Stochastic calculus

Itô and friends



Calculus that works, in a certain sense, for random objects, of certain types. Very popular for stochastic differential equations. This is a popular and well-explored tool; notes here are not supposed to be tutorial; I simply want to maintain a list of useful definitions because the literature gets messy and ambiguous sometimes.

Itô Integral

TBD.

Itô’s lemma

Specifically, let \(X=\left(X^{1}, \ldots, X^{n}\right)\) be a tuple of semimartingales and let \(f: \mathbb{R}^{n} \rightarrow\mathbb{R}\) have continuous second order partial derivatives. Then \(f(X)\) is also a semimartingale and the following formula holds:

\[\begin{aligned} f\left(X_{t}\right) - f\left(X_{0}\right) &= +\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \mathrm{d} X_{s}^{i} \\ &\quad +\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s-}\right) \mathrm{d}\left[X^{i}, X^{j}\right]_{s}^{c} \\ &\quad +\sum_{0<s \leq t}\left(f\left(X_{s}\right)-f\left(X_{s-}\right)-\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \Delta X_{s}^{i}\right) \end{aligned}\]

Here the bracket term is the quadratic variation, \[ [X,Y] := XY-\int X_{s-} \mathrm{d}Y(s)-\int Y_{s-} \mathrm{d}X(s) \]

For a continuous semimartingale, the jump terms are null, and the left limits are equal to the function itself \[\begin{aligned} f\left(X_{t}\right) - f\left(X_{0}\right) &= +\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s}\right) \mathrm{d} X_{s}^{i} \\ &+\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s}\right) \mathrm{d}\left[X^{i}, X^{j}\right]_{s}^{c} \end{aligned}\]

Some authors assume that in Itô calculus the driving noise is not a general semimartingale but a Brownian motion, which is a continuous driving noise. Others use the term Itô calculus to describe a more general version. Sometimes the distinction is made between an Itô diffusion, with a Brownian driving term and a Lévy SDE, which has no implication that the driving term is Brownian. However, this is generally messy and particularly in tutorials generated nby the applied finance people it can be quite hard to work out which set of definitions they are using.

Stratonovich Integral

Doss-Sussman transform

Reducing a stochastic integral to a deterministic one, i.e. replacing the Itô integral with a Lebesgue one. (Sussmann 1978; Karatzas and Ruf 2016).

Assumptions: \(\sigma \in C^{1,1}(\mathbb{R}), \sigma, \sigma^{\prime} \in L^{\infty}, b \in C^{0,1}\) \[ \mathrm{d} X(t)=b(X(s)) \mathrm{d} s+\frac{1}{2} \sigma(X(s)) \sigma^{\prime}(X(s)) \mathrm{d} s+\sigma(X(s)) \mathrm{d} W_{s} \] has a unique (strong) solution \(X=u(W, Y)\) for some \(u \in C^{2}(\mathbb{R})\) and \[ \mathrm{d} Y(t)=f(W(t), Y(t)) \mathrm{d} t \] for some \(f \in C^{0,1}\).

Rogers and Williams (1987) section V.28.

Paley-Wiener integral

There is a narrower, but lazier, version of the Itô integral. Jonathan Mattingly introduces it in Paley-Wiener-Zygmund Integral Assuming \(f\) continuous with continuous first derivative and \(f(1)\)=0.

We define the stochastic integral \(\int_{0}^{1} f(t) * \mathrm{d} W(t)\) for these functions by the standard Rieman integral, \[ \int_{0}^{1} f(t) * \mathrm{d} W(t)=-\int_{0}^{1} f^{\prime}(t) W(t) \mathrm{d} t \] Then \[ \mathbf{E}\left[\left(\int_{0}^{1} f(t) * \mathrm{d} W(t)\right)^{2}\right]=\int_{0}^{1} f^{2}(t) \mathrm{d} t. \] Paley, Wiener, and Zygmund then used this isometry to extend the integral to \(f\in L^{2}[0,1]\) as the limit of approximating continuous functions.

What does this get us in terms of SDEs?

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