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1. Lectures on Lévy Processes and Stochastic Calculus (Koc University) Lecture 3: The Lévy-Itô Decomposition David Applebaum School of Mathematics and Statistics, University of Shefﬁeld, UK 7th December 2011 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 1 / 44

2. Filtrations, Markov Processes and Martingales We recall the probability space (Ω, F, P) which underlies our investigations. F contains all possible events in Ω. When we introduce the arrow of time, its convenient to be able to consider only those events which can occur up to and including time t. We denote by Ft this sub-σ-algebra of F. To be able to consider all time instants on an equal footing, we define a filtration to be an increasing family (Ft , t ≥ 0) of sub-σ-algebras of F, , i.e. 0 ≤ s ≤ t < ∞ ⇒ Fs ⊆ Ft . Dave Applebaum (Sheffield UK) Lecture 3 December 2011 2 / 44

6. A stochastic process X = (X (t), t ≥ 0) is adapted to the given filtration if each X (t) is Ft -measurable. e.g. any process is adapted to its natural filtration, FtX = σ{X (s); 0 ≤ s ≤ t}. An adapted process X = (X (t), t ≥ 0) is a Markov process if for all f ∈ Bb (Rd ), 0 ≤ s ≤ t < ∞, E(f (X (t))|Fs ) = E(f (X (t))|X (s)) (a.s.). (0.1) (i.e. “past” and “future” are independent, given the present). The transition probabilities of a Markov process are ps,t (x, A) = P(X (t) ∈ A|X (s) = x), i.e. the probability that the process is in the Borel set A at time t given that it is at the point x at the earlier time s. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 3 / 44

11. Theorem If X is a Lévy process (adapted to its own natural ﬁltration) wherein each X (t) has law qt , then it is a Markov process with transition probabilities ps,t (x, A) = qt−s (A − x). Proof. This essentially follows from E(f (X (t))|Fs ) = E(f (X (s) + X (t) − X (s))|Fs ) = f (X (s) + y )qt−s (dy ). 2 Rd Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 4 / 44

14. Now let X be an adapted process defined on a filtered probability space which also satisfies the integrability requirement E(|X (t)|) < ∞ for all t ≥ 0. We say that it is a martingale if for all 0 ≤ s < t < ∞, E(X (t)|Fs ) = X (s) a.s. Note that if X is a martingale, then the map t → E(X (t)) is constant. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 5 / 44

18. An adapted Lévy process with zero mean is a martingale (with respect to its natural filtration) since in this case, for 0 ≤ s ≤ t < ∞ and using the convenient notation Es (·) := E(·|Fs ): Es (X (t)) = Es (X (s) + X (t) − X (s)) = X (s) + E(X (t) − X (s)) = X (s) Although there is no good reason why a generic Lévy process should be a martingale (or even have finite mean), there are some important examples: Dave Applebaum (Sheffield UK) Lecture 3 December 2011 6 / 44

21. e.g. the processes whose values at time t are σB(t) where B(t) is a standard Brownian motion, and σ is an r × d matrix. ˜ ˜ N(t) where N is a compensated Poisson process with intensity λ. Some important martingales associated to Lévy processes include: exp{i(u, X (t)) − tη(u)}, where u ∈ Rd is ﬁxed. |σB(t)|2 − tr(A)t where A = σ T σ. ˜ N(t)2 − λt. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 7 / 44

28. Càdlàg Paths A function f : R+ → Rd is càdlàg if it is continue à droite et limité à gauche, i.e. right continuous with left limits. Such a function has only jump discontinuities. Define f (t−) = lims↑t f (s) and ∆f (t) = f (t) − f (t−). If f is càdlàg, {0 ≤ t ≤ T , ∆f (t) = 0} is at most countable. If the filtration satisfies the “usual hypotheses” of right continuity and completion, then every Lévy process has a càdlàg modification which is itself a Lévy process. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 8 / 44

33. From now on, we will always make the following assumptions:- (Ω, F, P) will be a fixed probability space equipped with a filtration (Ft , t ≥ 0) which satisfies the “usual hypotheses”. Every Lévy process X = (X (t), t ≥ 0) will be assumed to be Ft -adapted and have càdlàg sample paths. X (t) − X (s) is independent of Fs for all 0 ≤ s < t < ∞. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 9 / 44

37. The Jumps of A Lévy Process - Poisson Random Measures The jump process ∆X = (∆X (t), t ≥ 0) associated to a Lévy process is defined by ∆X (t) = X (t) − X (t−), for each t ≥ 0. Theorem If N is a Lévy process which is increasing (a.s.) and is such that (∆N(t), t ≥ 0) takes values in {0, 1}, then N is a Poisson process. Proof. Define a sequence of stopping times recursively by T0 = 0 and Tn = inf{t > Tn−1 ; N(t + Tn−1 ) − N(Tn−1 )) = 0} for each n ∈ N. It follows from (L2) that the sequence (T1 , T2 − T1 , . . . , Tn − Tn−1 , . . .) is i.i.d. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 10 / 44

42. By (L2) again, we have for each s, t ≥ 0, P(T1 > s + t) = P(N(s) = 0, N(t + s) − N(s) = 0) = P(T1 > s)P(T1 > t) From the fact that N is increasing (a.s.), it follows easily that the map t → P(T1 > t) is decreasing and by a straightforward application of stochastic continuity (L3) we ﬁnd that the map t → P(T1 > t) is continuous at t = 0. Hence there exists λ > 0 such that P(T1 > t) = e−λt for each t ≥ 0. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 11 / 44

47. So T1 has an exponential distribution with parameter λ and P(N(t) = 0) = P(T1 > t) = e−λt , for each t ≥ 0. n Now assume as an inductive hypothesis that P(N(t) = n) = e−λt (λt) , n! then P(N(t) = n + 1) = P(Tn+2 > t, Tn+1 ≤ t) = P(Tn+2 > t) − P(Tn+1 > t). But Tn+1 = T1 + (T2 − T1 ) + · · · + (Tn+1 − Tn ) is the sum of (n + 1) i.i.d. exponential random variables, and so has a λn+1 sn gamma distribution with density fTn+1 (s) = e−λs for s > 0. n! The required result follows on integration. 2 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 12 / 44

53. The following result shows that ∆X is not a straightforward process to analyse. Lemma If X is a Lévy process, then for ﬁxed t > 0, ∆X (t) = 0 (a.s.). Proof. Let (t(n), n ∈ N) be a sequence in R+ with t(n) ↑ t as n → ∞, then since X has càdlàg paths, limn→∞ X (t(n)) = X (t−).However, by (L3) the sequence (X (t(n)), n ∈ N) converges in probability to X (t), and so has a subsequence which converges almost surely to X (t). The result follows by uniqueness of limits. 2 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 13 / 44

60. Much of the analytic difficulty in manipulating Lévy processes arises from the fact that it is possible for them to have |∆X (s)| = ∞ a.s. 0≤s≤t and the way in which these difficulties is overcome exploits the fact that we always have |∆X (s)|2 < ∞ a.s. 0≤s≤t We will gain more insight into these ideas as the discussion progresses. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 14 / 44

63. Rather than exploring ∆X itself further, we will find it more profitable to count jumps of specified size. More precisely, let 0 ≤ t < ∞ and A ∈ B(Rd − {0}). Define N(t, A) = #{0 ≤ s ≤ t; ∆X (s) ∈ A} = 1A (∆X (s)). 0≤s≤t Note that for each ω ∈ Ω, t ≥ 0, the set function A → N(t, A)(ω) is a counting measure on B(Rd − {0}) and hence E(N(t, A)) = N(t, A)(ω)dP(ω) is a Borel measure on B(Rd − {0}). We write µ(·) = E(N(1, ·)). Dave Applebaum (Sheffield UK) Lecture 3 December 2011 15 / 44

67. / ¯ We say that A ∈ B(Rd − {0}) is bounded below if 0 ∈ A. Lemma If A is bounded below, then N(t, A) < ∞ (a.s.) for all t ≥ 0. A Proof. Define a sequence of stopping times (Tn , n ∈ N) by T1A = inf{t > 0; ∆X (t) ∈ A}, and for A A n > 1, Tn = inf{t > Tn−1 ; ∆X (t) ∈ A}. A Since X has càdlàg paths, we have T1 > 0 (a.s.) and limn→∞ Tn = ∞A (a.s.). A Indeed suppose that T1 = 0 with non-zero probability and let N = {ω ∈ Ω : T1 A = 0}. Assume that ω ∈ Ω − N . Then given any u > 0, we can find 0 < δ, δ < u and > 0 such that |X (δ)(ω) − X (δ )(ω)| > and this contradicts the (almost sure) right continuity of X (·)(ω) at the origin. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 16 / 44

74. Similarly, we assume that limn→∞ Tn = T A < ∞ with non-zero A A probability and deﬁne M = {ω ∈ Ω : limn→∞ Tn = ∞}. If ω ∈ Ω − M then we obtain a contradiction with the fact that X has a left limit (almost surely) at T A (ω). Hence, for each t ≥ 0, N(t, A) = 1{Tn ≤t} < ∞ a.s. A 2 n∈N Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 17 / 44

77. Be aware that if A fails to be bounded below, then this lemma may no longer hold, because of the accumulation of large numbers of small jumps. The following result should at least be plausible, given Theorem 2 and Lemma 4. Theorem 1 If A is bounded below, then (N(t, A), t ≥ 0) is a Poisson process with intensity µ(A). 2 If A1 , . . . , Am ∈ B(Rd − {0}) are disjoint, then the random variables N(t, A1 ), . . . , N(t, Am ) are independent. It follows immediately that µ(A) < ∞ whenever A is bounded below, hence the measure µ is σ-ﬁnite. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 18 / 44

81. The main properties of N, which we will use extensively in the sequel, are summarised below:-. 1 For each t > 0, ω ∈ Ω, N(t, ·)(ω) is a counting measure on B(Rd − {0}). 2 For each A bounded below, (N(t, A), t ≥ 0) is a Poisson process with intensity µ(A) = E(N(1, A)). 3 ˜ The compensator (N(t, A), t ≥ 0) is a martingale-valued measure ˜ A) = N(t, A) − tµ(A), for A bounded below, i.e. where N(t, ˜ For ﬁxed A bounded below, (N(t, A), t ≥ 0) is a martingale. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 19 / 44

86. Poisson Integration Let f be a Borel measurable function from Rd to Rd and let A be bounded below, then for each t > 0, ω ∈ Ω, we may define the Poisson integral of f as a random finite sum by f (x)N(t, dx)(ω) := f (x)N(t, {x})(ω). A x∈A Note that each A f (x)N(t, dx) is an Rd -valued random variable and gives rise to a càdlàg stochastic process, as we vary t. Now since N(t, {x}) = 0 ⇔ ∆X (u) = x for at least one 0 ≤ u ≤ t, we have f (x)N(t, dx) = f (∆X (u))1A (∆X (u)). (0.2) A 0≤u≤t Dave Applebaum (Sheffield UK) Lecture 3 December 2011 20 / 44

91. In the sequel, we will sometimes use µA to denote the restriction to A of the measure µ. in the following theorem, Var stands for variance. Theorem Let A be bounded below, then A f (x)N(t, dx), t ≥ 0 is a compound Poisson process, with 1 characteristic function E exp i u, f (x)N(t, dx) = exp t (ei(u,x) − 1)µf ,A (dx) A Rd for each u ∈ Rd , where µf ,A (B) := µ(A ∩ f −1 (B)), for each B ∈ B(Rd ). 2 If f ∈ L1 (A, µA ), then E f (x)N(t, dx) =t f (x)µ(dx). A A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 21 / 44

95. Theorem 3 If f ∈ L2 (A, µA ), then Var f (x)N(t, dx) =t |f (x)|2 µ(dx). A A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 22 / 44

96. Theorem 3 If f ∈ L2 (A, µA ), then Var f (x)N(t, dx) =t |f (x)|2 µ(dx). A A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 22 / 44

97. Proof. - part of it! 1) For simplicity, we will prove this result in the case where f ∈ L1 (A, µA ). First let f be a simple function and write f = n cj 1Aj j=1 where each cj ∈ Rd . We can assume, without loss of generality, that the Aj ’s are disjoint Borel subsets of A. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 23 / 44

100. By Theorem 5, we ﬁnd that     n  E exp i u, f (x)N(t, dx) = E exp i u, cj N(t, Aj )  A  j=1  n = E exp i u, cj N(t, Aj ) j=1 n = exp t ei(u,cj ) − 1 µ(Aj ) j=1 = exp t (ei(u,f (x)) − 1)µ(dx) . A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 24 / 44

104. Now for an arbitrary f ∈ L1 (A, µA ), we can ﬁnd a sequence of simple functions converging to f in L1 and hence a subsequence which converges to f almost surely. Passing to the limit along this subsequence in the above yields the required result, via dominated convergence. (2) and (3) follow from (1) by differentiation. 2 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 25 / 44

107. It follows from Theorem 6 (2) that a Poisson integral will fail to have a finite mean if f ∈ L1 (A, µ). / For each f ∈ L1 (A, µA ), t ≥ 0, we define the compensated Poisson integral by ˜ f (x)N(t, dx) = f (x)N(t, dx) − t f (x)µ(dx). A A A A straightforward argument shows that ˜ A f (x)N(t, dx), t ≥ 0 is a martingale and we will use this fact extensively in the sequel. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 26 / 44

110. Note that by Theorem 6 (2) and (3), we can easily deduce the following two important facts: E exp i u, ˜ f (x)N(t, dx) A = exp t (ei(u,x) − 1 − i(u, x))µf ,A (dx) , (0.3) Rd for each u ∈ Rd , and for f ∈ L2 (A, µA ), 2 E ˜ f (x)N(t, dx) =t |f (x)|2 µ(dx). (0.4) A A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 27 / 44

111. Note that by Theorem 6 (2) and (3), we can easily deduce the following two important facts: E exp i u, ˜ f (x)N(t, dx) A = exp t (ei(u,x) − 1 − i(u, x))µf ,A (dx) , (0.3) Rd for each u ∈ Rd , and for f ∈ L2 (A, µA ), 2 E ˜ f (x)N(t, dx) =t |f (x)|2 µ(dx). (0.4) A A Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 27 / 44

112. Processes of Finite Variation We begin by introducing a useful class of functions. Let P = {a = t1 < t2 < · · · < tn < tn+1 = b} be a partition of the interval [a, b] in R, and define its mesh to be δ = max1≤i≤n |ti+1 − ti |. We define the variation VarP (g) of a càdlàg mapping g : [a, b] → Rd over the partition P by the prescription n VarP (g) = |g(ti+1 ) − g(ti )|. i=1 Dave Applebaum (Sheffield UK) Lecture 3 December 2011 28 / 44

117. If V (g) = supP VarP (g) < ∞, we say that g has finite variation on [a, b]. If g is defined on the whole of R (or R+ ), it is said to have finite variation if it has finite variation on each compact interval. It is a trivial observation that every non-decreasing g is of finite variation. Conversely if g is of finite variation, then it can always be written as the difference of two non-decreasing functions - to see this, just write g = V (g)+g − V (g)−g , where V (g)(t) is the variation of g on 2 2 [a, t]. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 29 / 44

122. Functions of finite variation are important in integration, for suppose that we are given a function g which we are proposing as an integrator, then as a minimum we will want to be able to define the Stieltjes integral I fdg, for all continuous functions f (where I is some finite interval). In fact a necessary and sufficient condition for obtaining such an integral as a limit of Riemann sums is that g has finite variation. A stochastic process (X (t), t ≥ 0) is of finite variation if the paths (X (t)(ω), t ≥ 0) are of finite variation for almost all ω ∈ Ω. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 30 / 44

125. The following is an important example for us. Example Poisson Integrals Let N be a Poisson random measure with intensity measure µ and let f : Rd → Rd be Borel measurable. For A bounded below, let Y = (Y (t), t ≥ 0) be given by Y (t) = A f (x)N(t, dx), then Y is of ﬁnite variation on [0, t] for each t ≥ 0. To see this, we observe that for all partitions P of [0, t], we have VarP (Y ) ≤ |f (∆X (s))|1A (∆X (s)) < ∞ a.s. (0.5) 0≤s≤t where X (t) = A xN(t, dx), for each t ≥ 0. Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 31 / 44

131. In fact, a necessary and sufficient condition for a Lévy process to be of finite variation is that there is no Brownian part (i.e. a = 0 in the Lévy-Khinchine formula) , and |x|<1 |x|ν(dx) < ∞. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 32 / 44

132. In fact, a necessary and sufficient condition for a Lévy process to be of finite variation is that there is no Brownian part (i.e. a = 0 in the Lévy-Khinchine formula) , and |x|<1 |x|ν(dx) < ∞. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 32 / 44

133. The Lévy-Itô Decomposition This is the key result of this lecture. First, note that for A bounded below, for each t ≥ 0 xN(t, dx) = ∆X (u)1A (∆X (u)) A 0≤u≤t is the sum of all the jumps taking values in the set A up to the time t. Since the paths of X are càdlàg, this is clearly a finite random sum. In particular, |x|≥1 xN(t, dx) is the sum of all jumps of size bigger than one. It is a compound Poisson process, has finite variation but may have no finite moments. Dave Applebaum (Sheffield UK) Lecture 3 December 2011 33 / 44

137. On the other hand it can be shown that X (t) − |x|≥1 xN(t, dx) is a Lévy process having ﬁnite moments to all orders. Now lets turn our attention to the small jumps. We study compensated integrals, which we know are martingales. Introduce the notation M(t, A) := ˜ x N(t, dx) A for t ≥ 0 and A bounded below. For each m ∈ N, let 1 1 Bm = x ∈ Rd , < |x| ≤ m+1 m n and for each n ∈ N, let An = m=1 Bm . Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 34 / 44

142. Deﬁne ˜ x N(t, dx) := L2 − lim M(t, An ), |x|<1 n→∞ which is a martingale. Moreover, on taking limits in (0.3), we get E exp i u, ˜ x N(t, dx) = exp t (ei(u,x) − 1 − i(u, x))µ(dx) |x|<1 |x|<1 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 35 / 44

143. Deﬁne ˜ x N(t, dx) := L2 − lim M(t, An ), |x|<1 n→∞ which is a martingale. Moreover, on taking limits in (0.3), we get E exp i u, ˜ x N(t, dx) = exp t (ei(u,x) − 1 − i(u, x))µ(dx) |x|<1 |x|<1 Dave Applebaum (Shefﬁeld UK) Lecture 3 December 2011 35 / 44

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