Stochastic calculus
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Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Albert Einstein and other physical diffusion processes in space of particles subject to random forces. More recently, the Wiener process has been widely applied in financial mathematics to model the evolution in time of stock and bond prices.
The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines.) The Stratonovich integral can readily be expressed in terms of the Itô integral. Another benefit of the Stratonovich integral is that it enables some problems to be expressed in a co-ordinate system invariant form and is therefore invaluable when developing stochastic calculus on manifolds other than <math>\mathbb{R}^n</math>. The Dominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itô form.
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[edit] Quadratic-variation process
The key to the construction of a stochastic integral is the definition of a quadratic-variation process; the quadratic variation of a general <math>L^2</math> (see L2 space) bounded martingale <math>X_t</math> may be defined as the increasing process <math>[X]_t</math> such that
- (i)<math>[X]_0 = 0</math>
- (ii)<math>\Delta [X]_t = (\Delta X_t ) ^2 \quad \forall t </math>
- (iii)<math>X_t^2 - [X]_t</math> is a uniformly integrable martingale.
The proof that such a process may be constructed and is unique is a major hurdle in the development of stochastic calculus. However for a process <math>X_t</math> with continuous sample paths it may be shown to be equivalent to the following definition for a partition
- <math>\pi_t = \{ 0 = t_0 < t_1 < \cdots < t_m=t\}</math>
whose mesh is defined by
- <math>\delta(\pi_t) = \max_{k \in [1,m]} | t_{k}-t_{k-1} | </math>
in terms of which the quadratic-variation process may be defined by
- <math>V_t = \lim_{\delta(\pi_t) \to 0} \sum_{\pi} | X_{t_k} - X_{t_{k-1}} | ^2.</math>
A related process <math>\langle X \rangle_t</math> is historically sometimes used as the basis of the integral; this process is defined as a previsible process satisfying the first and thirds conditions above. It can be shown that this process is the previsible projection of <math>[X]_t</math>. While much of the theory can be developed from this starting point, to approach the theory stochastic integration of discontinuous processes it proves the wrong starting point.
This definition is extended to semimartingales by defining
- <math> [X]_t = [X^{\mathrm{cm}}]_t + \sum_{0 \le s \le t} \Delta X_s^2</math>
where <math>X^{\mathrm{cm}}</math> is the canonical continuous martingale in the decomposition of <math>X</math> i.e.
- <math> X_t = X^{\mathrm{cm}}_t+ X^{\mathrm{dm}}_t + A_t</math>
where <math>A</math> is of finite variation.
The definition of the quadratic variation process gives rise immediately to the definition of the covariation process can be defined by polarization
- <math> [X,Y]_t := \frac{1}{4} \left ( [X+Y]_t - [X-Y]_t \right )</math>
[edit] Stochastic integral of simple process
For a sequence of stopping times satisfying <math>0 \le T_1 \le T_2 \le \cdots</math>, and for each <math>k</math>, <math>H_k</math> an <math>\mathcal{F}_{T_k}</math> measurable random variable, then a process <math>H</math> of the form
- <math> H_t = 1_{ \{0\}}(t) H_0 + \sum_k H_k 1_{(T_k, T_{k+1}]}(t)</math>
is said to be a simple process.
For <math>X</math> an L2 bounded local martingale define the Itô integral <math>(H \cdot X)</math> as
- <math> (H\cdot X)_t =\sum_k H_k (X_{T_{k+1}\wedge t} - X_{T_k\wedge t} )</math>
This process can be proved to be itself an <math>L^2</math> bounded martingale and thus by the usual <math>L^2</math> martingale convergence theorem it is only necessary to consider the limiting process <math>(H \cdot X)_\infty</math> which is consequently an element of <math>L^2 (\mathcal{F}_\infty)</math>.
[edit] Itô isometry
Given the quadratic-variation process, a seminorm may be introduced on the space of previsible stochastic processes
- <math>\|H\|^2_X = \int H^2_s \, d [X]_s</math>
where the integral is to be understood in the usual Lebesgue sense. This is not a norm, since <math>\|H\|_X=0</math> does not imply that <math>H</math> is the zero process. Let
- <math> L^2(X) = \{ H \mathrm{\ previsible\ such\ that\ } \|H\|_X < \infty \}</math>
The Itô isometry between <math>L^2(X)</math> and <math>L^2(\mathcal{F}_\infty)</math> is given by
- <math> \| (H \cdot X) \|^2_2 = \mathbb{E}(H \cdot X )^2 = \| H \|^2_X</math>
This can be shown to hold for simple processes following the definitions above and then via the usual Banach space arguments the isometry allows the definition of the Itô integral to be extended to the space of previsible processes <math>H \in L^2(X)</math>.
[edit] Semimartingales as integrators
The general Itô integration theory extends naturally to the semimartingales as integrators. For a semimartingale <math>Y</math> which has a Doob-Meyer decomposition
- <math>Y_t= M_t + A_t</math>
where <math>M</math> is a local martingale starting from zero, and <math>A</math> is a process of finite variation (this decomposition is unique for continuous process, but not in general). The Itô integral of a previsible process <math>X</math>with respect to <math>Y</math> is defined by
- <math>(X \cdot Y)_t = (X \cdot M)_t + (X \cdot A)_t</math>
where the first integral is defined by the natural extension of the Itô integral from martingale integrators to local martingale integrators, and the second integral is understood in the usual Lebesgue-Stieltjes sense. Because of the non-uniqueness of the semi-martingale decomposition it is necessary to prove that any result holds independently of the decomposition.
[edit] Itô's formula
One of the most powerful and frequently used theorems in Stochastic calculus states that if <math>f</math> is a <math>C^2</math> function from <math>\mathbb{R}^d \to \R</math> and <math>X_t=(X^{(1)}_t, \ldots, X^{(d)}_t)</math> is a d-dimensional semimartingale then
| <math> f(X_t) = </math> | <math>f(X_0) + \sum_{i=1}^d \int_0^t
\frac{\partial f}{\partial x_i} (X_{s-}) \, d X^{(k)}_s </math> |
| <math>+ \frac{1}{2} \sum_{i=1}^d \sum_{j=1}^d \int_0^t \frac{\partial^2 f}{\partial x_i\partial x_j} (X_{s-}) d [X^{(i)}, X^{(j)}]^{\mathrm{cm}}_s </math> | |
| <math>+ \sum_{0 \le s \le t} \Delta f(X_s) - \sum_{i=1}^m \frac{\partial f}{\partial x_i}(X_{s-})\Delta X^{(i)}_s </math> |
where the continuous martingale part of the quadratic covariation process of two semimartingales <math>X</math> and <math>Y</math> is defined by
- <math> [X,Y]^{\mathrm{cm}}_t = [X , Y]_t -\sum_{s \le t} \Delta X_s
\Delta Y_s.</math>
[edit] Discontinuous process
It might appear that the Itô integral defined here could be extended to discontinuous martingales by decomposing a square-integrable martingale X in the form
- <math>X_t = X^{\mathrm{cm}}_t + V_t</math>
where <math>X^{\mathrm{cm}}</math> is a square-integrable continuous martingale and V is a process of integrable variation. Unfortunately this is decomposition is not always possible, but the space of square-integrable processes of integrable variation is dense in the orthogonal complement of the space of continuous square-integrable martingales.
[edit] Stratonovich integral
The Stratonovich integral can be defined in terms of the Itô integral as
- <math> \int_0^t X_s \circ d Y_s : = \int_0^t X_s d Y_s + \frac{1}{2} \left [ X, Y\right]_t</math>
The alternative notation
- <math> \int_0^t X_s \partial Y_s </math>
is also used to denote the Stratonovich integral.
[edit] External links
- Notes on Stochastic Calculus A short elementary description of the basic Itô integral.fr:Calcul stochastique
