Ito integral for square-integrable predictable processes

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Fix a horizon $T>0$ and let ${\mathcal{P}}_T$ be the predictable sigma-algebra on $[0,T]\times \mathrm{\Omega }$ Progressively measurable and predictable processes. Let $H$ be a predictable process with $E{\int }_0^T{H}_s^{2}ds={\int }_{[0,T]\times \mathrm{\Omega }}{H}^{2}d(\mathrm{d}t\otimes P)<\mathrm{\infty },$ so that $H$ is an element of the quotient space $L^2(\mathrm{d}t\otimes P)$ over ${\mathcal{P}}_T$; its class is what is integrated. The Ito integral ${\int }_0^T{H}_{s}d{B}_s$, also written $H\cdot B$ or $I_T(H)$, is defined as follows.

1. Construction. By the density theorem Density of elementary predictable processes in predictable L2 there is a sequence $(H^n)$ of bounded elementary predictable integrands Elementary predictable Brownian integrands with $\mathrm{\parallel }{H}^n-H{\mathrm{\parallel }}_{L^2(\mathrm{d}t\otimes P)}\to 0$. For such a sequence the elementary integrals $I_T(H^n)$ Ito integral of an elementary predictable process form a Cauchy sequence in $L^2(P)$: by the elementary isometry Ito isometry for elementary integrands and the well-definedness of the elementary integral, $E(I_T(H^n)-I_T(H^m){)}^2=E{\int }_0^T(H^n-H^m{)}^{2}ds=\mathrm{\parallel }{H}^n-H^m{\mathrm{\parallel }}_{L^2(\mathrm{d}t\otimes P)}^2,$ and the right-hand side tends to $0$. Since $L^2(P)$ is complete Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, the classes $I_T(H^n)$ converge in $L^2(P)$; the limit is a class in $L^2(P)$, and ${\int }_0^T{H}_{s}d{B}_s:={\lim }_{n\to \mathrm{\infty }}{I}_T(H^n)\text{in }{L}^2(P)\mathrm{.}$ The limit is a random variable only up to almost-sure equality; statements about it are statements about that class. The limit does not depend on the chosen sequence: if another elementary sequence $K^n\to H$ is used, then the elementary isometry and linearity give $\mathrm{\parallel }{I}_T(H^n)-I_T(K^n){\mathrm{\parallel }}_{L^2(P)}=\mathrm{\parallel }{H}^n-K^n{\mathrm{\parallel }}_{L^2(\mathrm{d}t\otimes P)}\le \mathrm{\parallel }{H}^n-H{\mathrm{\parallel }}_2+\mathrm{\parallel }{K}^n-H{\mathrm{\parallel }}_2⟶0,$ so the two $L^2(P)$ limits agree.

2. Integrals at a time. For $t\in [0,T]$ put ${\int }_0^t{H}_{s}d{B}_s:={\int }_0^T{H}_s{1}_{[0,t]}(s)d{B}_s$, the construction of clause 1 applied to the truncated process $H{1}_{[0,t]}$. That process is predictable: $H$ is ${\mathcal{P}}_T$-measurable and $1_{[0,t]}$ is a deterministic predictable indicator, so the product is ${\mathcal{P}}_T$-measurable; and it is square-integrable because $\mathrm{\mid }H{1}_{[0,t]}\mathrm{\mid }\le \mathrm{\mid }H\mathrm{\mid }$ pointwise. The same construction applied to $H$ itself gives $t=T$. The family $({\int }_0^{t}HdB{)}_{t\in [0,T]}$ is a family of $L^2(P)$-classes indexed by $t$; its continuous version is supplied by item 13, and no path property is asserted by this definition.

3. Consistency with elementary integrands. If $H$ is itself elementary, then the constant sequence $H^n:=H$ is admissible in clause 1, so ${\int }_0^{T}HdB$ is the limit of the constant sequence of elementary sums, namely the elementary sum $I_T(H)$ of Ito integral of an elementary predictable process. The same argument with $H{1}_{[0,t]}$ in place of $H$, and the finite telescoping of the elementary sum over a refinement containing $t$, gives ${\int }_0^{t}HdB=I_t(H)$ for every $t\in [0,T]$. In particular the general definition extends, rather than replaces, the elementary definition.

4. Linearity in the integrand is not asserted here. Clause 1 defines each integral separately from an arbitrary approximating sequence; the linear and isometric properties of the extension are item 12. Until item 12 is proved, linearity may be used only for elementary integrands, where it is part of Ito integral of an elementary predictable process.

5. Comparison with deterministic Riemann integration. The integral is a stochastic integral: the integrand is paired with the Brownian path through left-endpoint sums and an $L^2(P)$ limit. This definition does not assert convergence of arbitrary tagged Riemann--Stieltjes sums for a general predictable integrand. Particular integrands can have a pathwise interpretation: for $H\equiv 1$ every tagged sum telescopes to $B_T-B_0$, which is also its Ito integral by clause 3 (the value at time zero is irrelevant to its product-measure class). The notation ${\int }_0^{T}HdB$ here always refers to the $L^2(P)$-class defined above.

The Axiom of Choice is declared because the construction selects an approximating sequence and uses the completeness and conditional-expectation interfaces that assume it; the inherited obligations are declared as dependencies of this item. An alternative construction that avoids selecting the sequence is not needed, because clause 1 shows every sequence gives the same class.