The W,Z/ν,δ paradigm for the first passage of strong Markov processes without positive jumps

Avram, Florin; Grahovac, Danijel; Vardar-Acar, Ceren Article The W,Z/ν,δ paradigm for the first passage of strong Markov processes without positive jumps Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Avram, Florin; Grahovac, Danijel; Vardar-Acar, Ceren (2019) : The W,Z/ν,δ paradigm for the first passage of strong Markov processes without positive jumps, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 7, Iss. 1, pp. 1-17, https://doi.org/10.3390/risks7010018 This Version is available at: https://hdl.handle.net/10419/257856 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article The W,Z/ν,δParadigm for the First Passage of Strong Markov Processes without Positive Jumps Florin Avram 1,*, Danijel Grahovac 2and Ceren Vardar-Acar 3 1Laboratoire de Mathématiques Appliquées, Université de Pau, 64012 Pau, France 2Department of Mathematics, University of Osijek, 31000 Osijek, Croatia; [email protected] 3Department of Statistics, Middle East Technical University, Ankara 06800, Turkey; cvar[email protected] *Correspondence: [email protected] Received: 21 November 2018; Accepted: 13 February 2019; Published: 19 February 2019   Abstract: As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes Xt , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective (33) for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Yt=sup0≤s≤tXs−Xt obtained by reflecting Xtat its maximum. This new variational problem has been solved in a parallel paper. Keywords: first passage; drawdown process; spectrally negative process; scale functions; dividends; de Finetti valuation objective; variational problem 1. A Brief Review of First Passage Theory for Strong Markov Processes without Positive Jumps and Their Draw-Downs Motivation. First passage times intervene in the control of reserves/risk processes. The rough idea is that when below low levels a , the reserves should be replenished at some cost, and when above high levels b , the reserves should be invested to yield dividends—see for example Albrecher and Asmussen (2010). There is a wide variety of first passage control problems (involving absorption, reflection and other boundary mechanisms), and it has been known for a long while that these problems are simpler in the “completely asymmetric” case when all jumps go in the same direction. In recent years it has become clearer that most first passage problems can be reduced to the two basic problems of going up before down, or vice versa, and that their answers may usually be ergonomically expressed in terms of two basic “scale functions” W , Z (Albrecher et al. (2016); Avram et al. (2004,2007,2015, 2017a,2017b,2018a,2018b); Avram and Zhou (2017); Bertoin (1997); Ivanovs and Palmowski (2012); Kyprianou (2014); Landriault et al. (2017b); Li et al. (2017); Li and Zhou (2018); Suprun (1976)). The proofs require typically not much more than the strong Markov property; it is natural, therefore, to develop extensions to strong Markov processes. This has been achieved already in particular spectrally negative cases such as random walks Avram and Vidmar (2017), Markov additive processes Ivanovs and Palmowski (2012), Lévy processes with Ω state-dependent killing Ivanovs and Palmowski (2012), certain Lévy processes with state-dependent drift Czarna et al. (2017), and is in fact possible in general. Risks 2019,7, 18; doi:10.3390/risks7010018 www.mdpi.com/journal/risks Risks 2019,7, 18 2 of 15 However, characterizing the functions W , Z is still an open problem, even for simple classic processes such as the Ornstein-Uhlenbeck and the Feller branching diffusion with jumps. Let Xt denote a one-dimensional strong Markov process without positive jumps, defined on a filtered probability space (Ω,{Ft}t≥0,P). Denote its first passage times above and below by Tb,+=Tb,+(X) = inf{t≥0 : Xt>b},Ta,−=Ta,−(X) = inf{t≥0 : Xt<a}, with inf ∅= +∞. Recall that first passage theory for diffusions and spectrally negative or spectrally positive Lévy processes is considerably simpler than that for processes which may jump both ways. For these two families, a large variety of first passage problems may be reduced to the computation of two monotone “scale functions” W , Z (by simple arguments such as the strong Markov property). See Albrecher et al. (2016); Avram et al. (2004,2007,2015,2017a,2018a); Avram and Zhou (2017); Bertoin (1997); Ivanovs and Palmowski (2012); Li and Zhou (2018); Suprun (1976) for the introduction and applications of W , Z in the Lévy case. For diffusions, the most convenient basic functions are the monotone solutions ϕ+ , ϕ− of the Sturm-Liouville equation—see Borovkov (2012). Finally, for spectrally negative or spectrally positive Lévy processes and diffusions, off-shelf computer programs could easily produce the answer to a large variety of problems, once approximations for the basic functions associated with the process have been produced. This continues to be true in principle for non-homogeneous Markov processes with one-sided jumps (by a simple application of the strong Markov property at the smooth crossing exit from an interval). However, there are very few papers proposing methods to compute W , Z for non-Lévy processes (see though Czarna et al. (2017), and Jacobsen and Jensen (2007), where the case of Ornstein-Uhlenbeck processes with phase-type jumps is studied). The two sided exit functions. The most important first passage functions are the solutions of the two-sided upward and downward exit problems from a bounded interval [a,b]:    Ψb q,θ(x,a):=Exhe−qTb,+−θ(XTb,+−b)1{Tb,+<Ta,−}i Ψb q,θ(x,a):=Exhe−qTa,−+θ(XTa,−−a)1{Ta,−<Tb,+}iq,θ≥0, a≤x≤b. (1) We will also call them killed survival and ruin first passage probabilities, respectively. Note that these are functions of five variables, very hard to compute in general. For processes with one-sided jumps, one of the exits must be smooth (without overshoot); in this case, the parameter θ is unnecessary and will be omitted. Also, when a=0, it will be omitted, to simplify the notation. For diffusions and Lévy processes with one-sided jumps, the two sided exit functions have well-known explicit formulas. For spectrally negative Lévy processes, the simplest is the smooth survival probability, whose factors are: Ψb q(x,a) = Wq(x−a) Wq(b−a)=e−Rb xνq(s−a)ds. (2) Wq(x)is called the scale function Bertoin (1998); Suprun (1976)1. We will assume throughout that Wq is differentiable (see Chan et al. (2011) for information on the smoothness of scale functions). Then, νq(s) = W0 q(s) Wq(s) is the logarithmic derivative of Wq , and may be interpreted as the “survival function of excursions lengths” Bertoin (1998). The non-smooth ruin probability has a more complicated explicit formula involving a second scale function ZqAvram et al. (2004)—see Remark 1below. 1 The fact that the survival probability has the multiplicative structure (2) is equivalent to the absence of positive jumps, by the strong Markov property. Risks 2019,7, 18 3 of 15 The draw-down/regret/loss/process. Motivated by applications in statistics, mathematical finance and risk theory, there has been increased interest recently in the study of the running maximum and of the draw-down/regret/loss/process reflected at the maximum, defined by Yt=Xt−Xt,Xt:=sup 0≤t0≤t Xs. Of equal interest is the infimum, and the draw-up/gain/process reflected at the infimum, defined by Yt=Xt−Xt,Xt=inf 0≤t0≤tXs. See Landriault et al. (2015,2017a); Mijatovic and Pistorius (2012) for references to the numerous applications of draw-downs and draw-ups. Draw-down and draw-up times are first passage times for the reflected processes: τd:=inf{t≥0 : Xt−Xt>d}, τd:=inf{t≥0 : Xt−Xt>d},d>0. (3) Such times turn out to be optimal in several stopping problems, in statistics Page (1954) in mathematical finance/risk theory—see for example Avram et al. (2004); Carr (2014); Lehoczky (1977); Shepp and Shiryaev (1993); Taylor (1975)—and in queueing. More specifically, they figure in risk theory problems involving capital injections or dividends at a fixed boundary, and idle times until a buffer reaches capacity in queueing theory. Remark 1. The second scale function Z Avram et al. (2004); Ivanovs and Palmowski (2012); Pistorius (2004) useful for solving the spectrally negative non-smooth ruin probability (and many other problems) is best defined via the solution of the non-smooth total discounted “regulation” problem. Let X[0 t=Xt+Lt denote the process Xt modified by Skorohod reflection at 0, with regulator Lt=−Xt , let E[0 xdenote expectation for this process and let T[0 b=Tb,+ 1 {Tb,+<T0,−}+τb 1 {T0,−<Tb,+}(4) denote the first passage to b of X[0 t. (a) The Laplace transform of the total regulation (“capital injections/bailouts”) into the process reflected non-smoothly at 0, until the first smooth up-crossing of a level b , may be factored as (Ivanovs and Palmowski 2012, Thm. 2): IE[0 x"e−qT[0 b−θLT[0 b#=       Zq,θ(x) Zq,θ(b),θ<∞ IExe−qT[0 b;Tb,+<T0,−=Wq(x) Wq(b),θ=∞ , (5) with Zq,θ(x)determined up to a multiplying constant. (b) Decomposing (5) at min(T+ b , T0,−) yields a formula (1) for the ruin probability Ivanovs and Palmowski (2012). Indeed: IE[0 x"e−qT[0 b−θLT[0 b#=Zq,θ(x) Zq,θ(b)=Wq(x) Wq(b)+IExhe−qT0,−+θXT0,−;T0,−<Tb,+iZq,θ(0) Zq,θ(b)=⇒(6) Ψb q,θ(x)Zq,θ(0) = IExhe−qT0,−+θXT0,−;T0,−<Tb,+iZq,θ(0) = Zq,θ(x)−Wq(x)Wq(b)−1Zq,θ(b). (7) To simplify this formula, it is customary to choose Zq,θ(0) = 1. Risks 2019,7, 18 4 of 15 For non-homogeneous spectrally negative Markov processes, it is possible Avram et al. (2017a) to extend the equalities (2), (7) to analogue expressions involving scale functions of two variables Ψb q(x,a) = Wq(x,a) Wq(b,a),Ψb q,θ(x,a) = Zq,θ(x,a)−Wq(x,a)Wq(b,a)−1Zq,θ(b,a). (8) However, it is simpler to start, following Landriault et al. (2017b), with differential versions, whose existence will be assumed throughout this paper. Assumption 1. For all q , θ≥ 0and y≤x fixed, assume that Ψb q(x , y) and Ψb q,θ(x , y) are differentiable in b at b=x, and in particular that the following limits exist: νq(x,y):=lim ε↓0 1−Ψx+ε q(x,y) ε(9) and δq,θ(x,y):=lim ε↓0 Ψx+ε q,θ(x,y) ε(10) Remark 2. A necessary condition for Assumption 1to hold is that X is upward regular and creeping upward at every x in the state space—see (Landriault et al. 2017b, Rem. 3.1). Within this class, it seems difficult to provide examples where Assumption 1is not satisfied. It turns out that the differentiability of the two-sided ruin and survival probabilities as functions of the upper limit provides a method for computing other first passage quantities; for example, (12) and (23) below may be computed by solving the first order ODE’s in Theorem 2. Informally, we may say that the pillar of first passage theory for spectrally negative Markov processes is proving the existence of ν,δ. In the Lévy case note that by (2)νq(x , y) = W0 q(x−y) Wq(x−y)=νq(x−y) , and δq,θ(x , y) = δq,θ(x−y) where Avram et al. (2017a) δq,θ(x):=Zq,θ(x)−Wq(x)Z0 q,θ(x) W0 q(x). (11) Remark 3. For diffusions, Wq(x , a) is a certain Wronskian–see for example Borovkov (2012). Also, for Langevin type processes with decreasing state-dependent drifts, Wq(x , a) solves a certain renewal equation Czarna et al. (2017). The case of Ornstein-Uhlenbeck/Segerdahl-Tichy processes with exponential jumps is currently under study in Avram and Garmendia (2019). Some information about the generalization to Ornstein-Uhlenbeck processes with phase-type jumps can be found in Jacobsen and Jensen (2007). Beyond that, computing Wq(x , a) or νq(x , a) is an open problem. This is an important problem, and we conjecture that the method of Jacobsen and Jensen (2007) may be extended, at least to affine diffusions with phase-type jumps, and possibly to all diffusions with phase-type jumps. The drawdown exit functions. Recently, control results with drawdown times τd replacing classic first passage times started being investigated—see for example Landriault et al. (2017a); Mijatovic and Pistorius (2012). Two natural objects of interest for studying τdare the two sided exit times Tb+,d=min(τd,Tb,+),Ta−,d=min(τd,Ta,−). In terms of the two-dimensional process t7→ (Xt , Yt) , these are the first exit times from the regions (−∞,b]×[0, d]and [a,∞)×[0, d]. Risks 2019,7, 18 5 of 15 Fundamental in the study of say Tb+,d are the following two Laplace transforms UbD/DbU (up-crossing before draw-down/draw-down before up-crossing), which are analogues of the killed survival and ruin probabilities : UbDb q,θ,d(x) = IExhe−qTb,+−θ(XTb,+−b);Tb,+<τdi=IExhe−qTb,+−θ(XTb,+−b);Xτd>bi DbUb q,θ,d(x) = IExhe−qτd−θ(Yτd−d);τd<Tb,+i=IExhe−qτd−θ(Yτd−d);Xτd<bi. (12) For spectrally negative Lévy processes, these have again simple formulas: 1. UbDb q,d(x):=IExhe−qTb,+;Tb,+≤τdi=e−(b−x)W0 q(d) Wq(d), (13) 2. The function DbU may be obtained by integrating the fundamental law (Mijatovic and Pistorius 2012, Thm 1), (Landriault et al. 2017a, Thm 3.1)2 δq,θ(d,x,s):=IExhe−qτd−θ(Yτd−d);Xτd∈dsi=νq(d)e−νq(d)(s−x)+dsδq,θ(d) ⇔IExhe−qτd−θ(Yτd−d)−ϑ(Xτd−x)i=νq(d) ϑ+νq(d)δq,θ(d)(14) where δq,θ(d)is given by (11). Integrating yields DbUb q,θ,d(x) = 1−e−(b−x)W0 q(d) Wq(d)!δq,θ(d). (15) Remark 4. The probabilistic interpretation of νq , the logarithmic derivative of Wq . Taking a= 0for simplicity, the last formula in (2)has the interesting interpretation as the probability that no arrival has occurred between times x and b , for a non-homogeneous Poisson process of rate νq(s) , s∈[x , b] . Alternatively, differentiating (2) yields d ds Ψb q(s)−νq(s)Ψb q(s) = 0, Ψb q(b) = 1. (16) This equation coincides the Kolmogorov equation for the probability that a deterministic process e Ys=s , killed at rate νq(s) , reaches b before killing, when starting at s . It turns out, by excursion theory, that such a process e Ys may be constructed by excising the negative excursions from Xt , and by taking the running maximum s as time parameter. The logarithmic derivative νq(s) will be needed below in the de Finetti problem (17) , where we will use the fact that the expected dividends vq(b) paid at a fixed barrier b , starting from b , equal the expected discounted time until killing, which is exponential with parameter νq(b) , being therefore simply the reciprocal of the killing parameter νq(b): vq(b):=IEb"ZTb] 0,− 0e−qtd(Xt−b)#=νq(b)−1. (17) 2 Please note that (Mijatovic and Pistorius 2012, Thm. 1) give a more complicated “sextuple law” with two cases, and that (Landriault et al. 2017a, Thm 3.1) use an alternative to the function Zq(x , θ) , so that some computing is required to get (11) and (14). Risks 2019,7, 18 6 of 15 We see in the equation above and others that νq may serve as a convenient alternative characteristic of a spectrally negative Markov process, replacing Wq . Just as Wq , it may be extended to the case of generalized drawdown killing introduced in Avram et al. (2017b); Li et al. (2017). Contents. We start in Section 2by presenting a pedagogic first passage example illustrating the W,Zparadigm: the first time TR=Ta,b,d=Ta,−∧Tb,+∧τd. (18) when (X,Y)with XLévy leaves a rectangular region R= [a,b]×[0, d]. Remark 5. Please note that letting a→ −∞ , b→∞ reduces Ta,b,d to τd , and letting d→∞ , b→∞ reduces Ta,b,d to Ta,− . Hence both classic first passage and drawdown times appear as special cases of Ta,b,d .For finite a,b,d, our region has two classic and one drawdown exit boundary.3 In Section 3we provide geometric considerations which reduce computations of the Laplace transforms of the “three-sided” exit times of (X , Y) to that of Laplace transforms of two-sided exit problems involving Ta,−,Tb,+and τd(like (1) and (12))—see Figure 1. Only the strong Markov property is used; however, for the sake of simple notations we restricted the exposition to the family of Lévy processes (which have also the convenient feature that the scale functions W , Z may be computed by inverting Laplace transforms Avram et al. (2004,2015); Bertoin (1998); Ivanovs and Palmowski (2012); Kyprianou (2014)). In Section 4we enlarge the framework to that of generalized drawdown times Avram et al. (2017b); Li et al. (2017). This immediately entails that ν , δ become functions of two variables defined in (9) and (10) , and the extension to the spectrally negative Markov case becomes natural. We turn therefore to exits from certain trapezoidal-type regions in Section 5, under the spectrally negative Markov model. In Section 6we consider processes reflected at an upper barrier and formulate a Finetti’s optimal dividends type objective with combined ruin and generalized drawdown stopping; this involves adding one reflecting vertex to our trapezoidal region. Included here is a new variational problem for de Finetti’s dividends with generalized drawdown stopping (33) ; since the solution is not immediate even in the Lévy case, this has been provided in the parallel paper Avram and Goreac (2018). 2. Geometric Considerations Concerning the Joint Evolution of a Lévy Process and Its Draw-Down in a Rectangle To study the process (Xt , Yt) , it is useful to start with its evolution in a rectangular region R:= [a,b]×[0, d]⊂R×R+, where a<band d>0. Define TR=Ta,b,d:=inf{t:(Xt,Yt)/∈R}=τd∧Ta,−∧Tb,+. A sample path of (X , Y) , where X is chosen to be a spectrally negative Lévy process, and the region Ris depicted in Figure 1. 3 Choosing a , b , d optimally in various control problems involving optimal dividends and capital injections should be of interest, and will be pursued in further work. Risks 2019,7, 18 7 of 15 -6 -4 -2 0 2 4 6 0 2 4 6 8 10 Figure 1. A sample path of (X , Y) with X a spectrally negative Lévy process. The region R has d= 10, a=− 6 and b= 7; the dark boundary shows the possible exit points of (X , Y) from R . The base of the red line separates Rin two parts with different behavior. As is clear from the figure and from its definition, the process (X , Y) has very particular dynamics on R : away from the boundary ∂1:={(x1 , x2)∈R×R+:x2= 0 } it oscillates during negative excursions from the maximum on line segments lXt where, for c∈R , lc:={(x1 , x2)∈R×R+: x1+x2=c}. As Xt increases, the line segment lXt on which (X , Y) oscillates advances to the right—continuously, in the spectrally negative case, and in general possibly with jumps. On ∂1 , we observe the Markovian upward ladder process, i.e., the maximum X with downward excursions excised, with extra spatial killing upon exiting R . If only time killing was present, with d=∞ , this would be a killed drift subordinator, with Laplace exponent κ(s) = s+Φq (as a consequence of the Wiener-Hopf decomposition Kyprianou (2014)). In the rectangle, in the spectrally negative case, the ladder process becomes a killed drift with generator Gϕ(s):=ϕ0(s)−νq(d)ϕ(s) Albrecher et al. (2014); Avram et al. (2017b) . Finally, with generalized drawdown (when the upper boundary is replaced by one determined by certain parametrizations (b d(s) , d(s)) —see below), the generator will have state-dependent killing: Gϕ(s):=ϕ0(s)−νq(d(s))ϕ(s). (19) Several functionals (ruin, dividends, tax, etc.) of the original process may be expressed as functionals of the killed ladder process. This explains the prevalence of first order ODE’s—see (25) for one example—when working with spectrally negative processes. Several implications for TR are immediately clear from these dynamics: for example, the process (X , Y) can leave R only through ∂R∩{(x1 , x2)∈R×R+:x1≤b−d} or through the point (b , 0 ) (see the shaded region in Figure 1). Also, 1. If b≤a+d , it is impossible for the process to leave R through the upper boundary of ∂R and for these parameter values TR reduces to Ta,−∧Tb,+ . Here it suffices to know the functions (1) to obtain the Laplace transform of TR. 2. If a+d≤x , it is impossible for the process to leave R through the left boundary of ∂R , and TR reduces to Tb,+∧τd . Here it suffices to apply the spectrally negative drawdown formulas provided in Landriault et al. (2017a); Mijatovic and Pistorius (2012). 3. In the remaining case x≤a+d≤b , both drawdown and classic exits are possible. For the latter case, see Figure 1. The key observation here is that drawdown [classic] exits occur iff Xt does [does not] cross the line x1=d+a. The final answers will combine these two cases. Risks 2019,7, 18 8 of 15 3. The Three Laplace Transforms of the Exit Time out of a Rectangle for Lévy Processes without Positive Jumps In this section we provide Laplace transforms of TR and of the eventual overshoot at TR . One can break down the analysis of TR to nine cases, depending on which of the three exit boundaries Ta,− , Tb,+or τdoccurred, and on the three relations between x,a,band ddescribed above. The following results are immediate applications of the strong Markov property and of known first passage and draw-down results. Theorem 1. Consider a spectrally negative Lévy process X with differentiable scale function Wq . Then, for fixed d ≥0and a ≤x≤b, letting UbD,DbU denote the functions defined in (13),(15), we have: a+d≤x≤b x ≤a+d≤b b ≤a+d IExe−qTb,+;Tb,+≤min(τd,Ta,−)=UbDb q,d(x)Ψ(a+d) q(x,a)UbDb q,d(a+d)Ψb q(x,a) IExhe−qTa,−+θ(XTa,−−a);Ta,−≤min(τd,Tb,+)i=0Ψ(a+d) q,θ(x,a)Ψb q,θ(x,a) IExhe−qτd−θ(Yτd−d);τd≤min(Tb,+,Ta,−)i=DbUb q,θ,d(x)Ψ(a+d) q(x,a)DbUb q,θ,d(a+d)0 (20) Proof. Please note that in the third column the d boundary is invisible and does not appear in the results, and in the first column the a boundary is invisible and does not appear in the results. These two cases follow therefore by applying already known results. The middle column holds by breaking the path at the first crossing of a+d . The main points here are that 1. the middle case may happen only if Xtvisits abefore a+d; 2. the first case (exit through b ) and the third case (drawdown exit) may happen only if Xt visits first a+d , with the drawdown barrier being invisible, and that subsequently the lower first passage barrier abecomes invisible. The results follow then due to the smooth crossing upward and the strong Markov property. Proof. Let us check the first and third row of the second column. Applying the strong Markov property at Ta+d,+yields IExhe−qTb,+;Tb,+≤min(τd,Ta,−)i=IExhe−qTb,+;Ta+d,+≤Ta,−iIEa+dhe−qTb,+;Tb,+≤τdi =Wq(x−a) Wq(d)e−(b−a−d)W0 q(d) Wq(d) and IExhe−qτd−θ(Yτd−d);τd≤min(Tb,+,Ta,−)i=IExhe−qτd−θ(Yτd−d);Ta+d,+≤Ta,−iIEa+dhe−qτd−θ(Yτd−d);τd≤Tb,+i =Wq(x−a) Wq(d)δq,θ(d) 1−e−(b−a−d)) W0 q(d) Wq(d)!. 4. 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