Optimal liquidation trajectories for the Almgren-Chriss model with Levy processes
OOptimal liquidation trajectories for the Almgren-Chriss modelwith L´evy processes
Arne Løkka ∗ Junwei Xu † September 16, 2020
Abstract
We consider an optimal liquidation problem with infinite horizon in the Almgren-Chriss framework, where the unaffected asset price follows a L´evy process. The temporaryprice impact is described by a general function which satisfies some reasonable conditions.We consider an investor with constant absolute risk aversion, who wants to maximisethe expected utility of the cash received from the sale of his assets, and show that thisproblem can be reduced to a deterministic optimisation problem which we are able tosolve explicitly. In order to compare our results with exponential L´evy models, whichprovides a very good statistical fit with observed asset price data for short time horizons,we derive the (linear) L´evy process approximation of such models. In particular we deriveexpressions for the L´evy process approximation of the exponential Variance-Gamma L´evyprocess, and study properties of the corresponding optimal liquidation strategy. We thenprovide a comparison of the liquidation trajectories for reasonable parameters between theL´evy process model and the classical Almgren-Chriss model. In particular, we obtain anexplicit expression for the connection between the temporary impact function for the L´evymodel and the temporary impact function for the Brownian motion model (the classicalAlmgren-Chriss model), for which the optimal liquidation trajectories for the two modelscoincide.
Keywords:
Almgren-Chriss model, algorithmic trading, optimal liquidation, optimalexecution, constant absolute risk aversion, market impact, L´evy processes, optimal control,Hamilton-Jacobi-Bellman equation,
The introduction of electronic trading platforms was followed by an increased interest inhow to split large orders into smaller orders in order to liquidate large asset positions. An ∗ Department of Mathematics Columbia House London School of Economics Houghton Street, LondonWC2A 2AE United Kingdom ([email protected]) † Department of Mathematics Columbia House London School of Economics Houghton Street, LondonWC2A 2AE United Kingdom ([email protected]) a r X i v : . [ q -f i n . T R ] S e p mportant question for large investors is how to sell a huge number of shares . Because of alack of liquidity in the market it is often not practical to sell all the shares immediately sincethis can result in too high an execution cost. By splitting a large block of orders into smallerones, the investor can often effectively reduce the cost substantially. The problem of findingthe optimal way to do this has therefore been the subject of considerable interest.When the investor determines the speed at which to sell the shares, the key componentsare execution cost and market risk. A slow execution speed will result in a low execution cost,but high market risk. On the other hand, a fast execution speed will result in a low marketrisk, but high execution cost. In most models dealing with optimal execution, Brownianmotion is driving the market risk. However, in reality observed stock price data demonstratethat Brownian motion is not a particularly good model for stock prices, especially for shortertime periods. For instance, sudden large price movements and the heavy-tailed distributionof log-returns can not be captured by Brownian motion. Also, observed logarithmic stockreturns over short-time horizons are not normally distributed. On the other hand, there hasbeen a lot of theoretical and empirical studies that show that L´evy processes provide a goodfit to market data. For detailed discussions, we refer to Madan and Seneta (1990), Eberleinand Keller (1995) and Barndorff-Nielsen (1997). Because of the reasons explained above, inparticular that in practise the time it takes to liquidate often is very short and that L´evyprocesses provide good statistical models for stock prices over short time periods, we will inthis paper consider models based on L´evy processes.We consider a continuous-time optimal liquidation problem of a single stock in the Almgren-Chriss framework with infinite time horizon. The permanent impact function is supposed tobe linear, and we describe the temporary market impact function in terms of general sufficientconditions ensuring that we are able to solve the problem explicitly. The unaffected shareprice is driven by a linear L´evy process. We assume that the large investor is not permittedto buy back shares during liquidation, but we can actually show by a dynamic programmingargument that any such strategy would be sub-optimal. The investor is supposed to haveconstant absolute risk aversion (CARA), and the aim is to maximise expected utility of thefinal cash position over a set of admissible liquidation strategies. Following an idea introducedin Schied et al. (2010), the optimisation problem is reduced to an optimisation problem over aset of deterministic strategies. Moreover, we show that for a general L´evy process, there is noimmediate relationship between the optimal strategy for the mean-variance criteria and theoptimal strategy for the expected exponential utility, which holds for the Brownian motioncase. We also show that when the L´evy process is a strict submartingale, our problem is ill-posed, and it is always optimal to hold on to the shares rather than sell. Then by solving theHamilton-Jacobi-Bellman equation, the optimal liquidation strategy is derived in an explicitform. After that, we provide some conditions which determines whether the optimal strategyhas a finite termination time.The standard way to analyse stock price data is to find the statistics of the log-returns.This naturally leads to exponential L´evy models, and most distributions for the driving L´evyprocess in relation to stock price data is of the exponential model type. Given a specific2xponential L´evy model, we therefore show how to linearise the model in order to get a modelof the form relevant to our paper. We then provide some examples where we assume thelog-returns of the share price satisfy the variance gamma distribution and where they satisfythe normal distribution. In the variance gamma case we find that the widely used powerlaw market impact function can result in optimal strategies which liquidate faster than whatseems practical. We point out that cost from large trading speeds may be underestimated bypower functions, and that a function with a bigger growth rate may better reflect the cost ofexecution.For an introduction to high-frequency trading and optimal execution, we refer the readerto Lehalle and Laruelle (2013), Cartea et al. (2015) and Gu´eant (2016), but below we providea brief review of the more relevant works in connection to this paper. Bertsimas and Lo (1998)introduced a discrete time stock price model with illiquidity effects and related problems. ThenAlmgren and Chriss (1999, 2000) classified the effects in terms of permanent and temporaryimpacts of trading. In this kind of market impact framework, various liquidation models weredeveloped. Almgren and Chriss (2000) introduced a discrete time model with linear permanentand temporary impact functions, a deterministic optimal trading strategy was derived bymean-variance optimisation. Almgren (2003) generalised the model by considering non-linearimpact functions. A single-asset continuous model with infinite time horizon was introducedin Schied and Sch¨oneborn (2009), a multi-asset finite horizon model was considered by Schiedet al. (2010) and Sch¨oneborn (2016) provides a multi-asset infinite horizon model. In thesepapers strategies were derived by maximising expected utilities instead of the mean-variancecriteria. Schied et al. (2010) explained the relationship between mean-variance criteria andthe expected exponential utility criteria in the Almgren-Chriss framework. They also provedthat in a finite time horizon, when the stock price is driven by a L´evy process and an investorwith exponential utility, the optimal strategy is deterministic. Gatheral (2010) suggestedthat instead of dealing with permanent and temporary impacts, the market impact shoulddecay over time. Moreover, Obizhaeva and Wang (2013) introduced a limit order book modeland calculated the optimal execution strategy for such a model. Afterwards, several authorsconsidered variations of this limit order book model, such as Alfonsi and Schied (2010), Alfonsiet al. (2010, 2012) and Løkka (2014). In the literature of continuous models of optimalexecution, price processes are often linear and impact additive. However, by considering anew optimisation criterion, a model in the Almgren-Chriss framework based on geometricBrownian motion with linear market impact was given in Gatheral and Schied (2011). Then,Schied (2013) extended this model to general square integrable semimartingales. Also, somemultiplicative impact models are introduced in Forsyth et al. (2012) and Guo and Zervos(2015); in particular, Forsyth et al. (2012) demonstrated that the linear model gives anexcellent approximation to models with prices modelled as a geometric Brownian motionand multiplicative impact in the Almgren-Chriss framework.The structure of this paper is the following. In Section 2 we introduce the model and theoptimal execution problem. We reduce the problem to a deterministic optimisation problemin Section 3, and solve it in Section 4. In Section 5 we show how to linearise exponential L´evy3odels, and illustrate with examples in Section 6. Section 7 contain proofs not covered in themain sections. Let (Ω , F , P ) be a complete probability space, equipped with a filtration F = ( F t ) t ≥ satisfyingthe usual conditions, which supports a one dimensional, non-trivial, F -adapted L´evy process L . We assume that the L´evy process L possesses the following properties. Assumption 2.1. L has finite second moment. Moreover, the set (cid:8) δ < | E (cid:2) e δL (cid:3) < ∞ (cid:9) isnon-empty.For future reference, we observe that this assumption ensures that L t has finite first andsecond moments, for all t ≥
0. Hence, L admits the decomposition L t = µt + σW t + (cid:90) R x (cid:0) N ( t, dx ) − tν ( dx ) (cid:1) , where µ ∈ R and σ ≥ W is a standard Brownian motion, N is a Poissonrandom measure which is independent of W with compensator tν ( dx ), and ν is the L´evymeasure associated with L (see e.g. Kyprianou, 2006). Set¯ δ = inf (cid:8) δ < | E (cid:2) e δL (cid:3) < ∞ (cid:9) < . (2.1)Then Assumption 2.1 also ensures that the cumulant generating function of L is finite onthe interval (¯ δ, t ≥
0, we denote by Y t the investor’s position in the stock at time t , and let y ≥ Definition 2.2.
Given an initial share position y ≥
0, the set of admissible strategies, denotedby A ( y ), consists of all F -adapted, absolutely continuous, non-increasing processes Y satisfying (cid:90) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ if µ (cid:54) = 0 , (2.2)and (cid:90) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ if µ = 0 . (2.3)Let A D ( y ) be the set of all deterministic strategies in A ( y ).4he reason for operating with different sets of admissibility depending on the drift pa-rameter µ is related to the asymptotic properties of the cumulant generating function of L around 0. If µ is 0 then the cumulant generating function is of order two around zero, while itis of order one if µ is different from zero (the importance of the cumulant generating functionof L will be explained later). The integrability conditions in (2.2) and (2.3) make sure thatthe investor’s finial cash position is well-defined (see Proposition 2.5), and are also necessaryin order for the optimisation problem to be well defined (see Remark 2.6).Let Y ∈ A ( y ). Then there exists an F -adapted, positive-valued process ξ such that Y admits the representation Y t = y − (cid:90) t ξ s ds, i.e. − ξ t is the time derivative of Y at time t . In the literature of optimal liquidation, thefunction t (cid:55)→ Y t is referred to as the liquidation trajectory and the associated process ξ as theliquidation speed (see Almgren and Chriss, 2000; Almgren, 2003, etc).It is common in the optimal liquidation literature to refer to the price process observed inthe market if the investor does not trade as the unaffected stock price process. Throughoutthis chapter we assume that the unaffected stock price process is modelled by s + L t , t ≥ , where s > t ≥ S t = s + L t + α ( Y t − Y ) − F ( ξ t ) , (2.4)where α ≥ F : [0 , ∞ ) → [0 , ∞ ) is a func-tion describing the temporary impact. We assume that F satisfies the following assumptions. Assumption 2.3.
The temporary impact function F : [0 , ∞ ) → [0 , ∞ ) satisfies that(i) F ∈ C ([0 , ∞ )) ∩ C ((0 , ∞ ));(ii) F (0) = 0;(iii) the function x (cid:55)→ xF ( x ) is strictly convex on [0 , ∞ );(iv) the function x (cid:55)→ x F (cid:48) ( x ) is strictly increasing, and it tends to infinity as x → ∞ .5n the above assumption, condition (iii) serves to ensure convexity of the objective functionin the optimisation problem we are going to solve (see (3.4)) and hence uniqueness of thesolution (see Theorem 4.2); condition (iv) ensures that the value function in our optimisationproblem is solved in an explicit form (see Proposition 4.1) and the optimal liquidation speedprocess can be expressed in a feedback form (see Theorem 4.2). Assumption 2.3 is satisfiedby a large class of functions, for example, F ( x ) = βx γ with β, γ > F ( x ) = (cid:26) β x . x ∈ [0 , ¯ x ] ,β e γ ( x − ¯ x +ˆ x ) − β e γ ˆ x + β ¯ x . x ∈ (¯ x, ∞ ) , (2.5)where β , β , γ and ¯ x are strictly positive constants and ˆ x is given byˆ x = ln (cid:16) β β γ (cid:17) − ln ¯ xγ . Under this assumption, we derive the following technical properties of F for future references. Lemma 2.4. F is strictly increasing and lim x → xF (cid:48) ( x ) = 0 . Hence lim x → x F (cid:48) ( x ) = 0 . For t ≥
0, let C Yt denote the cash position of the investor at time t associated with anadmissible strategy Y . Denote by c ∈ R the investor’s initial cash position. Then a directcalculation verifies that his cash position at some finite time T is given by C YT = c − (cid:90) T S t dY t = c − (cid:0) s − αy (cid:1)(cid:0) Y T − y (cid:1) + α (cid:0) y − Y T (cid:1) − L T Y T + (cid:90) T Y t − dL t − (cid:90) T ξ t F ( ξ t ) dt. (2.6)The next result states that the investor’s cash position at the end of time is well-defined. Proposition 2.5.
For any Y ∈ A ( y ) , we have(i) L T Y T → in L ( P ) , as T → ∞ ;(ii) (cid:82) ∞ Y t − dL t is well-defined in L ( P ) .Therefore, C Y ∞ = c + sy − αy + (cid:90) ∞ Y t − dL t − (cid:90) ∞ ξ t F ( ξ t ) dt, a.s., (2.7) for any Y ∈ A ( y ) . C Y ∞ , we can make a few observations. The term c + sy can be viewedas the initial mark-to-market wealth of the investor. His total loss due to the permanentimpact of trading is given by αy , which is deterministic and only depends on the initialliquidation size. In particular, it does not depend on the choice of liquidation strategy. Theterm (cid:82) ∞ ξ t F ( ξ t ) dt represents the total cost due to the temporary impact, and it does dependon the liquidation strategy. The term (cid:82) ∞ Y t − dL t represents the gain or loss due to marketvolatility. A relatively slow liquidation speed reduces the temporary impact, but provides asubstantial market volatility risk. The optimal liquidation strategy is therefore a compromisebetween the loss due to the temporary impact and the market volatility risk. We assumethat the investor has a constant absolutely risk aversion (CARA), thus his utility function U satisfies U ( x ) = − exp( − Ax ), for some constant A >
0. The investor aims to maximise theexpected utility of his cash position at the end of time, i.e. he wants to solvesup Y ∈A ( y ) E (cid:2) U (cid:0) C Y ∞ (cid:1)(cid:3) . (2.8)In view of (2.7), this problem takes the form ofinf Y ∈A ( y ) e − A (cid:101) C E (cid:20) exp (cid:18) − (cid:90) ∞ AY t − dL t + A (cid:90) ∞ ξ t F ( ξ t ) dt (cid:19)(cid:21) , (2.9)where (cid:101) C = c + sy − αy . To solve the above problem, it is sufficient to look atinf Y ∈A ( y ) E (cid:20) exp (cid:18) − (cid:90) ∞ AY t − dL t + A (cid:90) ∞ ξ t F ( ξ t ) dt (cid:19)(cid:21) . (2.10) Remark 2.6.
Suppose that we do not impose integrability conditions (2.2) and (2.3) on anadmissible strategy. The cash position at time infinity may then not be well-defined, but onemay consider to solve the problemsup Y ∈A ( y ) E (cid:104) − exp (cid:16) − A lim sup T →∞ C YT (cid:17)(cid:105) . However, without (2.2) and (2.3), our model admits an arbitrage in some week sense. To seethis, take for instance the L´evy process L to be a standard Brownian motion and considersome stock price p > s . Write τ p = inf { t ≥ | L t ≥ p } which is finite a.s. (see Rogersand Williams (2000), Lemma 3.6). Suppose Y is an absolutely continuous, non-increasingstrategy which consists of waiting until time τ p and then decreases to 0 in a deterministic wayduring a finite time, i.e. ( Y τ p + t ) t ≥ is a deterministic process starting from y . Such strategy7s admissible. Let ξ be the associated speed process. We calculate thatsup Y ∈A ( y ) E (cid:104) − exp (cid:16) − A lim sup T →∞ C YT (cid:17)(cid:105) ≥ E (cid:104) − exp (cid:16) − A lim sup T →∞ C YT (cid:17)(cid:105) ≥ E (cid:104) − exp (cid:16) − AC YT + τ p (cid:17)(cid:105) = − exp (cid:18) − A (cid:101) C + A (cid:90) T ξ t + τ p F (cid:0) ξ t + τ p (cid:1) dt (cid:19) E (cid:20) exp (cid:18) − (cid:90) T + τ p AY t dW t (cid:19)(cid:21) = − exp (cid:18) − A (cid:101) C + A (cid:90) T ξ pt + τ p F (cid:0) ξ pt + τ p (cid:1) dt (cid:19) E (cid:20) exp (cid:18) − AyW τ p − (cid:90) T + τ p τ p AY pt dW t (cid:19)(cid:21) = − exp (cid:18) − Ayp − A (cid:101) C + A (cid:90) T ξ pt + τ p F (cid:0) ξ t + τ p (cid:1) dt (cid:19) E (cid:20) exp (cid:18)(cid:90) T + τ p τ p A (cid:0) Y t (cid:1) dt (cid:19)(cid:21) = − exp (cid:18) − Ayp − A (cid:101) C + A (cid:90) T ξ t + τ p F (cid:0) ξ t + τ p (cid:1) dt + (cid:90) T A (cid:0) Y t + τ p (cid:1) dt (cid:19) , where (cid:101) C = c + sy − αy , and notice that the two integrals in the above line are two constants.Taking p to + ∞ gives lim p →∞ E (cid:104) − exp (cid:16) − AC YT + τ p (cid:17)(cid:105) = 0 , and hence that the associated value function is degenerate. Moreover, Jensen’s inequalityresults in lim p →∞ − exp (cid:16) − A E (cid:2) C YT + τ p (cid:3)(cid:17) ≥ lim p →∞ E (cid:104) − exp (cid:16) − AC YT + τ p (cid:17)(cid:105) = 0 , which implies that lim p →∞ E (cid:2) C YT + τ p (cid:3) = ∞ . However, Y clearly violates (2.2) and (2.3). This shows that (2.2) and (2.3) are not onlyconvenient from a mathematical point of view, but also necessary in order for the problem tobe well formulated. Throughout this section, we reduce problem (2.10) to a deterministic optimisation problem.Set ¯ δ A = − ¯ δ/A , where ¯ δ is the negative number appearing in (2.1) and A is the risk aversionparameter appearing in the utility function U . We make the following futher assumptions.8 ssumption 3.1. The initial stock position y is strictly less than ¯ δ A . Assumption 3.2.
The drift µ of the L´evy process L satisfies µ ≤ κ A : [0 , ¯ δ A ) → R by κ A ( x ) = κ ( − Ax ), where κ is the cumulant generatingfunction of L , that is κ ( x ) = ln (cid:0) E (cid:2) e xL (cid:3)(cid:1) , x ∈ R . This function will play an important role in the sequel.
Lemma 3.3.
The function κ A possesses the following properties(i) κ A (0) = 0 ;(ii) κ A is strictly convex;(iii) if µ = 0 , then lim x → κ A ( x ) x = K , for some constant K > ;(iv) if µ (cid:54) = 0 , then lim x → κ A ( x ) x = − Aµ . Lemma 3.4.
Let Y be a continuous process starting form y ∈ [0 , ¯ δ A ) . Then (cid:90) ∞ (cid:107) Y t (cid:107) iL ∞ ( P ) dt < ∞ if and only if (cid:90) ∞ κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) du < ∞ , where i = 1 if µ < , and i = 2 if µ = 0 . Moreover, with µ > , (cid:90) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ implies (cid:90) ∞ κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) du < ∞ . In order to reduce problem (2.10), we also require the following technical result.9 emma 3.5.
For any Y ∈ A ( y ) , the process M Y given by M Yt = exp (cid:18)(cid:90) t − AY u − dL u − (cid:90) t κ A ( Y u ) du (cid:19) , t ≥ , (3.1) is a uniformly integrable martingale. It follows from Lemma 3.4 and Lemma 3.5 that, for any Y ∈ A ( y ), the process M Y is astrictly positive martingale closed by M Y ∞ . We can therefore define a new probability measure Q Y by d Q Y d P = M Y ∞ . Based on the idea in Schied et al. (2010) Theorem 2.8, and with reference to (2.10) andLemma 3.5, we calculate thatinf Y ∈A ( y ) E (cid:20) exp (cid:18) − (cid:90) ∞ AY t − dL t + A (cid:90) ∞ ξ t F ( ξ t ) dt (cid:19)(cid:21) = inf Y ∈A ( y ) E (cid:20) exp (cid:18) − (cid:90) ∞ AY t − dL t − (cid:90) ∞ κ A ( Y t ) dt + (cid:90) ∞ (cid:16) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:17) dt (cid:19)(cid:21) = inf Y ∈A ( y ) E Q Y (cid:20) exp (cid:18)(cid:90) ∞ (cid:16) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:17) dt (cid:19)(cid:21) ≤ inf Y ∈A D ( y ) exp (cid:20)(cid:90) ∞ (cid:18) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:19) dt (cid:21) . (3.2)Now suppose that Y ∗ is a solution to probleminf Y ∈A D ( y ) exp (cid:20)(cid:90) ∞ (cid:18) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:19) dt (cid:21) . since A D ( y ) ⊂ A ( y ). Then it must also be a solution to problem (2.10), and hence equalityholds in (3.2). This is because otherwise there must be some ˜ Y ∈ A D ( y ) which coincides withsome sample path of some Y ∈ A ( y ) such thatexp (cid:20)(cid:90) ∞ (cid:18) κ A ( ˜ Y t ) + Aξ t F ( ˜ ξ t ) (cid:19) dt (cid:21) < E Q Y (cid:20) exp (cid:18)(cid:90) ∞ (cid:16) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:17) dt (cid:19)(cid:21) < exp (cid:20)(cid:90) ∞ (cid:18) κ A ( Y ∗ t ) + Aξ t F ( ξ ∗ t ) (cid:19) dt (cid:21) . This contradicts Y ∗ being a solution to problem (3.2). We conclude that it is sufficient tosolve the problem V ( y ) = inf Y ∈A D ( y ) J ( Y ) , y ∈ [0 , ¯ δ A ) (3.3)10here V denotes the value function and J is given by J ( Y ) = (cid:90) ∞ (cid:18) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:19) dt. (3.4)If we take Y ∈ A D ( y ) such that Y t = (cid:0) t − √ y (cid:1) , for t ∈ [0 , √ y ], and Y t = 0, for t > √ y ,then it can be checked that J ( Y ) = (cid:90) √ y (cid:18) κ A (cid:16)(cid:0) t − √ y (cid:1) (cid:17) + A (cid:0) √ y − t (cid:1) F (cid:0) √ y − t (cid:1)(cid:19) dt < ∞ , (3.5)which implies that V < ∞ . Lemma 3.3 implies κ A ≥
0, if µ ≤
0. Hence we have 0 ≤ V < ∞ ,for all µ ≤ µ >
0. Then Lemma3.3 (iv) implies that there exists some constant z > −∞ < κ A ( z ) <
0. Supposethat the investor’s initial stock position is z and consider the strategy Y ∈ A D ( z ) satisfying Y (cid:48) t = − ξ t = 0 for t ∈ [0 , s ] with some s >
0. Then V ( z ) ≤ (cid:90) s κ A ( z ) dt + V ( z ) = sκ A ( z ) + V ( z ) . This can happen only if V ( z ) = −∞ . Let ¯ Y ∈ A D ( y ) with y ≥ z and set t z = inf { t ≥ | ¯ Y t = z } < ∞ . Then V ( y ) ≤ (cid:90) t z (cid:18) κ A ( ¯ Y t ) + A ¯ ξ t F ( ¯ ξ t ) (cid:19) dt + V ( z ) , which implies that V ( y ) = −∞ . As z can be chosen to be arbitrarily close to zero, it followsthat V ( y ) = −∞ , for all y ∈ (0 , ¯ δ A ). We therefore conclude that the value function isdegenerate when µ >
0. Let y ∈ (0 , ¯ δ A ), and suppose (in order to get a contradiction) thatthere exists an optimal strategy Y ∗ ∈ A D ( y ). Define ˜ κ A to be the function which is identicalto κ A with µ = 0. Then with reference to the L´evy-Khintchine representation of L (see(7.5)), we have κ A ( x ) = − Aµx + ˜ κ A ( x ). By Assumption 2.3 and Lemma 3.3, we have that˜ κ A ( Y ∗ t ) + Aξ t F ( ξ ∗ t ) is positive. Thus, V ( y ) = (cid:90) ∞ (cid:18) − AµY ∗ t + ˜ κ A ( Y ∗ t ) + Aξ ∗ t F ( ξ ∗ t ) (cid:19) dt = −∞ , µ > , implies (cid:82) ∞ Y ∗ t dt = ∞ , which contradicts the definition of an admissible strategy. We concludethat if µ >
0, then there is no optimal admissible liquidation strategy.
Remark 3.6.
It is mentioned in Schied et al. (2010) that for the Almgren-Chriss model withBrownian motion describing the unaffected stock price, the problem of optimising the finalcost/reward for a CARA investor over a set of adapted strategies provides the same optimalsolution as for the problem of optimising for the same model over deterministic strategies,11ut with a mean-variance optimisation criterion. When the unaffected stock price is nota Brownian motion, but a general L´evy process, this relationship no longer holds. To seethis, we know that for our optimisation problem, the set of admissible strategies A ( y ) can bereplaced by A D ( y ). Then in view of (2.9), it suffices to considerinf Y ∈A D ( y ) E (cid:2) e − AC Y ∞ (cid:3) , where C Y ∞ = c + sy − αy + (cid:90) ∞ Y t − dL t − (cid:90) ∞ ξ t F ( ξ t ) dt. It can be calculated that E [ C Y ∞ ] = c + sy − αy + µ (cid:90) ∞ Y t dt − (cid:90) ∞ ξ t F ( ξ t ) dt and Var( C Y ∞ ) = σ (cid:90) ∞ Y t dt + (cid:90) ∞ (cid:18)(cid:90) R Y t x ν ( dx ) (cid:19) dt. Then, E (cid:2) exp (cid:0) − AC Y ∞ (cid:1)(cid:3) = exp (cid:20) − A E [ C Y ∞ ] + 12 A σ (cid:90) ∞ Y t dt + (cid:90) ∞ (cid:90) R (cid:16) e − AY t x − AY t x (cid:17) ν ( dx ) dt (cid:21) = exp (cid:20) − A E [ C Y ∞ ] + 12 A Var( C Y ∞ ) + (cid:90) ∞ (cid:90) R (cid:16) e − AY t x − AY t x − A Y t x (cid:17) ν ( dx ) dt (cid:21) . From the above expression, it is clear that the problem is equivalent tosup Y ∈A D ( y ) E [ C Y ∞ ] − A Var( C Y ∞ ) , if ν ( R ) ≡
0, i.e. the L´evy process L has no jumps. However, for any general L´evy process,this equivalence does not hold. Remark 3.7.
Suppose that the investor is also allowed to buy shares. Then in order for thefinal cash position to be well-defined, we need, in addition to the conditions in Definition 2.2,to assume that any admissible strategy Y satisfy lim t →∞ t (cid:107) Y t (cid:107) L ∞ ( P ) = 0 (see Lemma 7.1 andproof of Proposition 2.5 for more details). We also suppose Y is non-negative, that Y t < ¯ δ A for all t ≥
0, and that Y t = y + (cid:82) t ξ u du with ξ t ∈ R . Denote by A ± ( y ) the set of all suchadmissible strategies, and by A ± D ( y ) the collection of all deterministic admissible strategies.Then by similar arguments as previously, the liquidation problem can be reduced to V ( y ) = inf Y ∈A ± D ( y ) (cid:90) ∞ (cid:18) κ A ( Y t ) + A | ξ t | F (cid:0) | ξ t | (cid:1)(cid:19) dt. Y ∈ A ± D ( y ) be a strategy including intermediate buying. Then there exist times r and s with r < s such that Y r = Y s and Y t > Y r for all t ∈ ( r, s ). Consider an admissible strategy X such that X t = Y r for t ∈ ( r, s ) and X t = Y t for t ∈ [0 , r ] ∪ [ s, ∞ ). Then with reference toLemma 3.3, (cid:90) sr (cid:18) κ A ( X u ) + A | ξ Xu | F (cid:0) | ξ Xu | (cid:1)(cid:19) du = κ A ( X r )( s − r ) < (cid:90) sr (cid:18) κ A ( Y u ) + A | ξ u | F (cid:0) | ξ u | (cid:1)(cid:19) du, where ξ X is the speed process associated with X . Therefore, J ( X ) < J ( Y ). This shows Y is not optimal. So even if it is allowed, it is not optimal to do any intermediate buying ofshares. With reference to the previous section, recall that the original optimal liquidation problem(2.8) is equivalent to solving V ( y ) = inf Y ∈A D ( y ) (cid:90) ∞ (cid:18) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:19) dt, with dY t = − ξ t dt, Y = y ∈ [0 , ¯ δ A ) . According to the theory of optimal control, the corresponding Hamilton-Jacobi-Bellman equa-tion is given by κ A ( y ) + inf x ≥ (cid:8) AxF ( x ) − xv (cid:48) ( y ) (cid:9) = 0 , (4.1)with associated boundary condition v (0) = 0. Let G : [0 , ∞ ) → [0 , ∞ ) denote the inverse func-tion of x (cid:55)→ x F (cid:48) ( x ). Assumption 2.3 and Lemma 2.4 together imply that G is a continuous,strictly increasing function satisfying G (0) = 0. Proposition 4.1.
Equation (4.1) with boundary condition v (0) = 0 has a classical solutiongiven by v ( y ) = (cid:90) y (cid:26) κ A ( u ) G (cid:0) κ A ( u ) A (cid:1) + AF (cid:18) G (cid:18) κ A ( u ) A (cid:19)(cid:19)(cid:27) du, ≤ y < ¯ δ A . (4.2)The next result provides an expression for the optimal liquidation strategy, and statesthat the value function V identifies with the function v in (4.2). Theorem 4.2.
Let y ∈ [0 , ¯ δ A ) . Define τ = (cid:90) y G (cid:0) κ A ( u ) A (cid:1) du. (4.3)13 et Y ∗ satisfy (cid:90) yY ∗ t G (cid:0) κ A ( u ) A (cid:1) du = t, if t ≤ τ, and Y ∗ t = 0 , if t > τ. (4.4) Then Y ∗ ∈ A D ( y ) , and its associated speed process ξ ∗ satisfies ξ ∗ t = G (cid:18) κ A ( Y ∗ t ) A (cid:19) , for all t ≥ . (4.5) Moreover, V in (3.3) is equal to v in (4.2), for all y ∈ [0 , ¯ δ A ) , and Y ∗ is the unique optimalliquidation strategy for problem (2.8). Note that since is G continuous, (4.5) implies that the strategy Y ∗ in (4.4) is continuouslydifferentiable. Since the functions κ A and G are both strictly increasing, it follows from (4.5)that with a larger stock position at time t , the associated optimal liquidation speed at time t islarger. Moreover, it can be shown by the strict convexity of the cumulant generating functionof L that A (cid:55)→ κ A ( x ) /A is strictly increasing. Hence, the optimal liquidation speed at anytime is strictly increasing in the risk aversion parameter A . These two relations coincide withthe intuition that with a larger position of shares, the investor potentially encounters biggerrisk from the market volatility, as any tiny fluctuation of the stock price will be amplified bythe large number of shares held. It is therefore optimal to liquidate faster. Also if the investoris more risk averse, then he cares more about the volatility risk, which makes him employ aliquidation strategy with a larger speed of sale. Observe that given an initial stock position y ∈ [0 , ¯ δ A ), the quantity τ in (4.3) indicates the liquidation time for the optimal liquidationstrategy Y ∗ . Depending on the properties of the temporary impact function F , τ may or maynot be finite. The next result provides some sufficient conditions for the optimal liquidationstrategy Y ∗ to have a finite liquidation time. Proposition 4.3.
Under the condition that y > (i) suppose µ < and there exist constants p < and K > such that lim x → x p F (cid:48) ( x ) = K ,then τ < ∞ .(ii) suppose µ = 0 and there exist constants p < and K > such that lim x → x p F (cid:48) ( x ) = K .If p ∈ [0 , , then τ = ∞ . If p < , then τ < ∞ . In models for stock prices involving L´evy processes, it is common to consider exponential L´evyprocesses (see e.g. Madan and Seneta, 1990; Eberlein and Keller, 1995; Barndorff-Nielsen,1997, etc). However, it is common in the optimal liquidation literature to use linear model14s opposed to exponential models due to tractability and the short time horizons involved.For practical implementation of our model one could of course directly fit the data to a linearL´evy model. However families of distributions that fit observed stock market data well forthe exponential L´evy model are known and obviously the distribution of the jumps changewhen you take the exponential. We therefore investigate how to linearise exponential L´evymodels and how this affects the L´evy measure. To this end, we are going to derive a L´evyprocess which can be regarded as a linear approximation for a corresponding exponential L´evyprocess. We show that this L´evy process satisfies all of the assumptions of being the drivingprocess of the unaffected stock price in the liquidation model introduced in previous sections.Therefore, our optimal liquidation strategy derived in the previous section can be regardedas an approximation for the result of the corresponding exponential L´evy model. This linearapproximation argument is reasonable since (the majority of) liquidation is usually completedin a very short time.Consider a non-trivial, one dimensional, F -adapted L´evy process ˜ L which admits thecanonical decomposition˜ L t = ˜ µt + ˜ σ ˜ W t + (cid:90) | z |≥ z ˜ N ( t, dz ) + (cid:90) | z | < z (cid:0) ˜ N ( t, dz ) − t ˜ ν ( dz ) (cid:1) , t ≥ , (5.1)where ˜ µ ∈ R and ˜ σ ≥ W is a standard Brownian motion, ˜ N is a Poissonrandom measure which is independent of ˜ W with compensator t ˜ ν ( dz ), and ˜ ν is the L´evymeasure associated with ˜ L . We assume that ˜ L possesses the following properties. Assumption 5.1.
We assume that ˜ ν is absolutely continuous with respect to Lebesgue mea-sure, and that (cid:90) | z |≥ e z ˜ ν ( dz ) < ∞ . (5.2)Suppose the unaffected stock price is described by the process ˜ S u satisfying˜ S ut = ˜ s exp (cid:0) ˜ L t (cid:1) , t ≥ , where ˜ s > S ut is besquare integrable, for all t ≥ t ≥ S t = ˜ s exp (cid:0) ˜ L t (cid:1) + I t , where I t = α ( Y t − Y ) − F ( ξ t ) is the price impact at time t appearing in the previous liqui-dation model with function F satisfying Assumption 2.3 (Gatheral and Schied, 2011, studya liquidation model with the affected stock price in this form with a geometric Brownianmotion). By Itˆo’s formula, for all t ≥
0, ˜ S t can be rewritten as˜ S t = ˜ s + (cid:90) t ˜ S uu − ˜ m du + (cid:90) t ˜ S uu − ˜ σ d ˜ W u + (cid:90) t (cid:90) R ˜ S uu − (cid:0) e z − (cid:1) (cid:0) ˜ N ( t, dz ) − t ˜ ν ( dz ) (cid:1) + I t , m = ˜ µ + ˜ σ + (cid:82) R (e z − − z {| z | < } ) ˜ ν ( dz ). In order to approximate the exponentialL´evy model, consider the process ˆ S such thatˆ S t = ˜ s + ˜ s ˜ mt + ˜ s ˜ σ ˜ W t + (cid:90) R ˜ s (cid:0) e z − (cid:1) (cid:0) ˜ N ( t, dz ) − t ˜ ν ( dz ) (cid:1) + I t , t ≥ , which can be considered as a linear approximation of ˜ S . Recall that the affected stock pricein the preceding model is given by S t = s + L t + I t , t ≥ , where L t = µt + σW t + (cid:82) R x (cid:0) N ( t, dx ) − tν ( dx ) (cid:1) . Comparing this to the expression of ˆ S t , itcan be seen that if we take s = ˜ s and choose L to be such that L t = ˜ s ˜ mt + ˜ s ˜ σ ˜ W t + (cid:90) R ˜ s (cid:0) e z − (cid:1) (cid:0) ˜ N ( t, dz ) − t ˜ ν ( dz ) (cid:1) , t ≥ , (5.3)then it follows that ˆ S t = ˜ s + L t + I t , for all t ≥ . We may therefore consider ˆ S as the affected stock price process in the liquidation modelintroduced in previous sections. The next proposition verifies that L with the above expressionis a L´evy process satisfying Assumption 2.1. Proposition 5.2.
Let L be given by (5.3). Write ˆ L = L/ ˜ s . Then ˆ L is an F -adapted L´evyprocess whose L´evy measure, denoted by ˆ ν , satisfies ˆ ν ( dx ) = 1 x + 1 ˜ f (cid:16) ln( x + 1) (cid:17) dx, x > − , x (cid:54) = 0 . Therefore, L is an F -adapted L´evy process satisfying Assumption 2.1. Remark 5.3.
From equation (7.20) (in the proof of Proposition 5.2) we know that (cid:90) | x |≥ e ux ˆ ν ( dx ) < ∞ , for all u ≤ . This implies that ¯ δ given by (2.1) is equal to + ∞ , and therefore, Assumption 3.1 is satisfiedfor any initial stock position y >
0. In other words, if we consider an exponential L´evy modeland use the approximation scheme discussed above, we do not need to concern any restrictionon the maximum volume of liquidation.With L given by (5.3) and ˆ L defined in Proposition 5.2, in view of (3.3)-(3.4) we considerthe optimisation problem V ( y ) = inf Y ∈A D ( y ) (cid:90) ∞ (cid:18) ˆ κ ˜ A ( Y t ) + Aξ t F ( ξ t ) (cid:19) dt, y ≥ , (5.4)where A > A = A ˜ s and ˆ κ ˜ A : [0 , ∞ ) → [0 , ∞ ) is definedby ˆ κ ˜ A ( x ) = ˆ κ ( − ˜ Ax ) with ˆ κ being the cumulant generating function of ˆ L .16 heorem 5.4. The unique optimal liquidation speed for problem (5.4) is given by ξ ∗ t = G (cid:18) ˆ κ ˜ A ( Y ∗ t ) A (cid:19) , t ≥ , (5.5) where G : [0 , ∞ ) → [0 , ∞ ) is the inverse function of x (cid:55)→ x F (cid:48) ( x ) and Y ∗ is the associatedunique optimal admissible stock position process satisfying (cid:90) yY ∗ t G (cid:0) ˆ κ ˜ A ( u ) A (cid:1) du = t, if t ≤ τ, and Y ∗ t = 0 , if t > τ, with τ defined by τ = (cid:90) y G (cid:0) κ A ( u ) A (cid:1) du. The value function in (5.4) satisfies V ( y ) = (cid:90) y (cid:26) ˆ κ ˜ A ( u ) G (cid:0) ˆ κ ˜ A ( u ) A (cid:1) + AF (cid:18) G (cid:18) ˆ κ ˜ A ( u ) A (cid:19)(cid:19)(cid:27) du, y ≥ . In this section, we provide some examples following the approximation scheme discussed in theprevious section. We consider the process ˜ L in (5.1) as a variance gamma (VG) L´evy process,which is obtained by subordinating a Brownian motion using a gamma process. Precisely, weconsider ˜ L to be such that ˜ L t = θτ t + ρW τ t , t ≥ , where θ ∈ R and ρ > τ is agamma process such that τ t ∼ Γ (cid:0) tη , η (cid:1) , for some constant η >
0. Then ˜ L is a VG L´evyprocess whose L´evy density is given by˜ f ( z ) = 1 η | z | e Cz − D | z | , z ∈ R , where C = θρ and D = (cid:113) θ + ρ η ρ , Γ( a, b ) denotes a gamma distribution with shape parameter a > b >
0, for which theprobability density function is given by f ( x ) = b a Γ( a ) x a − e − bx , for x >
0, where Γ( · ) is the gamma function.For any X ∼ Γ( a, b ), E [ X ] = ab and Var[ X ] = ab . κ admits the expression˜ κ ( x ) = − η ln (cid:18) − x ρ η − θηx (cid:19) (6.1)(see e.g. Cont and Tankov, 2004). It can be shown that Assumption 5.1 is satisfied if D − C > ν of the process ˆ L satisfiesˆ ν ( dx ) = − η ln( x + 1) ( x + 1) C + D − dx, x ∈ ( − , , η ln( x + 1) ( x + 1) C − D − dx, x ∈ (0 , ∞ ) . Therefore, the function ˆ κ ˜ A : [0 , ∞ ) → [0 , ∞ ) in (5.4), denoting it by ˆ κ V G ˜ A in the example ofVG L´evy process, is given byˆ κ V G ˜ A ( u ) = − ˜ A ˜ mu + (cid:90) ∞− (cid:16) e − ˜ Aux − Aux (cid:17) ˆ ν ( dx ) , (6.2)where the drift parameter ˜ m = ˜ κ (1).The next result provides a lower bound for ˆ κ V G ˜ A , which later will be useful for decidingthe limit behaviour of the price impact function. Proposition 6.1.
For u ≥ , write ˆ κ V G ˜ A ( u ) = − ˜ A ˜ mu + e η (cid:20) − e ˜ Au ˜ AuC + D + 2 (cid:18) Au ∧ (cid:19) C + D +2 + e ˜ Au C + D + 1 (cid:18) Au ∧ (cid:19) C + D +1 + ˜ AuC + D + 2 − AuC + D + 1 (cid:21) , and in particular, for u ≥ A , ˆ κ V G ˜ A ( u ) = − ˜ A ˜ mu + e η (cid:20)(cid:18) Au (cid:19) C + D +1 e ˜ Au (cid:18) C + D + 1 − C + D + 2 (cid:19) + ˜ AuC + D + 2 − AuC + D + 1 (cid:21) . Then we have ˆ κ V G ˜ A ( u ) ≥ ˆ κ V G ˜ A ( u ) , for all u ≥ . In order to compare the optimal strategy for the model involving a VG L´evy process andthe optimal strategy for the corresponding model with a Brownian motion (i.e. when ˜ L is aBrownian motion), we derive that the function ˆ κ ˜ A : [0 , ∞ ) → [0 , ∞ ) in (5.4), denoting it byˆ κ BM ˜ A , is given by ˆ κ BM ˜ A ( u ) = − ˜ A (cid:16) ˜ µ + ˜ σ (cid:17) u + 12 ˜ A ˜ σ u , (6.3)18here ˜ µ ∈ R and ˜ σ > L ,respectively. In the case, Assumption 5.1 is always satisfied.Throughout this section we use the following parameters for our VG L´evy process: θ = − . ρ = 0 .
02 and η = 0 . ˜ L t match that of the VG model. Hence, ˜ µ and ˜ σ in (6.3) are taken to besuch that ˜ µ + ˜ σ = ˜ κ (1) and 2˜ µ + 2˜ σ = ˜ κ (2), where ˜ κ is given by (6.1). Therefore, throughoutthis section, ˜ µ = 2˜ κ (1) − ˜ κ (2)2 and ˜ σ = ˜ κ (2) − κ (1) . Moreover, we choose ˜ s = 100 for simplicity. Consider the power-law temporary impact function, i.e. F : [0 , ∞ ) → [0 , ∞ ) is given by F ( x ) = βx γ , where β > γ > F satisfies Assumption 2.3, and thefunction G appearing in (5.5) is given by G ( x ) = (cid:18) xβγ (cid:19) γ +1 , x ≥ . Applying Proposition 4.3, we see that if ˆ L is a strict supermartingale, then τ in (4.3) is finite,for all γ >
0; if ˆ L is a martingale, then τ = ∞ for γ ∈ (0 , τ < ∞ when γ >
1. Itfollows from (5.5) that the optimal liquidation speed takes the expression ξ ∗ t = (cid:18) ˆ κ ˜ A ( Y ∗ t ) Aβγ (cid:19) γ +1 , for all t ≥ . (6.4)We adopt the values of β and γ suggested in Almgren et al. (2005) where parameters of thepower-law temporary impact are studied empirically. In particular, we take γ = 0 . β = 4 . × − . With our notations, the temporary impact function F in Almgren et al. (2005) is given by F ( x ) = βx γ =˜ S − ˜ β ˜ σ (cid:0) x ˜ V (cid:1) γ , where ˜ V denotes the daily volume of a given stock, the value of exponent γ is argued to be0 . β is a constant which is suggested to be 0 . Optimal liquidation trajectories for variance gamma L´evy process model and Brownian motionmodel with 0.6 power-law temporary impact function. Thin curves are for A = 10 − , dashed curves are when A = 10 − and thick curves are for A = 10 − . × . Suppose the investor wants to liquidatea position of 2 × of this stock. Figure 1 shows the optimal liquidation trajectoriesfor both the VG L´evy process case and the Brownian motion case when the risk aversionparameter A takes values of 10 − , 10 − and 10 − . We see that when A = 10 − , theoptimal strategies for the two models are almost identical. As A increases, the optimal speedsincrease in both models, and in particular, speeds increase much faster in the VG model forbig positions. In each case, liquidation finishes in a short time period, which confirms thatthe linear approximation of the exponential model is reasonable.As shown in the first graph of Figure 1 when A = 10 − and A = 10 − , the stock positionsdrop immediately by a large proportion of its initial value. In order to get more details aboutthese two trajectories, we compute that when A = 10 − the time spent on liquidating 40%of 2 × shares is about 0 . . A = 10 − , then according to the optimal strategy for VGmodel, he spends roughly 1 . × − amount of time to liquidate 90% of his initial position.With a large stock position, due to the nature of jumps of the VG L´evy process we expectthat the investor would liquidate much faster than the optimal strategy for Brownian motionmodel. However, the above examples show that with the 0.6 power-law temporary impactfunction, in the VG case, optimal liquidation speeds can be too large for the optimal strategyto be practical, while speeds in the Brownian motion model stay in a reasonable range.Intuitively, an unrealistically high optimal liquidation speed can be due to price impact for alarge trading speed being underestimated. In other words, the cost resulting from large speedsis too small. This argument can be confirmed by the expression of the optimal liquidationspeed in (5.5) that if the temporary impact function F has a small growth rate, then growthrate of function G is large, and therefore optimal speed can be very high, when stock positionis large. It is mentioned in Ro¸su (2009); Gatheral (2010) that the impact function shouldbe concave for small trading speeds and convex for large speeds. Therefore, we next try toexplore a mode of growth of the price impact function for which the optimal liquidation speeds of parameters of the VG L´evy process that we have chosen, it can be calculated that the volatility ˜ σ in theBrownian motion case is roughly equal to 0 .
02. Comparing this number to the values of volatilities and dailyvolumes of stocks provided in examples in Almgren et al. (2005), we may take ˜ V = 2 × as a reasonablechoice. Moreover, we choose ˜ s = 100 for simplicity. Then β is calculated to be 4 . × − .Note that the empirical study in Almgren et al. (2005) is based on a model parametrised by the volume timewhich is defined as fractions of a daily volume. Therefore, any results of number regarding time derived froma model with power-law impact function in this section should be interpreted as volume time. Since the study of parameters of impacts in Almgren et al. (2005) is based on liquidating the amount ofshares that weighted as 10% of daily volume, in order to keep consistent with the values of parameters of thetemporary impact function that we have chosen, we let the initial stock position to be 2 × which is 10% ofthe daily volume that we have chosen as explained before. In view of the general literature on preferences, these values of A may seem small. However, in the contextof a liquidation model they can viewed as reasonable if the investor is not sensitive to any large costs whichare insignificant compared to his total wealth. We refer to Almgren and Chriss (2000) and Almgren (2003) formore details about the risk aversion parameter for the Almgren-Chirss liquidation model. In this section we derive a connection between a temporary impact function for the L´evyliquidation model and a temporary impact function for the Brownian motion liquidationmodel such that the two respective optimal strategies coincide with each other.Let F L : [0 , ∞ ) → [0 , ∞ ) and F BM : [0 , ∞ ) → [0 , ∞ ) be temporary impact functionssatisfying Assumption 2.3 considered in a L´evy model and a Brownian motion model, respec-tively. We denote by G L : [0 , ∞ ) → [0 , ∞ ) and G BM : [0 , ∞ ) → [0 , ∞ ) the inverse functionsof x (cid:55)→ x ( F L ) (cid:48) ( x ) and x (cid:55)→ x ( F BM ) (cid:48) ( x ), respectively. Then in view of (5.5), the optimalliquidation speed at time t for each model, denoted by ξ Lt and ξ BMt , are given by ξ Lt = G L (cid:18) ˆ κ L ˜ A ( Y Lt ) A (cid:19) and ξ BMt = G BM (cid:18) ˆ κ BM ˜ A ( Y BMt ) A (cid:19) , where ˆ κ L ˜ A and ˆ κ BM ˜ A are different versions for of ˆ κ ˜ A , and Y L and Y BM are correspondingoptimal liquidation strategies in each model. Suppose for all t ≥ Y ∗ t = Y Lt = Y BMt , then G L (cid:18) ˆ κ L ˜ A ( Y ∗ t ) A (cid:19) = G BM (cid:18) ˆ κ BM ˜ A ( Y ∗ t ) A (cid:19) , t ≥ . (6.5)Write z = G BM (cid:16) ˆ κ BM ˜ A ( Y ∗ t ) A (cid:17) . So by (6.3) we have Y ∗ t = ˜ u + (cid:112) ˜ u + 2 A ˜ σ z ( F BM ) (cid:48) ( z )˜ A ˜ σ , where ˜ u = ˜ µ + ˜ σ . Then from (6.5) we obtain that( F L ) (cid:48) ( z ) = 1 Az ˆ κ L ˜ A (cid:32) ˜ u + (cid:112) ˜ u + 2 A ˜ σ z ( F BM ) (cid:48) ( z )˜ A ˜ σ (cid:33) , which is equivalent to F L ( x ) = (cid:90) x Az ˆ κ L ˜ A (cid:32) ˜ u + (cid:112) ˜ u + 2 A ˜ σ z ( F BM ) (cid:48) ( z )˜ A ˜ σ (cid:33) dz. (6.6)It can be shown that Assumption 2.3 is satisfied by the above expression. We can thereforeconclude that if F L and F BM satisfy (6.6), then Y L = Y BM .Suppose F BM in (6.6) follows a power-law such that the optimal speed for the Brownianmotion case is practically realistic (this kind of model is indeed used in practice). Then itfollows from the relation in (6.6) that in order for the optimal speed in VG case to be prac-tically realistic, the function F L needs to increase to infinity faster than any power function.This is because for the VG L´evy process case, the lower bound of the function ˆ κ V G ˜ A given inProposition 6.1 tends to infinity faster than any power function.22 Proofs
Proof of Lemma 2.4 . For λ ∈ (0 ,
1) and x ∈ (0 , ∞ ), Assumption 2.3 (ii) and (iii) implythat F ( λx ) < λF ( x ) < F ( x ), which shows that F is strictly increasing.The derivative of x (cid:55)→ xF ( x ), together with the convexity of this function, implies thatlim x → xF (cid:48) ( x ) exists. As F (cid:48) ( x ) >
0, for all x >
0, it follows that lim x → xF (cid:48) ( x ) ≥
0. Supposelim x → xF (cid:48) ( x ) >
0. Then there exist constants ¯ x > c >
0, such that for all x ∈ (0 , ¯ x ), F (cid:48) ( x ) > cx . But then, F (¯ x ) = lim x → (cid:90) ¯ xx F (cid:48) ( u ) du ≥ lim x → (cid:90) ¯ xx cu du = ∞ , which contradicts the continuity of F . Hence, lim x → xF (cid:48) ( x ) = 0, and it therefore followsthat lim x → x F (cid:48) ( x ) = 0.The next lemma is used in the proof of Proposition 2.5. Lemma 7.1.
Let Z be a positive-valued, decreasing process satisfying (cid:82) ∞ Z t dt < ∞ . Then tZ t → , as t → ∞ .Proof. Suppose lim inf t →∞ tZ t >
0, then there exists some constant c such thatlim inf t →∞ tZ t > c > . This implies that we can find some s ≥ t ≥ s , Z t > ct . Hence (cid:90) ∞ s Z t dt ≥ (cid:90) ∞ s ct dt = ∞ , which contradicts (cid:82) ∞ Z t dt < ∞ . Thus, we have shown thatlim inf t →∞ tZ t = 0 . (7.1)We know that Z is a decreasing process, which is of finite variation. By Itˆo’s formula wecalculate that tZ t = (cid:90) t u dZ u + (cid:90) t Z u du. It can be observed that t (cid:55)→ (cid:82) t udZ u is negative and decreasing while t (cid:55)→ (cid:82) t Z u du is positiveand increasing. Then,0 ≤ sup t ≥ r tZ t ≤ sup t ≥ r (cid:90) t u dZ u + sup t ≥ r (cid:90) t Z u du = (cid:90) r u dZ u + (cid:90) ∞ Z u du. (7.2)23lso, inf t ≥ r tZ t ≥ inf t ≥ r (cid:90) t u dZ u + inf t ≥ r (cid:90) t Z u du = (cid:90) ∞ u dZ u + (cid:90) r Z u du. (7.3)Taking r to infinity in (7.3) and (7.2), and by (7.1) we have0 ≤ lim sup t →∞ tZ t = lim r →∞ sup t ≥ r tZ t ≤ (cid:90) ∞ u dZ u + (cid:90) ∞ Z u du, t →∞ tZ t = lim r →∞ inf t ≥ r tZ t ≥ (cid:90) ∞ u dZ u + (cid:90) ∞ Z u du. Therefore, we conclude that lim t →∞ tZ t = 0. Proof of Proposition 2.5 . (i) Let f be the characteristic function of L t , so f ( u ) = E [e i uL t ] = e tψ ( u ) , where ψ ( u ) is given by the L´evy-Khintchine representation of L . By Assumption 2.1 weknow that f , hence ψ , are twice differentiable at 0. Hence, we calculate that f (cid:48) (0) =i E [ L t ] = tψ (cid:48) (0) and f (cid:48)(cid:48) (0) = − E [ L t ], and therefore, E [ L t ] = ( µt ) − ψ (cid:48)(cid:48) (0) t. Then, E (cid:2) ( L t Y t ) (cid:3) ≤ E [ L t ] (cid:107) Y t (cid:107) L ∞ ( P ) = µ (cid:0) t (cid:107) Y t (cid:107) L ∞ ( P ) (cid:1) − ψ (cid:48)(cid:48) (0) t (cid:107) Y t (cid:107) L ∞ ( P ) . (7.4)If µ (cid:54) = 0, then for any Y ∈ A ( y ), (cid:0) (cid:107) Y t (cid:107) L ∞ ( P ) (cid:1) t ≥ and (cid:0) (cid:107) Y t (cid:107) L ∞ ( P ) (cid:1) t ≥ are continuous, pos-itive and decreasing. The integrability condition in (2.2) implies that (cid:82) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ . Therefore, according to Lemma 7.1 we havelim t →∞ t (cid:107) Y t (cid:107) L ∞ ( P ) = 0 and lim t →∞ t (cid:107) Y t (cid:107) L ∞ ( P ) = 0 . Hence, by (7.4) and the finiteness of µ and ψ (cid:48)(cid:48) (0) we conclude thatlim T →∞ E (cid:2) ( L t Y t ) (cid:3) = 0 . When µ = 0, we get (cid:82) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ directly as a condition of admissible strategies.Therefore, the same result follows. 24ii) Using Cauchy-Schwarz inequality and Itˆo isometry we obtain E (cid:20) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T Y t − dL t (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≤ | µ | E (cid:20) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T Y t − dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) + E (cid:20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T Y t − d (cid:18) σW t + (cid:90) R x (cid:16) N ( t, dx ) − tν ( dx ) (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≤ | µ | (cid:90) T (cid:107) Y t (cid:107) L ∞ ( P ) dt + E (cid:20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T Y t − d (cid:18) σW t + (cid:90) R x (cid:16) N ( t, dx ) − tν ( dx ) (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) = | µ | (cid:90) T (cid:107) Y t (cid:107) L ∞ ( P ) dt + (cid:18) σ + (cid:90) R \{ } x ν ( dx ) (cid:19) E (cid:20)(cid:90) T Y t dt (cid:21) ≤ | µ | (cid:90) T (cid:107) Y t (cid:107) L ∞ ( P ) dt + (cid:18) σ + (cid:90) R \{ } x ν ( dx ) (cid:19) E (cid:20)(cid:90) T (cid:107) Y t (cid:107) L ∞ ( P ) dt (cid:21) From the existence of the first and second moments of L , we know that µ , σ and (cid:82) R \{ } x ν ( dx ) are all finite. The result then follows from the integrability conditions in(2.2) and (2.3) of an admissible strategy. Proof of Lemma 3.3 . (i) Let ψ ( u ) be given by the L´evy-Khintchine representation of L . Then for all u ∈ [0 , ¯ δ A ),we have κ A ( u ) = ψ (i Au ) = − Aµu + 12 A u σ + (cid:90) R (cid:16) e − Aux − Aux (cid:17) ν ( dx ) . (7.5)Therefore, κ A (0) = 0 follows directly.(ii) Observe that − Aµu , A u σ and e − Aux − Aux are all convex in u , and in particularthat A u σ and e − Aux − Aux are strictly convex in u . Thus, with reference to (7.5),the strict convexity of κ A can be concluded from the assumption that L is non-trivial.(iii) Let µ = 0. In view of (7.5), in order to proof lim x → κ A ( x ) x = K >
0, it suffices to showthat lim u → (cid:90) R (cid:18) e − Aux − AuxA u (cid:19) ν ( dx ) = K (cid:48) , for some constant K (cid:48) >
0. Let 0 < A ¯ u < ¯ δ A . It can be checked that for all u ∈ (0 , ¯ u ), (cid:12)(cid:12)(cid:12)(cid:12) e − Aux − AuxA u (cid:12)(cid:12)(cid:12)(cid:12) < x , if x > , (cid:12)(cid:12)(cid:12)(cid:12) e − Aux − AuxA u (cid:12)(cid:12)(cid:12)(cid:12) < e − A ¯ ux − A ¯ uxA ¯ u , if x < . Because of the finite second moment of L and the fact that κ A (¯ u ) < ∞ , both x and e − A ¯ ux − A ¯ uxA ¯ u are ν -integrable. Thus, by the dominated convergence theorem, it followsthat lim u → (cid:90) R (cid:18) e − Aux − AuxA u (cid:19) ν ( dx ) = (cid:90) R x ν ( dx ) = K (cid:48) , where K (cid:48) is some strictly positive constant.(iv) Let µ (cid:54) = 0. Then lim x → κ A ( x ) x = − Aµ follows from (7.5) as well as (iii). Proof of Lemma 3.4 . Let µ = 0. Then Lemma 3.3 (iii) implies that there exists strictlypositive constants ¯ x , C and C such that C x < κ A ( x ) < C x , for all x ∈ (0 , ¯ x ). Supposethat (cid:82) ∞ (cid:107) Y t (cid:107) L ∞ ( P ) dt < ∞ . Then Y t tends to zero as t tends to infinity. Hence, there exists s >
0, such that (cid:107) Y t (cid:107) L ∞ ( P ) ∈ (0 , ¯ x ), for all t > s . Then C (cid:90) ∞ s (cid:107) Y t (cid:107) L ∞ ( P ) dt < (cid:90) ∞ s κ A (cid:0) (cid:107) Y t (cid:107) L ∞ ( P ) (cid:1) dt < C (cid:90) ∞ s (cid:107) Y t (cid:107) L ∞ ( P ) dt, (7.6)from which it follows that (cid:82) ∞ s κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) du < ∞ . Since (cid:107) Y t (cid:107) L ∞ ( P ) is bounded for t ∈ [0 , s ], we have (cid:82) s κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) du < ∞ . A similar argument together with the inequality(7.6) also establishes the reverse implication. The proofs regarding the cases of µ < µ > Proof of Lemma 3.5 . By Itˆo’s formula and using the expression of κ A in (7.5) we calculatethat M Yt =1 − (cid:90) t M Yu − AY u − (cid:16)(cid:0) µ − (cid:90) R x ν ( dx ) (cid:1) du + σ dW u (cid:17) − (cid:90) t M Yu − (cid:16) κ A (cid:0) Y u − (cid:1) − A Y u − σ (cid:17) du + (cid:90) t (cid:90) R M Yu − (cid:16) e − AY u − x − (cid:17) (cid:16)(cid:0) N ( du, dx ) − ν ( dx ) du (cid:1) + ν ( dx ) du (cid:17) =1 − (cid:90) t M Yu − AY u − σ dW u + (cid:90) t (cid:90) R M u − (cid:16) e − AY u − x − (cid:17) (cid:0) N ( du, dx ) − ν ( dx ) du (cid:1) , which shows M is a local martingale. Define X t = (cid:90) t − AY u − d ˜ L u and K ( θ ) t = (cid:90) t ˜ κ A ( θY u ) du, θ ∈ [0 , Y ∈ A ( y ) with y ∈ [0 , ¯ δ A ), ˜ L is the martingale part of L and ˜ κ A is equal to κ A with µ = 0. It can be checked that the process M Y in (3.1) can be rewritten as M Y = exp (cid:0) X − K (1) (cid:1) . With reference to Definition 3.1 and Theorem 3.2 in Kallsen and Shiryaev (2002), in order toshow M Y is a uniformly integrable martingale, it is sufficent to check thatlim δ ↓ sup t ∈ R + δ log (cid:32) E (cid:20) exp (cid:18) δ (cid:16) (1 − δ ) K (1) t − K (1 − δ ) t (cid:17)(cid:19)(cid:21)(cid:33) = 0 , (7.7)for δ ∈ (0 , δ ↓ sup t ∈ R + δ log (cid:32) E (cid:20) exp (cid:18) δ (cid:16) (1 − δ ) K (1) t − K (1 − δ ) t (cid:17)(cid:19)(cid:21)(cid:33) ≤ lim δ ↓ sup t ∈ R + δ log (cid:32) exp (cid:32)(cid:13)(cid:13)(cid:13)(cid:13) δ (cid:16) (1 − δ ) K (1) t − K (1 − δ ) t (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( P ) (cid:33)(cid:33) = lim δ ↓ sup t ∈ R + (cid:13)(cid:13)(cid:13) (1 − δ ) K (1) t − K (1 − δ ) t (cid:13)(cid:13)(cid:13) L ∞ ( P ) = lim δ ↓ sup t ∈ R + (cid:13)(cid:13)(cid:13)(cid:13) (1 − δ ) (cid:90) t ˜ κ A ( Y u ) du − (cid:90) t ˜ κ A (cid:0) (1 − δ ) Y u (cid:1) du (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( P ) ≤ lim δ ↓ sup t ∈ R + (cid:90) t (cid:13)(cid:13)(cid:13) (1 − δ )˜ κ A ( Y u ) − ˜ κ A (cid:0) (1 − δ ) Y u (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ( P ) du ≤ lim δ ↓ (cid:90) ∞ (cid:13)(cid:13)(cid:13) (1 − δ )˜ κ A ( Y u ) − ˜ κ A (cid:0) (1 − δ ) Y u (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ( P ) du. (7.8)For δ ∈ (0 , (cid:13)(cid:13) (1 − δ )˜ κ A ( Y u ) − ˜ κ A (cid:0) (1 − δ ) Y u (cid:1)(cid:13)(cid:13) L ∞ ( P ) ≤ (cid:13)(cid:13) (1 − δ )˜ κ A ( Y u ) (cid:13)(cid:13) L ∞ ( P ) + (cid:13)(cid:13) ˜ κ A (cid:0) (1 − δ ) Y u (cid:1)(cid:13)(cid:13) L ∞ ( P ) = (1 − δ )˜ κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) + ˜ κ A (cid:0) (1 − δ ) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) ≤ κ A (cid:0) (cid:107) Y u (cid:107) L ∞ ( P ) (cid:1) . The last two steps are because ˜ κ A ( x ) is positive and non-decreasing for x ≥
0, which followfrom Lemma 3.3 (i), (ii) and (iii). According to (2.2) or (2.3) as well as Lemma 3.4, we have (cid:90) ∞ ˜ κ A (cid:0) (cid:107) Y t (cid:107) L ∞ ( P ) (cid:1) dt < ∞ . δ ↓ sup t ∈ R + δ log (cid:32) E (cid:20) exp (cid:18) δ (cid:16) (1 − δ ) K (1) t − K (1 − δ ) t (cid:17)(cid:19)(cid:21)(cid:33) ≤ (cid:90) ∞ lim δ ↓ (cid:13)(cid:13)(cid:13) (1 − δ )˜ κ A ( Y u ) − ˜ κ A (cid:0) (1 − δ ) Y u (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ( P ) du = 0 . (7.9)On the other hand, the convexity of ˜ κ A ( x ) and ˜ κ A (0) = 0 imply(1 − δ )˜ κ A ( x ) ≥ ˜ κ A (cid:0) (1 − δ ) x (cid:1) , for δ ∈ (0 , , hence, (1 − δ ) K (1) t − K (1 − δ ) t ≥ . Combining this with (7.9), we get (7.7).The next lemma is used in the proofs of Proposition 4.1 and Theorem 4.2.
Lemma 7.2.
Let the function F satisfy Assumption 2.3. Then x (cid:55)→ xG ( x ) is continuous on [0 , ∞ ) , where G : [0 , ∞ ) → [0 , ∞ ) is the inverse function of x (cid:55)→ x F (cid:48) ( x ) .Proof. Assumption 2.3 and Lemma 2.4 imply that G is continuous and G (0) = 0. Therefore,it is sufficient to check that lim x → xG ( x ) < ∞ . Let x = u F (cid:48) ( u ). Then it follows that xG ( x ) = uF (cid:48) ( u ). Hence, the result follows from the fact that u →
0, as x →
0, and lim u → uF (cid:48) ( u ) = 0(see Lemma 2.4). Proof of Proposition 4.1 . We first show that the function v given by (4.2) is continuouslydifferentiable, and note that it is sufficient to show that v (cid:48) ( y ) = κ A ( y ) G (cid:0) κA ( y ) A (cid:1) + AF (cid:0) G (cid:0) κ A ( y ) A (cid:1)(cid:1) is continuous on [0 , ¯ δ A ). This is the case if x (cid:55)→ xG ( x ) is continuous for x ≥
0. But this isdemonstrated by Lemma 7.2.Recall that the Hamilton-Jacobi-Bellman equation in our problem is κ A ( y ) + inf x ≥ (cid:8) AxF ( x ) − xv (cid:48) ( y ) (cid:9) = 0 . In order to prove that v in (4.2) is a solution to this equation, because AxF ( x ) − xv (cid:48) ( y ) isstrictly convex in x , it is enough to show that for all y ∈ [0 , ¯ δ A ), there exists x ∗ ≥ Ax ∗ F (cid:48) ( x ∗ ) + AF ( x ∗ ) − v (cid:48) ( y ) = 0 (7.10)and κ A ( y ) + Ax ∗ F ( x ∗ ) − x ∗ v (cid:48) ( y ) = 0 , (7.11)28here the equality in (7.10) comes from the first-order condition of optimality of the expression AxF ( x ) − xv (cid:48) ( y ). But with v (cid:48) ( y ) = κ A ( y ) G (cid:0) κA ( y ) A (cid:1) + AF (cid:0) G (cid:0) κ A ( y ) A (cid:1)(cid:1) , it can be checked that x ∗ = G (cid:0) κ A ( y ) A (cid:1) satisfies both (7.10) and (7.11). The boundary condition v (0) = 0 is a consequenceof the expression of v ( y ) and the continuity of v ( y ) at y = 0. Proof of Theorem 4.2 . We know that when t ≤ τ , (cid:90) yY ∗ t G (cid:0) κ A ( u ) A (cid:1) du = t, from which it follows that ξ ∗ t = − dY ∗ t dt = G (cid:18) κ A ( Y ∗ t ) A (cid:19) , t ≤ τ. On the other hand, when t > τ , Y ∗ t = 0. Hence, ξ ∗ t = 0 = G (cid:18) κ A ( Y ∗ t ) A (cid:19) , t > τ. We next prove that Y ∗ ∈ A D ( y ). It is clear that Y ∗ is deterministic and absolutelycontinuous. The non-negativity of G implies that Y ∗ is non-increasing. It remains to showthat if µ <
0, then (cid:82) ∞ Y ∗ t dt < ∞ ; and if µ = 0, then (cid:82) ∞ (cid:0) Y ∗ t (cid:1) dt < ∞ . However, withreference to Lemma 3.4, it is enough to check that (cid:90) ∞ κ A (cid:0) Y ∗ t (cid:1) dt = (cid:90) τ κ A (cid:0) Y ∗ t (cid:1) dt < ∞ . By a change of variable, we have that (cid:90) τ κ A (cid:0) Y ∗ t (cid:1) dt = (cid:90) y − κ A (cid:0) Y ∗ t (cid:1) G (cid:16) κ A ( Y ∗ t ) A (cid:17) dY ∗ t < ∞ , where the finiteness is du to the continuity of the integrand on the compact interval [0 , y ],which is implied by Lemma 7.2.With reference to (7.10) and (7.11), the function v in (4.2) satisfies κ A ( y ) + AξF ( ξ ) − ξv (cid:48) ( y ) ≥ , for all ξ ≥ , (7.12)and equality holds only when ξ = G (cid:0) κ A ( y ) A (cid:1) . Let Y ∈ A D ( y ). Observe that v ( Y T ) = v ( y ) − (cid:90) T v (cid:48) ( Y t ) ξ t dt. T to infinity and using the boundary condition v (0) = 0, it follows that v ( y ) = (cid:90) ∞ v (cid:48) ( Y t ) ξ t dt. Then by (7.12) we have v ( y ) ≤ (cid:90) ∞ (cid:16) κ A ( Y t ) + Aξ t F ( ξ t ) (cid:17) dt. (7.13)Now consider the strategy Y ∗ in (4.4), which has a speed process ξ ∗ satisfying ξ ∗ t = G (cid:0) κ A ( Y ∗ t ) A (cid:1) ,for all t ≥
0. Then, κ A (cid:0) Y ∗ t (cid:1) + Aξ ∗ t F (cid:0) ξ ∗ t (cid:1) − ξ ∗ t v (cid:48) (cid:0) Y ∗ t (cid:1) = 0 , t ≥ , hence, v ( y ) = (cid:90) ∞ (cid:16) κ A (cid:0) Y ∗ t (cid:1) + Aξ t F (cid:0) ξ ∗ t (cid:1)(cid:17) dt. This together with (7.13) implies that V ( y ) = v ( y ), for all y ∈ [0 , ¯ δ A ). Therefore, withreference to the analysis after equation (3.2), we get that Y ∗ is the unique optimal strategyto problem (2.8). Proof of Proposition 4.3 . (i) Suppose µ < p < x → x p F (cid:48) ( x ) = K , with K being somestrictly positive constant. Write u = x F (cid:48) ( x ). Then we have u − p G ( u ) = (cid:0) x p F (cid:48) ( x ) (cid:1) − p . By letting x tend to 0, so u tends to 0 as well, it follows thatlim u → u − p G ( u ) = K − p . (7.14)Lemma 3.3 (iv) together with (7.14) giveslim x → x − p G (cid:0) κ A ( x ) A (cid:1) = K (cid:48) , for some other constant K (cid:48) >
0. Therefore, there exist strictly positive constants K , K and ¯ x such that for all x ∈ (0 , ¯ x ), K x − p < G (cid:0) κ A ( x ) A (cid:1) < K x − p . x → (cid:90) ¯ xx K u − p du ≤ lim x → (cid:90) ¯ xx G (cid:0) κ A ( u ) A (cid:1) du ≤ lim x → (cid:90) ¯ xx K u − p du. Observe that p < − p <
1, and therefore (cid:82) ¯ x u − p du < ∞ . Hence,lim x → (cid:90) ¯ xx G (cid:0) κ A ( u ) A (cid:1) du < ∞ . Then the required result follows from (4.3) and the fact that (cid:82) y ¯ x G (cid:0) κA ( u ) A (cid:1) du < ∞ , if theinitial stock position y > ¯ x .(ii) Suppose µ = 0. Observe that (7.14) implieslim x → x − p G ( x ) = C, for some constant C >
0. Combining this with Lemma 3.3 (iii), we obtainlim x → x − p G (cid:0) κ A ( x ) A (cid:1) = C (cid:48) , for some other constant C (cid:48) >
0. Then there exist strictly positive constants C , C and¯ x such that for all x ∈ (0 , ¯ x ), C x − p < G (cid:0) κ A ( x ) A (cid:1) < C x − p . Therefore, lim x → (cid:90) ¯ xx C u − p du ≤ lim x → (cid:90) ¯ xx G (cid:0) κ A ( u ) A (cid:1) du ≤ lim x → (cid:90) ¯ xx C u − p du. If p <
0, then − p <
1. Hence τ < ∞ is obtained by the same argument as in (i) of thisproof. If p ∈ [0 , − p ≥
1. It follows that (cid:82) ¯ x G (cid:0) κA ( u ) A (cid:1) du = ∞ , and therefore τ = ∞ . 31 roof of Proposition 5.2 . We show that ˆ L given byˆ L t = ˜ mt + ˜ σ ˜ W t + (cid:90) t (cid:90) R (cid:0) e z − (cid:1) (cid:0) ˜ N ( dt, dz ) − ˜ ν ( dz ) dt (cid:1) , t ≥ , (7.15)is a L´evy process. Define a random measure ˆ N : Ω × B (cid:0) [0 , ∞ ) (cid:1) ⊗ B ( R ) → Z + and a measureˆ ν : B ( R ) → Z + to be such that if B ∈ B ( R ) and B ∩ ( − , ∞ ) (cid:54) = ∅ , thenˆ N ( ω, A, B ) = ˜ N (cid:16) A , ln (cid:0) B ∩ ( − , ∞ ) + 1 (cid:1)(cid:17) ( ω ) , ˆ ν ( B ) = ˜ ν (cid:16) ln (cid:0) B ∩ ( − , ∞ ) + 1 (cid:1)(cid:17) ; (7.16)otherwise, they are both equal 0, where Z + is the set of all positive integers and ln( B ∩ ( − , ∞ ) + 1) = { ln( x + 1) | x ∈ B ∩ ( − , ∞ ) } ( we have for all A ∈ B ([0 , ∞ )) and ω ∈ Ω,˜ N ( A, { } )( ω ) = ˜ ν ( { } ) = 0 ). Write ˆ N ( · , · ) = ˆ N ( ω, · , · ). Then by writing x = e z −
1, it followsfrom (7.15) that ˆ L t = ˜ mt + ˜ σ ˜ W t + (cid:90) t (cid:90) R x (cid:0) ˆ N ( dt, dx ) − ˆ ν ( dx ) dt (cid:1) , t ≥ . (7.17)With reference to Kallenberg (2001) Corollary 15.7, to prove ˆ L is a L´evy process, it suffices toshow that for any B ∈ B ( R ), (cid:0) ˆ N ( t, B ) (cid:1) t ≥ is a Poisson process with intensity ˆ ν ( B ) satisfying (cid:90) R (cid:0) x ∧ (cid:1) ˆ ν ( dx ) < ∞ . (7.18)But from the definition of ˆ N , it is clear that (cid:0) ˆ N ( t, B ) (cid:1) t ≥ is a Poisson process. Observe that E [ ˆ N ( t, B )] = E (cid:2) ˜ N (cid:0) t, ln( B ∩ ( − , ∞ ) + 1) (cid:1)(cid:3) = t ˜ ν (cid:0) ln( B ∩ ( − , ∞ ) + 1) (cid:1) = t ˆ ν ( B ) , which proves that ˆ ν ( B ) is the intensity of (cid:0) ˆ N ( t, B ) (cid:1) t ≥ . From the Taylor expansion of (e z − ,it can be shown that there exist constants ¯ z > C > z ∈ ( − ¯ z, ¯ z ), (cid:0) e z − (cid:1) ≤ Cz . For (cid:15) ∈ (0 , S = (cid:0) ln(1 − (cid:15) ) , ln( (cid:15) + 1) (cid:1) . Then using (7.16) we calculatethat for (cid:15) close enough to 0 so that S ⊆ ( − ¯ z, ¯ z ), we have (cid:90) ( − (cid:15),(cid:15) ) x ˆ ν ( dx ) = (cid:90) S (cid:0) e z − (cid:1) ˜ ν ( dz ) ≤ C (cid:90) S z ˜ ν ( dz ) ≤ C (cid:90) ( − ¯ z, ¯ z ) z ˜ ν ( dz ) < ∞ , (7.19)where the finiteness follows since ˜ ν is a L´evy measure. Again by (7.16), we obtain (cid:90) R \ ( − (cid:15) , (cid:15) ) ˆ ν ( dx ) = (cid:90) R \S ˜ ν ( dz ) < ∞ , ν is a L´evy measure . This implies that ˆ ν (cid:0) R \ ( − , (cid:1) < ∞ and ˆ ν (cid:0) ( − , − (cid:15) ] ∪ [ (cid:15), (cid:1) < ∞ . Since x is bounded on ( − , − (cid:15) ] ∪ [ (cid:15), (cid:90) ( − , x ˆ ν ( dx ) < ∞ . Combining this with ˆ ν (cid:0) R \ ( − , (cid:1) < ∞ , we get (7.18). We therefore conclude that ˆ N and ˆ ν are Poisson random measure and L´evy measure associated with the L´evy process ˆ L ,respectively. Moreover, we calculate from (7.16) that for x > − x (cid:54) = 0,ˆ ν ( dx ) = ˜ ν (cid:0) d (cid:0) ln( x + 1) (cid:1) (cid:1) = ˜ f (cid:0) ln( x + 1) (cid:1) d (cid:0) ln( x + 1) (cid:1) = 1 x + 1 ˜ f (cid:0) ln( x + 1) (cid:1) dx. The relation L = ˜ s ˆ L shows that L is also a L´evy process. The expression of L in (5.3)shows the adaptedness. Now we check Assumption 2.1 is satisfied by L , but it suffices tocheck for ˆ L . According to Assumption 5.1, we know (cid:82) | z |≥ e z ˜ ν ( dz ) < ∞ , and since for any (cid:15) >
0, ˜ ν (cid:0) R \ ( − (cid:15), (cid:15) ) (cid:1) < ∞ , it follows that on [ln 2 , ∞ ), e z and e z are both ˜ ν -integrable and˜ ν (cid:0) [ln 2 , ∞ ) (cid:1) < ∞ . Therefore, (cid:90) | x |≥ x ˆ ν ( dx ) = (cid:90) [ln 2 , ∞ ) (cid:0) e z − (cid:1) ˜ ν ( dz ) < ∞ , which implies that ˆ L has finite second moment (see e.g. Kyprianou, 2006, Theorem 3.8).Observe that when u ≤
0, we haveexp (cid:0) u (e z − (cid:1) ≤ , for all z ≥ . Hence, (cid:90) | x |≥ e ux ˆ ν ( dx ) = (cid:90) [ln 2 , ∞ ) exp (cid:0) u (e z − (cid:1) ˜ ν ( dz ) < ∞ , (7.20)from which it follows that E [e u ˆ L ] < ∞ , for all u ≤ Proof of Theorem 5.4 . This is a direct consequence of Theorem 4.2.33 roof of Proposition 6.1 . For u ≥
0, we calculate that (cid:90) − (cid:16) e − ˜ Aux − Aux (cid:17) ˆ ν ( dx )= (cid:90) − (cid:16) e − ˜ Aux − Aux (cid:17) − η ln( x + 1) ( x + 1) C + D − dx ≥ e η (cid:90) − (cid:16) e − ˜ Aux − Aux (cid:17) ( x + 1) C + D dx = e η (cid:90) (cid:16) e − ˜ Au ( x − x C + D (cid:17) dx + e η (cid:90) (cid:16) ˜ Aux C + D +1 (cid:17) dx + e η (cid:90) (cid:16) − (1 + ˜ Au ) x C + D (cid:17) dx (7.21)where the first inequality is due to − x +1) ln( x +1) ≥ e, for all − < x <
0, since ( x + 1) ln( x + 1)is convex with minimum value − e − . Observe that (cid:90) (cid:16) e − ˜ Au ( x − x C + D (cid:17) dx ≥ e ˜ Au (cid:90) Au ∧ (cid:16)(cid:0) − ˜ Aux + 1 (cid:1) x C + D (cid:17) dx = − ˜ Au e ˜ Au C + D + 2 (cid:18) Au ∧ (cid:19) C + D +2 + e ˜ Au C + D + 1 (cid:18) Au ∧ (cid:19) C + D +1 (7.22)and (cid:90) (cid:16) ˜ Aux C + D +1 (cid:17) dx + (cid:90) (cid:16) − (1 + ˜ Au ) x C + D (cid:17) dx = ˜ AuC + D + 2 − AuC + D + 1 , (7.23)where we have C + D > − ˜ Aux ≥ − ˜ Aux + 1 on interval (cid:2) , Au ∧ (cid:3) . Therefore, the required result follows from (7.21)-(7.23) and the expression ofˆ κ V G ˜ A in (6.2) as well as the fact that e − ˜ Aux − Aux and ˆ ν are positive for all u ≥ x ∈ R . References
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