put β ( k z k + k z ′ k ) / for some β > . If we stop here, the test function may or may not satisfy B i , i ∈ { , } . To be more precise, for z ∈ ∂ O or z ′ ∈ ∂ O , we can not guarantee that ∇ h α k z − z ′ k + β k z k + k z ′ k ) i · q i > , i ∈ { , } . (4.34)To eliminate the possibility to satisfy B i , i.e., to focus on F , we seek to remedy the test functionto meet (4.34). To this end, pick any ν i > , i ∈ { , } , and choose z ν := ( R, ν , ν ) . Then for any z = ( x, , y ) ∈ ∂ O , we have ( z − z ν ) · q = ( x − R ) q + ν (1 + q ) > . Likewise, we also have ( z − z ν ) · q > for any z ∈ ∂ O . Therefore, instead of β ( k z k + k z ′ k ) / , we put χ β ( z , z ′ ) := β k z − z ν k + β k z ′ − z ν k . However, the effect of (4.34) is offset by the derivative of k z − z ′ k /α . Thus, to remove the derivative,we add additional terms and define ζ α ( z , z ′ ) := k z − z ′ k α + X i ∈{ , } n C iα ( z , z ′ )[ d i ( z ) − d i ( z ′ )] + k q i k α ( n i · q i ) [ d i ( z ) − d i ( z ′ )] o , + q q α (1 + q )(1 + q ) (cid:2) X i ∈{ , } { d i ( z ) − d i ( z ′ ) } (cid:3) ,C iα ( z , z ′ ) :=( z − z ′ ) · q i / ( α n i · q i ) ,d i ( z ) := dist ( z , ∂ O i ) , n :=[0 , − , ⊤ , n := [0 , , − ⊤ . Note that ∇ d i = − n i , i ∈ { , } . Then we define Ψ α,β : R × R → R as Ψ α,β ( z , z ′ ) := u ( z ) − v ( z ′ ) − ψ α,β ( z , z ′ ) , (4.35) ψ α,β ( z , z ′ ) := ζ α ( z , z ′ ) + χ β ( z , z ′ ) . (4.36) Now, we check some properties of ψ by straightforward calculations. First, we can derive that ∇ z ψ α,β ( z , z ′ ) = α − ( z − z ′ ) + X i ∈{ , } n − C iα ( z , z ′ ) n i + q i ( α n i · q i ) − [ d i ( z ) − d i ( z ′ )] − k q i k α ( n i · q i ) [ d i ( z ) − d i ( z ′ )] n i o + β ( z − z ν ) − q q α (1 + q )(1 + q ) (cid:2) X i ∈{ , } { d i ( z ) − d i ( z ′ ) } (cid:3) [ n + n ] , (4.37) ∇ z ′ ψ α,β ( z , z ′ ) = α − ( z ′ − z ) + X i ∈{ , } n C iα ( z , z ′ ) n i − q i ( α n i · q i ) − [ d i ( z ) − d i ( z ′ )]+ k q i k α ( n i · q i ) [ d i ( z ) − d i ( z ′ )] n i o + β ( z ′ − z ν )+ q q α (1 + q )(1 + q ) (cid:2) X i ∈{ , } { d i ( z ) − d i ( z ′ ) } (cid:3) [ n + n ] . (4.38)Then we can observe that ∇ z ζ α ( z , z ′ ) = − ∇ z ′ ζ α ( z , z ′ ) , ∇ z ψ α,β ( z , z ′ ) = − ∇ z ′ ψ α,β ( z , z ′ ) + β ( z − z ν ) + β ( z ′ − z ν ) . Moreover, for any z ∈ O , i = j , z j ∈ ∂ O j , ∇ z ψ α,β ( z j , z ) · q j = β ( z j − z ν ) · q i + q q α (1 + q i ) d j ( z ) > , (4.39) ∇ z ′ ( − ψ α,β )( z j , z ) · q j = − β ( z j − z ν ) · q − q q α (1 + q i ) d j ( z ) < . (4.40)(4.39)-(4.40) will be used later in the proof of Proposition 3.1. In addition, from (4.37) - (4.38) , thesecond order derivative of ψ is obtained. Let A := I + X i ∈{ , , } n k q i k n i ( n i ) ⊤ ( n i · q i ) − n i ( q i ) ⊤ + q i ( n i ) ⊤ ( n i · q i ) o + q q (1 + q )(1 + q ) [ n + n ][ n + n ] ⊤ , where I is the × -identity matrix. If q i , i ∈ { , } , are not too big, we clearly have A (cid:23) . Thenwe can write ∇ ψ α,β ( z , z ′ ) = 1 α (cid:20) A − A − A A (cid:21) + β (cid:20) I I (cid:21) . We are ready to prove the comparison principle. Proof of Proposition 3.1. We argue by contradiction. To this end, we suppose that for some z e ∈ cl( O ) , u ( z e ) − v ( z e ) = δ > . Let us choose β small enough such that δ > χ β ( z e , z e ) , and choose { α n } n ∈ N such that α n ↓ . Denote Ψ n := Ψ α n ,β . As u and v are bounded, χ β dominates u − v outside a compactset. Therefore, for each n ∈ N , Ψ n has its maximum on cl( O ) × cl( O ) in a compact set and we denotethe maximal point by ( z n , z ′ n ) , i.e., Ψ n ( z n , z ′ n ) = sup ( z , z ′ ) ∈ cl( O ) × cl( O ) Ψ n ( z , z ′ ) . ifetime Ruin under HWM Fees and Drift Uncertainty 21 The maximal point ( z n , z ′ n ) actually depends on β but we drop it for simplicity. As { ( z n , z ′ n ) } n ≥ liein a compact set, we choose a convergent subsequence, still denoted by ( z n , z ′ n ) , such that ( z n , z ′ n ) → ( z , z ′ ) = ( x, y, x ′ , y ′ ) . As u ≤ v on ∂ O R ∪ ∂ O c/r by the definition of viscosity sub/super solution, ( z , z ′ ) must be in O × O .The previous assumption yields that Ψ n ( z n , z ′ n ) ≥ sup z ∈ cl( O ) [ u ( z ) − v ( z ) − χ β ( z , z )] ≥ δ − χ β ( z e , z e ) > . Therefore, it follows that ζ α n ( z n , z ′ n ) ≤ u ( z n ) − v ( z ′ n ) − χ β ( z n , z ′ n ) − sup z ∈ cl( O ) [ u ( z ) − v ( z ) − χ β ( z , z )] . In view that the right hand side is bounded above but α n → as n → ∞ , ( x, y ) = ( x ′ , y ′ ) . Moreover,the fact that u − v is u.s.c implies that ≤ lim sup n →∞ ζ α n ( z n , z ′ n ) ≤ u ( z ) − v ( z ′ ) − χ β ( z , z ′ ) − sup z ∈ cl( O ) [ u ( z ) − v ( z ) − χ β ( z , z )] ≤ . Hence, lim n →∞ ζ n ( z n , z ′ n ) = 0 .By Crandall-Ishii’s lemma, for large n ∈ N , there exist A n , B n ∈ S such that ( ∇ z ψ α n ,β ( z n , z ′ n ) , A n ) ∈ J , + O u ( z n ) , ( −∇ z ′ ψ α n ,β ( z n , z ′ n ) , B n ) ∈ J , −O v ( z ′ n ) and that − α n (cid:20) I I (cid:21) ≺ (cid:20) A n − B n (cid:21) ≺ α n (cid:20) I − I − I I (cid:21) + 2 β (cid:20) I I (cid:21) . (4.41)We can calculate that ∇ z ψ α n ,β ( z n , z ′ n ) = ∇ z ζ α n ( z n , z ′ n ) + β ( z n − z ν ) −∇ z ′ ψ α n ,β ( z n , z ′ n ) = ∇ z ζ α n ( z n , z ′ n ) − β ( z ′ n − z ν ) . Let F be the function such that F [ ϕ ]( z ) = F ( z , ϕ ( z ) , ∇ ϕ ( z ) , ∇ ϕ ( z )) . Then we have λ D ( u ( z n ) − v ( z ′ n )) = F ( z n , u ( z n ) , ∇ z ψ α n ,β ( z n , z ′ n ) , A n ) − F ( z n , v ( z ′ n ) , ∇ z ψ α n ,β ( z n , z ′ n ) , A n ) ≤ F ( z ′ n , v ( z n ) , ∇ z ψ α n ,β ( z n , z ′ n ) , B n ) − F ( z n , v ( z ′ n ) , ∇ z ψ α n ,β ( z n , z ′ n ) , A n ) ≤ F (cid:0) z ′ n , v ( z n ) , ∇ z ζ α n ,β ( z n , z ′ n ) , B n + 2 β I (cid:1) − F (cid:0) z n , v ( z ′ n ) , ∇ z ζ α n ,β ( z n , z ′ n ) , A n − β I (cid:1) + c ( β ) , (4.42)where c ( β ) is the modulus of continuity. The last inequality of (4.42) is obtained by the compactnessof K . By (4.41), we moreover, have A n − β I ≺ B n + 2 β I . Therefore, we obtain F (cid:0) z ′ n , v ( z n ) , ∇ z ζ α n ,β ( z n , z ′ n ) , B n − β I (cid:1) ≤ F (cid:0) z n , v ( z ′ n ) , ∇ z ζ α n ,β ( z n , z ′ n ) , A n + 2 β I (cid:1) . (4.43)By (4.42) and (4.43), taking n ↑ ∞ leads to λ D δ ≤ c ( β ) . Again taking β ↓ , we have the desiredcontradiction, which completes the proof. Proof of Theorem 3.1. Theorem 4.1, Theorem 4.2, together with Proposition 3.1 imply that v + ≤ v − .Therefore, it suffices to show v − ≤ V ≤ v + . To show the first inequality, let us consider an arbitrary φ ∈ V − . It is obvious that φ ≤ V on ∂ O R ∪ ∂ O c/r . Let ( x, y ) ∈ O and take the random initialcondition as τ = 0 and ξ = ( x, y ) . We fix some π ∈ R and the hitting time defined by τ τ,ξ,πc/r := inf { t ≥ X τ,ξ,πt ≥ c/r } . As there exists θ -generated measure Q for φ to satisfy (SB2), it follows that φ ( x, y ) ≤ E Q (cid:20) φ ( Z τ,ξ,πτ τ,ξ,πE ) − a Z τ τ,ξ,πE τ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) = E Q (cid:20) τ τ,ξ,πE = τ x,y,πR − a Z τ τ,ξ,πE τ e − λ D s k θ s k d s (cid:12)(cid:12)(cid:12)(cid:12) G τ (cid:21) . (4.44)Moreover, we have E Q [ τ τ,ξ,πE = τ x,y,πR ] = Q [ τ τ,ξ,πR < τ D ∧ τ x,y,πc/r ] ≤ Q [ τ x,y,πR < τ D ] . (4.45)By combining (4.44) and (4.45), we have φ ( x, y ) ≤ V ( x, y ) , together with (4.10) yield v − ≤ V .In a similar fashion, we can show V ≤ v + as well. Because v − is a viscosity super-solution, byProposition 3.1, we have v + ≤ v − . It follows that v − ≤ V ≤ v + ≤ v − , which readily implies ourdesired equality v − = V = v + and hence the value function is the unique viscosity solution of theHJB equation (3.3). 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