Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 1

Let $\mathcal{P}$ be the predictable sigma-algebra on $\Omega \times [0,\infty )$, generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,u]$ with $A\in {\mathcal{F}}_s$. A stochastic process $H$ is previsible precisely when $(\omega ,t)\mapsto H_t(\omega )$ is $\mathcal{P}$-measurable.

Every left-continuous adapted process is previsible. Indeed, on intervals $(k{2}^{-n},(k+1)2^{-n}]$, put $H_t^{(n)}=H_{k{2}^{-n}}$, and put $H_0^{(n)}=H_0$. Each approximation is a predictable process, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable; left continuity gives $H_t^{(n)}\to H_t$ for every $t$, so measurability passes to the limit.

An adapted càdlàg process need not be previsible. For a concrete example, let $\xi$ take values $0,1$ with equal probabilities, let ${\mathcal{F}}_t$ be trivial for $t<1$ and ${\mathcal{F}}_t=\sigma (\xi )$ for $t\ge 1$, and set $H_t=\xi 1_{[1,\infty )}(t)$. This is an adapted càdlàg process, but $H_1=\xi$ is not ${\mathcal{F}}_{1-}$-measurable, where ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{s<1}{\mathcal{F}}_s)$. Sections at time $1$ of the generators of $\mathcal{P}$ all belong to ${\mathcal{F}}_{1-}$, so every predictable process has an ${\mathcal{F}}_{1-}$-measurable value there; this proves the counterexample directly.

Use the standard convention that the finite-variation process $X$ has càdlàg paths and that the stochastic integral starts from zero. Write $V(X)$ for its total-variation process. The necessary admissibility condition is ${\int }_{(0,t]}|H_s|d{V}_s(X)<\infty$ almost surely for each finite $t$; it holds, for example, for locally bounded $H$. Define the pathwise Lebesgue-Stieltjes integralAn atom at $s>0$ contributes $H_s\Delta X_s$, so this convention also fixes the treatment of jumps.

Fix $t$. For a generator $H_s=1_A{1}_{(a,b]}(s)$ of the predictable sigma-algebra, with $A\in {\mathcal{F}}_a$, the Lebesgue-Stieltjes integral isIt vanishes when $t\le a$ and is ${\mathcal{F}}_t$-measurable otherwise. The total-variation process is adapted: for a càdlàg path its value at $t$ is the supremum over a countable collection of partitions of an interval using rational interior points and endpoint $t$. Thus the positive and negative Lebesgue-Stieltjes measures $d{X}^{\pm }=(dV(X)\pm dX)/2$ have ${\mathcal{F}}_t$-measurable masses on the generator intervals.

The Monotone class theorem, applied separately to these positive random measures on $[0,t]$, extends this measurability from the generators of the predictable sigma-algebra to all nonnegative predictable processes. Truncation and positive/negative decomposition then cover every admissible $H$. Consequently $H\cdot X$ is adapted.

The integrability condition is essential. Taken literally, an arbitrary previsible process cannot always be integrated against a finite-variation process: with $X_t=t$ and deterministic $H_s=s^{-1}$ for $s>0$, $H_0=0$, the proposed value at every $t>0$ diverges. The usual càdlàg convention is also what identifies $dX$ by $dX((a,b])=X_b-X_a$.

For deterministic step functions, define directly from Brownian motion incrementswith $s_0$ the left endpoint of the first interval and $f=0$ outside the listed intervals. The printed endpoint list omits $s_0$; the intended convention is $0\le s_0<s_1<\cdots <s_n$, usually $s_0=0$.

Given finitely many observation times, refine the deterministic partition of an interval to include all observation times and all $s_i$. Every $Z_{t_j}$ is a linear combination of the same finite family of independent centered normal Brownian motion increments. Thus every linear combination of $(Z_{t_1},\dots ,Z_{t_m})$ is normal, including possibly a degenerate constant: $Z$ is a Gaussian process. By independence, only increments in the common interval $[0,s\wedge t]$ contribute to the covariance, givingThe same calculation for two step functions $f,g$ proves, rather than assumes, the Itô isometryFor $f\in L^2({\mathbb{R}}_{+})$, choose deterministic step functions $f_n\to f$ in the L2 space. This identity makes $Z_t(f_n)$ a Cauchy sequence in the L2 space; the Riesz-Fischer theorem defines $Z_t(f)$ by mean-square convergence, independently of the approximation. The covariance formula passes to the limit by the Cauchy-Schwarz inequality. For finitely many times and coefficients $a_j$, the characteristic functions converge toso the extension remains a Gaussian process.

For completeness, the construction has a continuous version. Each approximating $Z(f_n)$ is a martingale, directly by the centered independent increments of Brownian motion; the Doob L2 maximal inequality givesTo prove the inequality used here without assuming Itô integral properties, a continuous martingale $N$ satisfies $\lambda \mathbb{P}(N_T^{\ast }\ge \lambda )\le \mathbb{E}(|N_T|1_{\{N_T^{\ast }\ge \lambda \}})$, by conditioning at the first crossing on finite deterministic grids and then using continuity; integrating in $\lambda$ and using the Cauchy-Schwarz inequality yields $\Vert N_T^{\ast }{\Vert }_2\le 2\Vert N_T{\Vert }_2$. A sufficiently fast subsequence therefore converges uniformly almost surely on every compact time interval, giving the continuous version of the stochastic integral.