Abstract
In this chapter, we focus on the essential aspects of stochastic calculus , a theory of integration with respect to Brownian-motion-type processes and their transformation properties.
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Notes
- 1.
We use the standard notation \(X(\omega )\), where \(\omega \in \varOmega \) and \(X(\omega ) \in \mathbb {R}\), for a (real-valued) random variable on \((\varOmega ,\mathscr {F},\mathsf {P})\).
- 2.
Remember that X is normally distributed with mean \(\mu \) and variance \(\sigma ^2\) when \(\mathsf {P}[ X \in A ] = \int _A \frac{1}{\sqrt{2\pi \sigma ^2}} \exp \left( -\frac{(x-\mu )^2}{2\sigma ^2}\right) \mathrm {d}x\) for any Borel subset A of \(\mathbb {R}\).
- 3.
The Brownian bridge construction of the Brownian motion: Consider a probability space with a countably infinite sequence of independent standard Gaussian random variables.
Suppose that we have constructed \(B^{[n]}=(B_{k \, T \,2^{-n}})_{k=0,1,2,\ldots ,2^n}\). Since the law of \(B^{[n+1]}=(B_{k \, T \,2^{-n-1}})_{k=0,1,2,\ldots ,2^{n+1}}\) is multivariate Gaussian, we can write the conditional law \((B_{k \, T \,2^{-n-1}})_{k=1,3,5,\ldots ,2^{n+1}-1}\) given \(B^{[n]}\) explicitly as multivariate Gaussian which we can construct using the given sequence of standard Gaussians. Continue this iteration ad infinitum. We leave as an exercise to check that, if \(B^{[n]}\) is linearly interpolated to all \(t \in [0,T]\), then the sequence \(B^{[n]}\) converges uniformly almost surely as \(n \rightarrow \infty \). It is sufficient to show that the uniform norms of \(B^{[n+1]}-B^{[n]}\) are summable over n.
- 4.
Hölder continuity of a Brownian motion follows as a side product from the Brownian bridge construction of the Brownian motion.
- 5.
We use the standard notations \(\sigma (A,B,\ldots )\) and \(\sigma (A_i, i \in I)\) for the \(\sigma \)-algebra generated by the random variables \(A,B,\ldots \) and \(A_i\), \(i \in I\), respectively.
- 6.
Convergence in probability means here that that for each \(\varepsilon > 0\) there exist \(\delta >0\) such that \(\mathsf {P}[ | \sum _{k=0}^{m(\pi )-1} |X_{t_{k+1}} - X_{t_{k}}|^p - V_X^{(p)} (t) | \ge \varepsilon ] < \varepsilon \) when \(\text {mesh}(\pi ) < \delta \).
- 7.
The class \(\mathscr {L}^2\) could be instead called \(\mathscr {L}^2(T)\) and then we could set \(f \in \mathscr {L}^2\) if and only if \(f \in \mathscr {L}^2(T)\) for any \(T > 0\). For simplicity, we use the notation \(\mathscr {L}^2\) for both classes.
- 8.
The minimum of two stopping times is a stopping time.
- 9.
If \(\mathsf {E}[ \exp (\mathrm {i}\theta _1 X) | \mathscr {G}]=\psi (\theta _1)\) is a constant as a function of \(\omega \in \varOmega \), then for any \(\mathscr {G}\)-measurable random variable Z, it holds that \(\phi _{(X,Z)}(\theta _1,\theta _2)=\mathsf {E}[ \exp (\mathrm {i}\theta _1 X + \mathrm {i}\theta _2 Z) ]=\mathsf {E}[ \exp ( \mathrm {i}\theta _2 Z) \, \mathsf {E}[ \exp (\mathrm {i}\theta _1 X) | \mathscr {G}] ]=\psi (\theta _1) \phi _Z(\theta _2)\) by properties of conditional expected value. Here \(\phi _{\underline{\theta }}=\mathsf {E}[ \exp ( \mathrm {i}\underline{\theta }\cdot \underline{Y})]\), \(\underline{\theta } \in \mathbb {R}^n\) is the characteristic function of a \(\mathbb {R}^n\)-valued random variable \(\underline{Y}\). Thus by the uniqueness theorem of the characteristic function, it follows that X and Z are independent and that the law of X is the unique probability measure on \(\mathbb {R}\) such that \(\phi _X=\psi \). Since this holds in particular for any random variable \(Z=\mathbbm {1}_E\), \(E \in \mathscr {G}\), it follows that X is independent from \(\mathscr {G}\).
- 10.
Define as usual the following partial differential operators \(\partial =\frac{1}{2} (\partial _x - \mathrm {i}\partial _y)\) and \(\overline{\partial }=\frac{1}{2} (\partial _x + \mathrm {i}\partial _y)\).
- 11.
By the Cauchy–Riemann equations, \(u_{xx} + u_{yy} = v_{xy} - v_{xy} = 0\) and similarly for v.
- 12.
The notation \(\mathsf {Q}(f)\) denotes the expected value of f with respect to the measure \({\mathsf {Q}}\).
- 13.
There are many equivalent definitions of the weak convergence—by a result named the Portmanteau theorem [1]. The above definition is equivalent to any of the following statements (i) \(\limsup \mathsf {Q}_n (C) \le {\hat{\mathsf {Q}}} (C)\) for all closed sets C of the space \(\mathscr {X}\), (ii) \(\liminf \mathsf {Q}_n(U) \ge {\hat{\mathsf {Q}}} (U)\) for all open sets U of the space \(\mathscr {X}\), (iii) \(\lim \mathsf {Q}_n(A) = {\hat{\mathsf {Q}}}(A)\) for all continuity sets A of the measure \({\hat{\mathsf {Q}}}\). A set A is a continuity set if \({\hat{\mathsf {Q}}} (\partial A)=0\).
- 14.
A set is sequentially precompact if every sequence in the set contains a converging subsequence.
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Kemppainen, A. (2017). Introduction to Stochastic Calculus. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_2
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