§ 03 / integration

Stochastic Integration

Why pathwise integrals fail, the Itô isometry, and the continuous martingale integral.

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Section 1 / 5

Stochastic Integration

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The stochastic integral can also be motivated as a continuous-time generalization of the martingale transform in discrete time. The prototype of a discrete-time martingale is the symmetric random walk

Sn=∑k=1nXk,S_n=\sum_{k=1}^n X_k,

where (Xk)k∈N(X_k)_{k\in\mathbb{N}} are independent random variables taking the values ±1\pm1 with probability 1/21/2 each.

The martingale SnS_n can be interpreted as the gain or loss from a fair game after nn rounds. For example, XnX_n may represent a coin toss: on heads, player A pays one euro to player B, and on tails, player B pays one euro to player A.

Now allow player A to choose the stake freely in every round, up to an absolute bound KK, and to switch between heads and tails at will. This can be described by a game strategy (Ak)k∈N(A_k)_{k\in\mathbb{N}}, where each AkA_k is Fk−1\F_{k-1}-measurable. Thus the strategy does not have to be fixed before the game begins, but the stake for round kk may use only the information from rounds 1,…,k−11,\ldots,k-1.

The gain from the game with strategy AA is

(A∙S)n=∑k=1nAk(Sk−Sk−1),Ak∈Fk−1.(A\bullet S)_n = \sum_{k=1}^n A_k(S_k-S_{k-1}), \qquad A_k\in\F_{k-1}.

It is easy to show that (A∙S)(A\bullet S) is also a martingale. Thus, no matter how sophisticated the strategy is, a fair game cannot be transformed into a favorable game.

In continuous time, Brownian motion BB takes the place of the random walk SS, and a stochastic process f(ω,t)f(\omega,t) takes the place of the strategy AA. The goal is to define, as a continuous analogue of (3.1), a stochastic integral

I(f)(ω)=∫0Tf(ω,t) dBt(ω).I(f)(\omega) = \int_0^T f(\omega,t)\,dB_t(\omega).

As minimal requirements for a meaningful integral, we impose:

  • Linearity:
    I(αf+βg)=αI(f)+βI(g),α,β∈R.I(\alpha f+\beta g)=\alpha I(f)+\beta I(g), \qquad \alpha,\beta\in\R.
  • Indicator functions: for
    f(ω,t)=1{a<t≤b},f(\omega,t)=\1_{\{a<t\le b\}},
    we require
    I(f)=∫0T1{a<t≤b} dBt=∫abdBt=Bb−Ba.I(f) = \int_0^T\1_{\{a<t\le b\}}\,dB_t = \int_a^b dB_t = B_b-B_a.
  • At least every bounded, adapted, piecewise continuous stochastic process ff should be integrable with respect to Brownian motion.

It follows from the first two requirements that, for an integrand of the form

f(ω,t)=∑i=0n−1ai(ω)1{ti<t≤ti+1},f(\omega,t) = \sum_{i=0}^{n-1} a_i(\omega)\1_{\{t_i<t\le t_{i+1}\}},

we must have

I(f)(ω)=∑i=0n−1ai(ω)(Bti+1−Bti).I(f)(\omega) = \sum_{i=0}^{n-1} a_i(\omega) \bigl(B_{t_{i+1}}-B_{t_i}\bigr).

If we allow only adapted integrands, then necessarily

ai∈Fti,a_i\in\F_{t_i},

and (3.3) is exactly the discrete-time martingale transform (3.1).

The question is how to extend II to general integrands. It turns out that the most natural pathwise approach causes fundamental problems.