Andreas E Kyprianou Introductory Lectures on Fluctuations of L´evy Processes with Applications July 29, 2005 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo Preface In 2003 I began teaching a course entitled L´evy processes on the AmsterdamUtrecht masters programme in stochastics and financial mathematics Quite naturally, I wanted to expose to my students my own interests in L´evy processes That is the role that certain subtle behaviour concerning their fluctuations explain different types of phenomena appearing in a number of classical models of applied probability (as a general rule that does not necessarily include mathematical finance) Indeed, recent developments in the theory of L´evy processes, in particular concerning path fluctuation, has offered the clarity required to revisit classical applied probability models and improve on well established and fundamental results Results which were initially responsible for the popularity of the models themselves Whilst giving the course I wrote some lecture notes which have now matured into this text Given the audience of students, who were either engaged in their ‘afstudeerfase’ (equivalent to masters for the U.S or those European countries which have engaged in that system) or just starting a Ph.D., these lecture notes were originally written with the restriction that the mathematics used would not surpass the level that they should in principle have reached That roughly means the following Experience to the level of third year or fourth year university courses delivered by a mathematics department on - foundational real and complex analysis, elementary functional analysis (specifically basic facts about Lp spaces), measure theory, integration theory and measure theoretic probability theory, elements of the classical theory of Markov processes, stopping times and the Strong Markov Property, Poisson processes and renewal processes, - an understanding of Brownian motion as a Markov process and elementary martingale theory in continuous time For the most part this affected the way in which the material was handled compared to the classical texts and research papers from which almost all of the results and arguments in this text originate Likewise the exercises are VI Preface pitched at a level appropriate to this audience Indeed several of the exercises have been included in response to some of the questions that have been asked by students themselves concerning curiosities of the arguments given in class Arguably some of the exercises are at times quite long Such exercises reflect some of the other ways in which I have used preliminary versions of this text A small number of students in Utrecht also used the text as an individual reading programme contributing to their ‘kleinescripite’ (extended mathematical essay) or ’onderzoekopdracht’ (research option) The extended exercises were designed to assist with this self-study programme In addition, some exercises were used as examination questions There can be no doubt, particularly to the more experienced reader, that the current text has been heavily influenced by the outstanding books of Bertoin (1996) and Sato (1999); especially the former which also takes a predominantly pathwise approach to its content It should be reiterated however that, unlike the latter two books, this text is not aimed at functioning as a research monograph nor a reference manual for the researcher Andreas E Kyprianou Edinburgh 2005 Contents L´ evy processes and applications 1.1 L´evy processes and infinitely divisibility 1.2 Some examples of L´evy processes 1.3 L´evy processes in classical applied probability models 14 Exercises 21 The L´ evy-Itˆ o decomposition and path structure 2.1 The L´evy-Itˆ o decomposition 2.2 Poisson point processes 2.3 Functionals of Poisson point processes 2.4 Square integrable martingales 2.5 Proof of the L´evy-Itˆ o decomposition 2.6 L´evy processes distinguished by their path type 2.7 Interpretations of the L´evy-Itˆo decomposition Exercises 27 27 29 34 38 45 47 50 55 More distributional and path related properties 3.1 The Strong Markov Property 3.2 Duality 3.3 Exponential moments and martingales Exercises 61 61 66 68 75 General storage models and paths of bounded variation 4.1 Idle times 4.2 Change of variable and compensation formulae 4.3 The Kella-Whitt martingale 4.4 Stationary distribution of the workload 4.5 Small-time behaviour and the Pollaczeck-Khintchine formula Exercises 79 80 81 88 91 93 96 VIII Contents Subordinators at first passage and renewal measures 101 5.1 Killed subordinators and renewal measures 101 5.2 Overshoots and undershoots 110 5.3 Creeping 112 5.4 Regular variation and Tauberian theorems 115 5.5 Dynkin-Lamperti asymptotics 120 Exercises 124 The Wiener-Hopf factorization 129 6.1 Local time at the maxiumum 130 6.2 The ladder process 136 6.3 Excursions 143 6.4 The Wiener-Hopf factorization 145 6.5 Examples of the Wiener-Hopf factorization 158 Exercises 163 L´ evy processes at first passage and insurance risk 169 7.1 Drifting and oscillating 169 7.2 Cram´er’s estimate of ruin 175 7.3 A quintuple law at first passage 179 7.4 The jump measure of the ascending ladder height process 184 7.5 Creeping 186 7.6 Regular variation and infinite divisibility 189 7.7 Asymptotic ruinous behaviour with regular variation 192 Exercises 195 Exit problems for spectrally negative processes 201 8.1 Basic properties reviewed 201 8.2 The one- and two-sided exit problems 203 8.3 The scale functions W (q) and Z (q) 209 8.4 Potential measures 212 8.5 Identities for reflected processes 216 Exercises 220 Applications to optimal stopping problems 227 9.1 Sufficient conditions for optimality 227 9.2 The McKean optimal stopping problem 229 9.3 Smooth fit versus continuous fit 233 9.4 The Novikov-Shiryaev optimal stopping problem 237 9.5 The Shepp-Shiryaev optimal stopping problem 244 9.6 Stochastic games 249 Exercises 257 References 259 L´ evy processes and applications In this chapter we define a L´evy process and attempt to give some indication of how rich a class of processes they form To illustrate the variety of processes captured within the definition of a L´evy process, we shall explore briefly the relationship of L´evy processes with infinitely divisible distributions We also discuss some classical applied probability models which are built on the strength of well understood path properties of elementary L´evy processes We hint at how generalizations of these models may be approached using more sophisticated L´evy processes At a number of points later on in this text we shall handle these generalizations in more detail The models we have chosen to present are suitable for the course of this text as a way of exemplifying fluctuation theory but are by no means the only applications 1.1 L´ evy processes and infinitely divisibility Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process A real valued process B = {Bt : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold (i) The paths of B are P-almost surely continuous (ii) P(B0 = 0) = (iii) For each t > 0, Bt is equal in distribution to a normal random variable with variance t (iv) For ≤ s ≤ t, Bt − Bs is independent of {Bu : u ≤ s} (v) For ≤ s ≤ t, Bt − Bs is equal in distribution to Bt−s A process valued on the non-negative integers N = {Nt : t ≥ 0}, defined on a probability space (Ω, F , P) is said to be a Poisson process with intensity λ > if the following hold (i) The paths of N are P-almost surely right continuous with left limits L´evy processes and applications (ii) P(N0 = 0) = (iii) For each t > 0, Nt is equal in distribution to a Poisson random variable with parameter λt (iv) For ≤ s ≤ t, Nt − Ns is independent of {Nu : u ≤ s} (v) For ≤ s ≤ t, Nt − Ns is equal in distribution to Nt−s On first encounter, these processes would seem to be considerably different from one another Firstly, Brownian motion has continuous paths whereas Poisson processes not Secondly, a Poisson processes is a non-decreasing process and thus has paths of bounded variation over finite time horizons, whereas a Brownian motion does not have monotone paths and in fact its paths are of unbounded variation over finite time horizons However, when we line up their definitions next to one another, we see that they have a lot in common Both processes have right continuous paths with left limits, are initiated from the origin and both have stationary and independent increments; that is properties (i), (ii), (iv) and (v) We may use these common properties to define a general class of stochastic processes which are called L´evy processes Definition 1.1 (L´ evy Process) A process X = {Xt : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a L´evy processes if it possesses the following properties (i) The paths of X are right continuous with left limits P-almost surely (ii) P(X0 = 0) = (iii) For ≤ s ≤ t, Xt − Xs is independent of {Xu : u ≤ s} (iv) For ≤ s ≤ t, Xt − Xs is equal in distribution to Xt−s Unless otherwise stated, from now on, when talking of a L´evy process, we shall always use the measure P (with associated expectation operator E) to be implicitly understood as its law The name ‘L´evy process’ honours the work of the French mathematician Paul L´evy who, although not alone in his contribution, played an instrumental role in bringing together an understanding and characterization of processes with stationary independent increments In earlier literature, L´evy processes can be found under a number of different names In the 1940s, L´evy himself referred them as a sub class of processus additif (additive processes), that is processes with independent increments For the most part however, research literature through the 1960s and 1970s refers to L´evy proceses simply as processes with stationary independent increments One sees a change in language through the 1980s and by the 1990s the use of the term L´evy process had become standard From Definition 1.1 alone it is difficult to see just how rich a class of processes the class of L´evy processes forms De Finetti (1929) introduced the 1.1 L´evy processes and infinitely divisibility notion of an infinitely divisible distribution and showed that they have an intimate relationship with L´evy processes This relationship gives a reasonably good impression of how varied the class of L´evy processes really is To this end, let us now devote a little time to discussing infinitely divisible distributions Definition 1.2 We say that a real valued random variable Θ has an infinitely divisible distribution if for each n = 1, 2, there exist a sequence of i.i.d random variables Θ1,n , , Θn,n such that d Θ = Θ1,n + + Θn,n d where = is equality in distribution Alternatively, we could have expressed this relation in terms of probability laws That is to say, the law µ of a real valued random variable is infinitely divisible if for each n = 1, 2, there exists another law µn of a real valued random variable such that µ = µ∗n n A third equivalent defintion can be reworded in terms of characteristic exponents Suppose that Θ has characteristic exponent Ψ (u) := − log E(eiuΘ ) for all u ∈ R Then Θ has an infintely divisible distribution if for all n ≥ there exists a characterisitc exponent of a probability distribution, say Ψn such that Ψ (u) = nΨn (u) for all u ∈ R The full extent to which we may characterize infinitely divisible distributions is done via their characteristic exponent Ψ and an expression known as the L´evy-Khintchine formula Theorem 1.3 (L´ evy-Khintchine formula) A probability law µ of a real valued random variable is infinitely divisible with characteristic exponent Ψ, R eiθx µ (dx) = e−Ψ (θ) for θ ∈ R, if and only if there exists a triple (a, σ, Π), where a ∈ R, σ ≥ and Π is a measure on R\{0} satisfying R\{0} ∧ x2 Π(dx) < ∞, such that Ψ (θ) = iaθ + σ θ2 + R\{0} (1 − eiθx + iθx1(|x| 0, Xt is a random variable belonging to the class of infinitely divisible distributions This follows from the fact that for any n = 1, 2, Xt = Xt/n + (X2t/n − Xt/n ) + + (Xt − X(n−1)t/n ) (1.1) together with the fact that X has stationary independent increments Suppose now that we define for all θ ∈ R, t ≥ Ψt (θ) = − log E eiθXt then using (1.1) twice we have for any two positive integers m, n that mΨ1 (θ) = Ψm (θ) = nΨm/n (θ) and hence for any rational t > Ψt (θ) = tΨ1 (θ) (1.2) If t is an irrational number, then we can choose a decreasing sequence of rationals {tn : n ≥ 1} such that tn ↓ t as n tends to infinity Almost sure right continuity of X implies right continuity of exp{−Ψt (θ)} (by dominated convergence) and hence (1.2) holds for all t ≥ In conclusion, any L´evy process has the property that E eiθXt = e−tΨ (θ) where Ψ (θ) := Ψ1 (θ) is the characteristic exponent of X1 which has an infinitely divisible distribution Definition 1.5 In the sequel we shall also refer to Ψ (θ) as the characteristic exponent of the L´evy process It is now clear that each L´evy process can be associated with an infinitely divisible distribution What is not clear is whether given an infinitely divisible distribution, one may construct a L´evy process X, such that X1 has that distribution This latter issue is resolved by the following theorem which gives the L´evy-Khintchine formula for L´evy processes Theorem 1.6 (L´ evy-Khintchine formula for L´ evy processes) Suppose that a ∈ R, σ ≥ and Π is a measure on R \{0} such that R\{0} (1 ∧ |x|2 )Π(dx) < ∞ From this triple define for each θ ∈ R 252 Applications to optimal stopping problems is to say, it has the same structure as in (9.23) with d > E(S1 ) > Consider the stochastic game (9.29) when G(x) = (K − ex )+ and H(x) = G(x) + δ for fixed δ > This stochastic game is very closely related to the McKean optimal stopping problem We may think of it as modeling the optimal time to sell to an agent a risky asset whose dynamics follow those of an exponential L´evy process for a fixed price K However we have in addition now that the purchasing agent may also demand a forced purchase; in which case the agent must pay a supplementary penalty δ for forcing the purchase Note however, if the constant δ is too large (for example greater than K) then it will never be optimal for the agent to force a purchase in which case the stochastic game will have the same solution as the analogous optimal stopping problem That is to say the optimal stopping problem (9.4) with q = Suppose however that δ is a very small number In this case, based on the experience of the above optimal stopping problems, it is reasonable to assume that one may optimize the expected gain by opting to sell the risky asset once the value of X drops below a critical threshold where the gain function G is large The level of the threshold is determined by the fact that X drifts to infinity (and hence may never reach the threshold) as well as the behaviour of purchasing agent The latter individual can minimize the expected gain by stopping X in a region of R where the gain function H is small This is clearly the half line (log K, ∞) As the gain G is identically zero on this interval, it turns out to be worthwhile to stop in the aforementioned interval, paying the penalty δ, rather than waiting for the other player to request a purchase requiring a potentially greater pay out by the agent These ideas are captured in the following theorem Theorem 9.14 Under the assumptions above, the stochastic game (9.29) has the following two regimes of solutions (a) Define x∗ by ∗ ex = K ψ ′ (0) ψ(1) (9.38) and note by strict convexity of ψ that x∗ < log K Suppose that log(ψ(1)/ψ ′ (0)) δ ≥ Kψ(1) e−y W (y)dy then v(x) = K − ex + ψ(1)ex x−x∗ e−y W (y)dy (9.39) where the saddle point strategies are given by τ ∗ = inf{t > : Xt < x∗ } and σ ∗ = ∞ That is to say, (v, τ ∗ ) is also the solution to the optimal stopping problem 9.6 Stochastic games 253 v(x) = sup Ex ((K − eXτ )+ ) τ ∈T (b) Suppose that log(ψ(1))/ψ ′ (0) δ < Kψ(1) e−y W (y)dy then v(x) = x−z ∗ K − ex + ψ(1)ex δ e−y W (y)dy for x < log K for x ≥ log K (9.40) where z ∗ ∈ (0, K) is the unique solution of the equation log K−z Kψ(1) e−y W (y)dy = δ (9.41) and the saddle point strategies are given by τ ∗ = inf{t > : Xt < z ∗ } and σ ∗ = inf{t > : Xt > log K} Proof For both parts (a) and (b) we shall resort to checking conditions given in Lemma 9.13 Note that for the stochastic game at hand, Conditions (9.30) and (9.31) are both satisfied The first condition is trivially satisfied and the second follows from the fact that limt↑∞ Xt = ∞ and H(x) = δ for all x > log K (a) As indicated in the statement of the Theorem, the proposed solution necessarily solves the McKean optimal stopping problem Under the present circumstances of spectral negativity and bounded variation, the solution to this optimal stopping problem is given in Corollary 9.3 It is left to the reader to check that this solution corresponds to the function given on the right hand side of (9.39) Let v ∗ be the expression given on the right hand side of (9.39) We now proceed to show that v ∗ , τ ∗ and σ ∗ fulfill the conditions (i) – (vi) of Lemma 9.13 Bounds (i) and (ii) Since by definition of W , we may apply L’Hˆ opital’s rule and compute lim v ∗ (x) = K − lim x↑∞ x↑∞ − ψ(1) ∗ ∞ −y W (y)dy e x−x∗ −y e W (y)dy e−x = K − lim ψ(1)ex W (x − x∗ ) x↑∞ =K− =0 ψ(1) x∗ e ψ ′ (0) = 1/ψ(1) 254 Applications to optimal stopping problems where the final equality follows from (8.12) and (9.38) Hence, with the assumption on δ, which reworded says δ ≥ v ∗ (log K), and the convexity of v ∗ (cf Theorem 9.4) we have (K − ex )+ ≤ v ∗ (x) ≤ (K − ex )+ + δ Hence conditions (i) and (ii) of Lemma 9.13 are fulfilled Support properties (iii) and (iv) Property (iii) is trivially satisfied by virtue of the fact that v ∗ solves (9.4) with q = As σ ∗ = ∞ the statement (iv) is empty Supermartingale and subartingale properties (v) and (vi) From the proof of Theorem 9.2 we have seen that {v ∗ (Xt ) : t ≥ 0} is a supermartingale with right continuous paths For the submartingale property, we actually need to show that {v ∗ (Xt∧τ ∗ ) : t ≥ 0} is a martingale However this follows by the Strong Markov Property since for all t ≥ Ex ((K − eXτ ∗ )|Ft∧τ ∗ ) = Ex′ ((K − eXτ ∗ )) where x′ = Xt∧τ ∗ = v ∗ (Xt∧τ ∗ ) (b) Technically speaking, one reason why we may not appeal to the solution to (9.4) with q = as the solution to the problem at hand is because the condition on δ now implies that the solution to part (a) is no longer bounded above by (K − ex )+ + δ, at least at the point x = log K The right hand side of (9.40) can easily be checked using the fluctuation identities in Chapter to be equal to v ∗ (x) = Ex ((K − e Xτ − z∗ )1(τ −∗ τ + z log K ) ) (9.42) and we check now whether the triple (v ∗ , τ ∗ , σ ∗ ) fulfill the conditions (i) – (vi) of Lemma 9.13 Bounds (i) and (ii) The lower bound is easily established since by (9.42) we have that v ∗ (x) ≥ and from (9.40) v ∗ (x) ≥ K − ex Hence v ∗ (x) ≥ (K − ex )+ For the upper bound, note that v ∗ (x) = H(x) for x ≥ log K On x < log K we also have from (9.40) that ex v ∗ (x) − (K − ex ) = Kψ(1) K x−z ∗ log K−z ∗ ≤ Kψ(1) =δ thus confirming that v ∗ (x) ≤ H(x) e−y W (y)dy e−y W (y)dy 9.6 Stochastic games 255 Support properties (iii) and (iv) This is clear by inspection Supermartingale properties (v) By considering the representation (9.42) the martingale property is affirmed by a similar technique to the one used in part (a) and applying the Strong Markov Property to deduce that v ∗ (Xt∧τ ∗ ∧σ∗ ) = Ex ((K − e Xτ − z∗ )1(τ −∗ τ + z log K ) |Ft ) Note that in establishing the above equality, it has been used that v ∗ (Xτ −∗ ) = (K − e Xτ − z z∗ ) and v ∗ (Xτ + log K ) = δ Noting that v ∗ is a continuous function which is also C function on R\{log K} we may apply the change of variable formula in the spirit of Exercise 4.1 and obtain ∗ t ∗ v (Xt ) = v (x) + Hv ∗ (Xs− )ds + Mt , t ≥ (9.43) Px almost surely where {Mt : t ≥ 0} is a right continuous martingale, Hv ∗ (x) = d dv ∗ (x) + dx (−∞,0) (v ∗ (x + y) − v ∗ (x))Π(dy) and Π is the L´evy measure of X The details of this calculation are very similar in nature to those of a related calculation appearing in the proof of Theorem 9.11 and hence are left to the reader Since {v ∗ (Xt ) : t < τ ∗ ∧ σ ∗ } is a martingale we deduce from Exercise 9.28 that Hv ∗ (x)ds = for all x ∈ (z ∗ , log K) A straightforward calculation also shows that for x < z ∗ , where v ∗ (x) = K − ex , Hv ∗ (x) = −ex d+ (−∞,0) (ey − 1)Π(dy) = −ex ψ(1) < Reconsidering (9.43) we see then that on {t < σ ∗ } the Lebesgue integral is a non-increasing adpated process and hence {v ∗ (Xt ) : t < σ ∗ } is a right continuous supermaringale Submartingale property (vi) For x > log K, where v ∗ (x) = δ, our aim is to show that Hv ∗ (x) = (−∞,0) (v ∗ (x + y) − δ)Π(dy) ≥ a consequence of the fact that v ∗ (x) ≥ δ on (0, log K) (9.44) If this is the case then in (9.43) we have on {t < τ ∗ } that the Lebesgue integral is a non-decreasing adapted process and hence {v ∗ (Xt ) : t < τ ∗ } is a right 256 Applications to optimal stopping problems continuous submartingale Showing that (9.44) holds turns out to be quite difficult and we prove this in the lemma immediately below In conclusion, the conditions of Lemma 9.13 are satisfied by the triple (v , τ ∗ , σ ∗ ) thus establishing part (b) the theorem ∗ Lemma 9.15 The function v ∗ in the proof of part (b) of the above theorem is strictly greater than δ for all x < log K Proof We assume the notation from the proof of part (b) of the above theorem As a first step we show that v ∗ together with the stopping time τ ∗ solves the optimal stopping problem f (x) = sup Ex (G(Xτ∗ )) (9.45) τ ∈T where {Xt∗ : t ≥ 0} is the process X stopped on first entry to (log K, ∞), so that Xt∗ = Xt∧σ∗ , and G(x) = (K − ex )+ 1(x≤log K) + δ1(x>log K) Recalling the remark at the end of Section 9.1, we note from the lower bound (i), support property (iii), martingale property (v) and supermartingale property (vii) established in the proof of part (b) of the above theorem that (v, τ ∗ ) solves the optimal stopping problem (9.45) Next define for each q ≥ the solutions to the McKean optimal stopping problem g (q) (x) = sup Ex (e−qτ (K − eXτ )+ ) τ ∈T for x ∈ R The associated optimal stopping time for each q ≥ is given by τq∗ = inf{t > : Xt < x∗p } From Theorem 9.2 and Corollary 9.3 we know that ∗ q Φ(q) − X exq = KE(e eq ) = K Φ(q) q − ψ(1) Hence the constant x∗q is continuous and strictly increasing to log K as q tends to infinity It may also be verified from Corollary 9.3 that g (q) log K−x∗ q (1−e (log K) = Kq −y )W (q) log K−x∗ q (y)dy+Kψ(1) e−y W (q) (y)dy 0 We see then that g (q) (log K) is continuous in q From Theorems 9.2 we also know that g (q) is decreasing Thanks to these facts we may deduce that there exits a solution to the equation g (q) (log K) = δ Henceforth we shall assume that q solves the latter equation The Strong Markov Property implies that ∗ Xτq∗ )) (K − e Xτq∗ )1(τ ∗ τ + )+e q log K ) ) log K ) ) 9.6 Exercises 257 Hence as v ∗ solves (9.45) we have v ∗ (x) = sup Ex ((K − eXτ )+ 1(τ ψ(1) the above optimal stopping problem can be rewritten τ x v(x) = sup E1 e−pτ +Yτ + λ τ ∈T x e−(p+λ)t+Yt dt (9.46) where for t ≥ 0, Ytx = (x ∨ X t ) − Xt (ii) With the help of a potential measure, show that va (x) := E e −(p+λ)τ +Yσxx a σx a +λ x e−(p+λ)t+Yt dt = λ η − λ x (η) e Z (a − x) + ez η η (η) (a) − ηW (η) (a)) + λ (η) x (η − λ)(Z W (a − x) +e η(W (η)′ (a) − W (η) (a)) where η = q − ψ(1) + λ (iii) Show that the solution to (9.46) is given by the pair (va∗ , σ xa∗ ) where λ a∗ = inf{x ≥ : Z (η) (x) − ηW (η) (x) ≤ − } η (iii) Show that there is only continuous fit at the optimal threshold 258 Applications to optimal stopping problems 9.2 Suppose that X is any L´evy process and either q > or 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