Stochastic Integration

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_3

Mathematical Finance pp 171-248 | Cite as

Stochastic Integration

Authors
Authors and affiliations

Ernst Eberlein
Jan Kallsen

Chapter

First Online: 04 December 2019

1.2k Downloads

Part of the Springer Finance book series (FINANCE)

Abstract

Traditional stochastic calculus is based on stochastic integration. It constitutes the basis of modern Mathematical Finance. The most important notions and results from the theory are presented in this chapter.

This is a preview of subscription content, to check access.

Appendix 1: Problems

The exercises correspond to the section with the same number.

3.1

Suppose that M, N are square-integrable martingales. Show that M, N are strongly orthogonal if and only if M^τ and N − N(0) are orthogonal in the Hilbert space sense for any stopping time τ.

3.2

For a semimartingale X and $z\in \mathbb R$ set $Z(t):=e^{-\lambda t}z+\int _0^te^{-\lambda (t-s)}dX(s)$. Show that Z(t) = z − λZ₋•I + X.

3.3

Show that a predictable function ξ is (μ^X − ν^X)-integrable and ξ ∗ (μ^X − ν^X) is a locally square-integrable martingale if ξ² ∗ ν^X(t) < ∞ for any t ≥ 0.

3.4

Let X be a Lévy–Itō process as in (3.57) and $f:\mathbb R\to \mathbb R$ twice continuously differentiable. Show that f(X) is a Lévy–Itō process and determine its coefficients.

3.5

Let W = (W₁, W₂) be a standard Brownian motion in $\mathbb R^2$. Show that $X:=W_1^2+W_2^2$ solves the stochastic differential equation $dX(t)=2dt+2\sqrt {X(t)}d\widetilde W(t)$ with some univariate Wiener process $\widetilde W$.

Hint: you may use Lévy’s characterisation of Brownian motion in Example 3.28.

3.6

Suppose that X is a semimartingale which does not have jumps of size − 1. Show that $1/\mathscr {E}(X)=\mathscr {E}(Y)$ for

$$\displaystyle \begin{aligned}Y:=-X+[X^c,X^c]+{x^2\over 1+x}*\mu^X.\end{aligned}$$

3.7

Let S be an $\mathbb R^d$-valued exponential Lévy process and X = 1 + φ•S with $\varphi _i(t)={X(t-)\over S_i(t-)}\gamma _i, i=1,\dots ,d$ for some $\gamma _1,\dots ,\gamma _d\in \mathbb R _+$ satisfying γ₁ + ⋯ + γ_d < 1. Show that X is an exponential Lévy process. In Mathematical Finance, X corresponds to the wealth of a trader who invests constant fractions γ₁, …, γ_d of her wealth in assets S₁, …, S_d.

3.8

Let X be a time-inhomogeneous Lévy process with d = 1 and ν = 0 in Theorem 3.51. Show that X is Gaussian and determine the autocovariance Cov(X(s), X(t)) for any s, t ≥ 0. (From Proposition 4.7 for PIIACs or [154, Theorem II.4.15] for general time-inhomogeneous Lévy process it follows that ν = 0 holds if and only if X is continuous.)

3.9

Determine the solution to the system of SDEs

$$\displaystyle \begin{aligned} dX_1(t)&=-\lambda_1X_2(t)dt+dW_1(t),\quad X_1(0)=x_1,\\ dX_2(t)&=-\lambda_2X_1(t)dt+dW_2(t),\quad X_2(0)=x_2 \end{aligned} $$

for $x_1,x_2,\lambda _1,\lambda _2\in \mathbb R$ and some two-dimensional Wiener process W = (W₁, W₂).

3.10

Let W = (W₁, W₂) be a Brownian motion in $\mathbb R^2$ with E(W(t)) = 0 and

$$\displaystyle \begin{aligned}\mathrm{Cov}(W(t))=\left(\begin{array}{cc} \sigma_1^2 &\varrho\sigma_1\sigma_2 \\ \varrho\sigma_1\sigma_2 & \sigma_2^2 \end{array}\right)\end{aligned}$$

for some σ₁, σ₂ > 0, ϱ ∈ [−1, 1]. Determine the Galtchouk–Kunita–Watanabe decomposition of $M:=\mathscr {E}(W_2)$ relative to $X:=\mathscr {E}(W_1)$.

3.11 (Linear BSDE)

Consider the BSDE (3.107) with

$$\displaystyle \begin{aligned}f(\omega,t,x,y,z)=\alpha(t)+\beta(t)x+\gamma(t)c^Ly+\int\delta(t,\xi)z(\xi)K^L(d\xi)\end{aligned}$$

for some bounded measurable functions $\alpha ,\beta ,\gamma :[0,T]\to \mathbb R$ and a function $\delta :[0,T]\times \mathbb R\to (-1,\infty ]$ such that $t\mapsto \int \delta (t,\xi )^2K^L(d\xi )$ is bounded on [0, T]. Define $Z:=\mathscr {E}(\beta \bullet I+\gamma \bullet L^c+\delta *(\mu ^L-\nu ^L))$ and

$$\displaystyle \begin{aligned}X(t):=E\bigg(H{Z(T)\over Z(t)}+\int_t^T\alpha(s){Z(s-)\over Z(t)}ds\bigg|\mathscr{F}_t\bigg).\end{aligned}$$

Verify that there are integrands φ, ψ such that (X, φ, ψ) solves (3.107). In particular, we have φ(t) = d〈X^c, L^c〉(t)∕(c^Ldt).

3.12

Suppose that the $\mathbb R^d$-valued Itō process X is of the form X(t) = X(0) + μ•I + σ•W in the sense

$$\displaystyle \begin{aligned}X_i(t)=X_i(0)+\mu_i\bullet I+\sigma_{i\cdot}\bullet W,\quad i=1,\dots,d\end{aligned}$$

for some $\mathbb R^d$-valued standard Brownian motion W, some $\mathbb R^d$-valued process μ ∈ L(I) and an $\mathbb R^{d\times d}$-valued process σ such that σ₁, …, σ_d ∈ L(W). Moreover, assume that σ⁻¹μ ∈ L(W) and that Q ∼ P is a probability measure with density process $Z:=\mathscr {E}((\sigma ^{-1}\mu )\bullet W)$. Show that X(t) = X(0) + σ•W^Q(t) for some Q-standard Brownian motion W^Q in $\mathbb R^d$.

Hint: you my use the multivariate extension of Example 3.28, namely that W is an $\mathbb R^d$-valued standard Brownian motion if it is a continuous local martingale with W(0) = 0 and [W_i, W_j](t) = t as well as [W_i, W_j](t) = t for i, j = 1, …, d with i ≠ j.

Appendix 2: Notes and Comments

Stochastic integration is covered in [2, 76, 154, 186, 238, 241, 252] for stochastic processes. For integration relative to random measures we refer to [152, 154].

Example 3.28 follows [154, Theorem II.4.4]. Example 3.33 is part of [154, Corollary II.4.19]. Proposition 3.36 is based on [154, Proposition II.1.28]. For the integral relative to compensated random measures we refer to [154, Definition II.1.27]. Most of Proposition 3.37 can be found in [154, Section II.1d]. Proposition 3.44 is stated in [152, Proposition 6.4]. For Proposition 3.46 see [182, Lemma 2.4]. Proposition 3.48 is a special case of [152, Théorème 6.8]. Theorem 3.49 is based on [125, Lemma A.8]. Equations (3.88–3.90) can be found, for example, in [60, Section 15.3.1]. Proposition 3.54 is stated in [60, Lemma 15.1]. For Proposition 3.55, Examples 3.57, 3.58, Proposition 3.59, (3.97, 3.98) we refer to [263, Sections 5.2.2, 5.3.3, 5.3.4, 5.5.1, 5.5.2]. Proposition 3.70 is similar to [154, Theorem III.3.11]. Proposition 3.73 is related to [169, Proposition 2.6]. Problem 3.1 can be found in [154, Proposition I.4.15]. For Problem 3.11 we refer to [239, Theorem 3.3].

References

2.
D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge Univ. Press, Cambridge, 2009)CrossRefGoogle Scholar
8.
G. Barles, R. Buckdahn, E. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60(1–2), 57–83 (1997)MathSciNetCrossRefGoogle Scholar
43.
P. Brockwell, Lévy-driven CARMA processes. Ann. Inst. Stat. Math. 53(1), 113–124 (2001)MathSciNetCrossRefGoogle Scholar
58.
E. Çinlar, J. Jacod, P. Protter, M. Sharpe, Semimartingales and Markov processes. Z. Wahrsch. verw. Gebiete 54(2), 161–219 (1980)MathSciNetCrossRefGoogle Scholar
60.
R. Cont, P. Tankov, Financial Modelling with Jump Processes (Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar
76.
C. Dellacherie, P.-A. Meyer, Probabilities and Potential B: Theory of Martingales (North-Holland, Amsterdam, 1982)zbMATHGoogle Scholar
114.
H. Föllmer, Yu. Kabanov, Optional decomposition and Lagrange multipliers. Finance Stochast. 2(1), 69–81 (1998)MathSciNetzbMATHGoogle Scholar
125.
T. Goll, J. Kallsen, Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89(1), 31–48 (2000)MathSciNetCrossRefGoogle 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
163.
J. Kallsen, Semimartingale Modelling in Finance. PhD thesis, University of Freiburg, 1998Google 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
182.
J. Kallsen, A. Shiryaev, The cumulant process and Esscher’s change of measure. Finance Stochast. 6(4), 397–428 (2002)MathSciNetCrossRefGoogle Scholar
186.
I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)zbMATHGoogle Scholar
196.
D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105(4), 459–479 (1996)MathSciNetCrossRefGoogle Scholar
204.
D. Lamberton, B. Lapeyre, Stochastic Calculus Applied to Finance (Chapman & Hall, London, 1996)zbMATHGoogle Scholar
230.
E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)MathSciNetCrossRefGoogle Scholar
238.
P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar
239.
M.-C. Quenez, A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123(8), 3328–3357 (2013)MathSciNetCrossRefGoogle Scholar
241.
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
252.
C. Rogers, D. Williams, Diffusions, Markov processes, and Martingales: Volume 2, Itô Calculus (Cambridge Univ. Press, Cambridge, 1994)zbMATHGoogle Scholar
259.
K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge Univ. Press, Cambridge, 1999)zbMATHGoogle Scholar
263.
W. Schoutens, Lévy Processes in Finance (Wiley, New York, 2003)CrossRefGoogle Scholar
284.
S. Tang, X. Li, Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

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

Stochastic Integration

Abstract

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite chapter