Stochastic calculus

Still WIP

We define a (1d)-stochastic process to be a function \(X_t\) given by

\[ dX_t = a(t,X_t) dt + b(t,X_t) dW_t \]

where \(a\), \(b\) are real-functions and \(W_t\) is a Wiener process.

Equivalently, we can write it as

\[ X_t = X_0 + \int_0^t a(t,X_t) dt + \int_0^t b(t,X_t) dW_t \]

for some suitable linear definition of an integral \(\int\). The first part is the standard RIemannian integral, so the definition of a particular stochastic calculus is determined by the definition of the stochastic integral \(\int f dW_t\).

Each alternatively definition of the stochastic integral leads to an alternative solution of any stochastic SDE. In reality, there are only two viable choices, namely the Ito integral, denoted \(\int f dW_t\), and the Stratonovich integral, denoted \(\int f \circ d W_t\), besides the deterministic choice, namely the Riemann integral, denoted \(\int f dt\).

Ito integral

write down definition of Ito integral and Lemma it is martingale: definition

Ito Lemma

The general Ito lemma for a \(m\)-dimensional twice differentiable continuous semi-martingale \(X = \langle X^1, ... ,X^m \rangle\) states that

\[ f(X_t) = f(X_0) + \sum_{i=1}^m \int_0^t \frac{\partial}{\partial x_i} f(X_s) d X_s^i + \frac{1}{2} \sum_{i , j = 1}^m \int_0^t \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} f(X_s) d [X^i, X^j]_s \]

where \([X^i, X^j]_s\) is the quadratic covariantion.

TODO: explain chain-rule application

Stratonovich integral

write down definition of Stratonovich integral

Taylor expansion

The standard multi-dimensional Taylor expansion of a real function \(f : \mathbb{R}^n \rightarrow \mathbb{R}\) around a point \(x_0\) is defined as

\[ T(f(x),x_0) = \sum_{i \in \mathbb{N}^n} a_i (x_1 - (x_0)_1)^{i_1}\cdots (x_n - (x_0)_n)^{i_n} \]

where

\[ a_i = \frac{f^{(i)}(x_0)}{i!} = \frac{\frac{\partial^{i_1}}{\partial x_{1}^{i_1}} \cdots \frac{\partial^{i_n}}{\partial x_{n}^{i_n}} f (x_0)}{i_1 ! \cdots i_n!} \]

check the above...

Ito-Taylor expansion

The multi-dimensional Ito-Taylor expansion of a stochastic function \(f : \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}\) around a point \(x_0\) is defined as

\[ f(X_\tau) = \sum_{\alpha \in A} I_\alpha[f_\alpha(X_\rho)]_{\rho,\tau} + \sum_{\alpha \in B(A)} I_\alpha [f_\alpha(\cdot, X\cdot)]_{\tau,\rho} \]

where the second term is the remainder term.

The truncated expansion of degree \(m \in \mathbb{N}\) then becomes

\[ T(f, t_0)(t) = \sum_{\alpha \in M, l(\alpha)+n(\alpha)\leq m} I_\alpha[f_\alpha(X_{t_0})]_{t_0,t} \]

Where the Ito-index function is defined as

\[ I_\alpha[] \]

Stratonovich-Taylor expansion

Stratonovich versus Ito Calculus

As described below we can for any problem always switch between the Stratonovich and Ito calculi as our framework. Whichever one might the right setup depends on which properties are needed by the particular situation.

The Ito integral is nicely martingale, linear, scalar and such. However, it does not follow the classical transformation rules of the Riemannian integral. Instead it has the Ito Lemma as its chain-rule. On the plus side it allows for the use of nice moment equations and the Fokker-Planck equation.

The Stratonovich integral upholds all of the rules and techniques that we know from classical/Riemannian calculus. However, it is not Martingale, does not uphold the Fokker-Planck equation or comes with nice moment equations.

As mentioned in Kloeden and Platen, when dealing with approximations of truely continuous systems, in both time and state, i.e. systems which can essentially be viewed as classical Riemannian systems with a added random noise, the Stratonovich calculus is usually the best fit, since it follows the rules of the Riemann calculus. However, if we are trying to approximate a discrete system, i.e. if the discrete deltas are model by a discrete noisy system, then the Ito calculus is the better fit, since for discrete simulations the Euler scheme coincides with the 0.5 strong Ito-Taylor expansion and similar with regards to the Milstein scheme. Also, of course, the Stratonovich integral is forward looking and non-Martingale, which makes it difficult to work with - this should also matter when it comes to simulations.

The Stratonovich calculus is also particular nice for simulations, since the Riemann integral of a sample path of a Brownian bridge process converges towards the Stratonovich integral. This means that we can simulate a Stratonvich SDE by replacing the Stratonovich integral by the limit of the Riemann integral of the sample paths of a Brownian bridge process, where the limit is both with regards to the paths and the Brownian bridge linearisation.

Another thing: It seems that the Stratonovich integral is nice when it comes to the Taylor expansion... although both are generalisations of the Riemann expansion

Transformations

The Ito and Stratonovich integrals are related in the following way:

\[ \int_0^T h(W_t) \circ dW_t = \int_0^T h(W_t) dW_t + \frac{1}{2} \int_0^T h'(W_t) dt \]

Likewise SDEs involving Ito and Stratonovich integrals are related in the following way:

\[ \begin{array}{lcl} X_t & = & X_0 + \int_0^T a(t,X_t) dt + \int_0^T b(t,X_t)\circ dW_t \\ & = & X_0 + \int_0^T \bar{a}(t,X_t) dt + \int_0^T b(t,X_t) dW_t \end{array} \]

where

\[ \bar{a}(t,X) = a(t,X) + \frac{1}{2}b(t,X)\frac{db}{dX}(t,X) \]