Stochastic calculus

Categories: Stochastic processes | Financial mathematics

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consisent 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 modelling 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 intregral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines) and the integrals can readily be expressed in terms of the Itô integral. 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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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> 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 UI 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 an 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 cannonical 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>
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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>.

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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>.

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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 martingal 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 decompostion.

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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>

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