Semimartingale Characteristics

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_4

Mathematical Finance pp 249-298 | Cite as

Semimartingale Characteristics

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

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Part of the Springer Finance book series (FINANCE)

Abstract

The stochastic calculus in Chap. 3 is based on integration. Small Lévy-like bits of processes are pieced together to yield something that behaves differently from any Lévy process on a global scale.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

4.1

Show that X is a Poisson process with intensity 1 if and only if it is a counting process with X(0) = 0 and such that (X(t) − t)_t≥0 is a local martingale.

4.2

Let X be an $\mathbb R^d$-valued semimartingale with characteristic exponent ψ(t, u). Show that

$$\displaystyle \begin{aligned} \big\langle e^{iu^\top X},e^{iv^\top X}\big\rangle(t)= \int_0^te^{i(u+v)^\top X(s)} \big(\psi(s,u+v)-\psi(s,u)-\psi(s,v)\big)ds \end{aligned} $$

(4.68)

for any $u,v\in \mathbb R^d$.

Remark: Provided that $e^{z^\top X}$ is a special semimartingale for z = Re(u∕i), Re(v∕i), Re((u + v)∕i), both (4.68) and the local martingale property of (4.11) can be extended to complex $u,v\in \mathbb C^d$.

4.3

Let h(x) = x1_{|x|≤1}. Show that the terms on the right-hand sides of (4.20), (4.21), and (4.22) are independent Lévy processes if X is a Lévy process.

4.4

Let W be a Wiener process. Show that X = |W| is a semimartingale which does not allow for absolutely continuous characteristics. Determine its integral characteristics B^h, C, ν.

Hint: Show first that {t ≥ 0 : W(t) = 0} is almost surely a Lebesgue-null set. Moreover, recall ( 3.40).

4.5

Suppose that X is a variance-gamma Lévy process under both P and Q ∼ P with parameters α, λ, β, μ and $\widetilde \alpha , \widetilde \lambda , \widetilde \beta , \widetilde \mu $, respectively. What can be said about the possible values of $\widetilde \alpha , \widetilde \lambda , \widetilde \beta , \widetilde \mu $ in terms of α, λ, β, μ?

4.6

Let $x\in \mathbb R$ and suppose that (β^h, γ, κ) = (β^h, 1, 0) for some bounded function $\beta ^h:\mathbb R _+\times \mathbb R\to \mathbb R$. Show that uniqueness holds for the martingale problem related to P⁰ = ε_x and (β^h, γ, κ) = (β^h, 1, 0).

Hint: show that X is a Wiener process after an appropriate change of measure.

4.7 (Heston Limit of Hawkes Processes)

For fixed n consider two counting processes $N_+^{(n)},N_-^{(n)}$, not jumping at the same time, with intensities $\lambda _+^{(n)},\lambda _-^{(n)}$ of the form

$$\displaystyle \begin{aligned}{\lambda_+^{(n)} \choose \lambda_-^{(n)}}={\mu\choose\mu} + \int_0^t a_n\left(\begin{array}{cc} \alpha_1e^{-\eta(t-s)}& \alpha_2e^{-\eta(t-s)}\\ \alpha_2e^{-\eta(t-s)}& \alpha_1e^{-\eta(t-s)} \end{array}\right) d {N_+^{(n)} \choose N_-^{(n)}}(s)\end{aligned}$$

with constants $\mu ,a_n,\alpha _1,\alpha _2,\eta \in \mathbb R _+$ such that ξ :=lim_n→∞(1 − a_n)n exists, is positive, and α₁ + α₂ = η. Define bivariate processes (X⁽ⁿ⁾, V⁽ⁿ⁾) by

$$\displaystyle \begin{aligned}\big(X^{(n)}(t),V^{(n)}(t)\big)=\left({\lambda_+^{(n)}(nt)+\lambda_-^{(n)}(nt) \over n} ,{N_+^{(n)}(nt)-N_-^{(n)}(nt) \over n} \right).\end{aligned}$$

Show that (X⁽ⁿ⁾, V⁽ⁿ⁾) converges in law as n →∞ to some bivariate process (X, V ) solving the SDE

$$\displaystyle \begin{aligned} dX(t)&=\eta(2\mu-\xi X(t))dt+\eta\sqrt{X(t)}dW_1(t),\\ dV(t)&=\sqrt{X(t)}dW_2(t) \end{aligned} $$

with two independent Wiener processes W₁, W₂.

In Mathematical Finance the events in $N_+^{(n)}, N_-^{(n)}$ can be interpreted as price moves. Since

$$\displaystyle \begin{aligned} d\lambda_+^{(n)}(t)&=-\eta\big(\lambda_+^{(n)}(t)-\mu\big)dt+a_n\alpha_1dN_+(t)-a_n\alpha_2dN_-(t), \end{aligned} $$

(4.69)

$$\displaystyle \begin{aligned} d\lambda_-^{(n)}(t)&=-\eta\big(\lambda_-^{(n)}(t)-\mu\big)dt+a_n\alpha_2dN_+(t)-a_n\alpha_1dN_-(t),{} \end{aligned} $$

(4.70)

each price change increases the probability of subsequent moves but this increased intensity fades after some time. The limiting process is an instance of the popular Heston model, which is discussed in Sect. 8.2.4.

Hint: Use Theorem 4.36. To this end, express the compensator of V⁽ⁿ⁾ as an integral

$$\displaystyle \begin{aligned} A^{V^{(n)}}(t) &=\int_0^tf^{(n)}(s)dV^{(n)}(s)\\ &=f^{(n)}(t)V^{(n)}(t)-f^{(n)}(0)V^{(n)}(0)-\int_0^tV^{(n)}(s)(f^{(n)})'(s)ds \end{aligned} $$

of the past values of V⁽ⁿ⁾. Show that f⁽ⁿ⁾, (f⁽ⁿ⁾)′ and hence A⁽ⁿ⁾ converge to 0 as n →∞. The uniqueness of the law of (X, V ) need not to be shown. It follows from Theorem 6.6.

4.8

Consider S = (S₁, …, S_d) of the form $S_i(t)=S_i(0)e^{X_i}$, i = 1, …, d for S₁(0), …, S_d(0) > 0 and some $\mathbb R^d$-valued Lévy process or, more generally, semimartingale X. For φ ∈ L(S) write V_φ(t) := 1 + φ•S(t). Determine φ such that 1∕V_φ and S_i∕V_φ, i = 1, …, d are local martingales—provided that it exists. In Mathematical Finance, φ is called the numeraire portfolio for the asset price process S.

Appendix 2: Notes and Comments

Semimartingale characteristics are covered in [138, 152, 154]. The equivalence in Definition 4.3 follows from [154, Theorem II.2.42]. Proposition 4.6 is stated in [154, Corollary II.4.19]. For Proposition 4.7 we refer to [154, Theorem II.4.15]. Proposition 4.14 is stated in [181, Lemma 5]. For Theorem 4.15 and Proposition 4.16 see [154, Theorem III.3.24], [169, Proposition 2.6], [168, Lemma 5.1]. Theorem 4.24 goes back to [207]. For Proposition 4.31 we refer to [154, Theorem II.2.42]. Convergence of Hawkes processes as in Example 4.39 and Problem 4.7 is studied in [155]. The general approach of Sect. 4.8 is due to Aleš Černý. The Černý–Ruf representation is introduced in [57], where a thorough treatment can be found. For a more general statement of Problem 4.6 see [241, Corollary IX.1.14]. Problem 4.1 goes back to [291].

References

57.
A. Černý, J. Ruf, Stochastic modelling without Brownian motion: Simplified calculus for semimartingales (2018). PreprintGoogle Scholar
100.
S. Ethier, T. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986)CrossRefGoogle Scholar
125.
T. Goll, J. Kallsen, Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89(1), 31–48 (2000)MathSciNetCrossRefGoogle Scholar
138.
S. He, J. Wang, J. Yan, Semimartingale Theory and Stochastic Calculus (CRC Press, Boca Raton, 1992)zbMATHGoogle Scholar
152.
J. Jacod, Calcul Stochastique et Problèmes de Martingales, volume 714 of Lecture Notes in Math (Springer, Berlin, 1979)CrossRefGoogle Scholar
154.
J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar
155.
T. Jaisson, M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25(2), 600–631 (2015)MathSciNetCrossRefGoogle Scholar
168.
J. Kallsen, σ-localization and σ-martingales. Teor. Veroyatnost. i Primenen. 48(1), 177–188 (2003)MathSciNetCrossRefGoogle Scholar
169.
J. Kallsen, A didactic note on affine stochastic volatility models. From Stochastic Calculus to Mathematical Finance (Springer, Berlin, 2006), pp. 343–368CrossRefGoogle Scholar
181.
J. Kallsen, A. Shiryaev, Time change representation of stochastic integrals. Teor. Veroyatnost. i Primenen. 46(3), 579–585 (2001)MathSciNetCrossRefGoogle Scholar
207.
D. Lépingle, J. Mémin, Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42(3), 175–203 (1978)MathSciNetCrossRefGoogle Scholar
238.
P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar
241.
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
291.
S. Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process. Jpn. J. Math. 34, 53–70 (1964)CrossRefGoogle Scholar

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Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Semimartingale Characteristics

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