Mathematical Finance pp 171-248 | Cite as
Stochastic Integration
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.
Suppose that M, N are square-integrable martingales. Show that M, N are strongly orthogonal if and only if Mτ and N − N(0) are orthogonal in the Hilbert space sense for any stopping time τ.
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.
Show that a predictable function ξ is (μX − νX)-integrable and ξ ∗ (μX − νX) is a locally square-integrable martingale if ξ2 ∗ νX(t) < ∞ for any t ≥ 0.
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.
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.
Suppose that X is a semimartingale which does not have jumps of size − 1. Show that \(1/\mathscr {E}(X)=\mathscr {E}(Y)\) for
$$\displaystyle \begin{aligned}Y:=-X+[X^c,X^c]+{x^2\over 1+x}*\mu^X.\end{aligned}$$
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 + ⋯ + γd < 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, …, γd of her wealth in assets S1, …, Sd.
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.)
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).
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〈Xc, Lc〉(t)∕(cLdt).
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, …, σd ∈ 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•WQ(t) for some Q-standard Brownian motion WQ 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 [Wi, Wj](t) = t as well as [Wi, Wj](t) = t for i, j = 1, …, d with i ≠ j.
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].
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