Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 1

On a probability space $(\Omega ,\mathcal{F},\mathbb{P})$, a standard one-dimensional Brownian motion is a real-valued stochastic process $B=(B_t{)}_{t\ge 0}$ with $B_0=0$ almost surely, continuous sample paths almost surely, and independent increments, such thatHere the second parameter is the variance. Thus its Brownian increments have the normal distribution with mean zero and variance equal to elapsed time. Equivalently, relative to a specified filtration $({\mathcal{F}}_t)$, require $B$ to be an adapted process and $B_t-B_s$ to be independent of ${\mathcal{F}}_s$ with this normal distribution. Taking the natural Brownian filtration recovers the first definition.

In the Lévy construction of Brownian motion, start with $B_0=0$ and $B_1=Z_0$, where $Z_0$ has the standard normal distribution. Suppose values are defined at the endpoints $u=k{2}^{-n}$ and $v=(k+1)2^{-n}$. Setusing fresh independent standard normal random variables $Z_{n,k}$, also independent of everything previously used. Join the values by straight segments to obtain a continuous piecewise linear process $B^{(n)}$ on the dyadic grid.

To verify the Brownian increments, write $D=B_v-B_u$ and $\ell =v-u$. The two new increments are $D/2+\sqrt{\ell }{Z}_{n,k}/2$ and $D/2-\sqrt{\ell }{Z}_{n,k}/2$. If $D\sim N(0,\ell )$, their variances are $\ell /2$ and their covariance is zero. They are Jointly Gaussian, so they are independent. Induction proves the required Brownian increments on every dyadic grid.

For convergence, successive interpolants differ by tents of disjoint interiors, soThe Gaussian tail bound and the union bound giveThese probabilities are summable. The First Borel-Cantelli lemma and ${\sum }_{n}n{2}^{-n/2}<\infty$ therefore prove uniform convergence of the interpolants almost surely. Their limit has continuous sample paths. Approximating any finite set of times by dyadic times passes the Jointly Gaussian increment laws to the limit, giving a standard Brownian motion on $[0,1]$.

Equivalently, with ${\psi }_{n,k}(t)={\int }_0^t{H}_{n,k}(s)ds$, these tents give the Lévy-Ciesielski expansion

Let $S_N={\sum }_{n=0}^N{\sum }_{k=0}^{2^n-1}{a}_{n,k}{H}_{n,k}$. The assumed orthonormality of the Haar wavelets givesEnumerate the coefficients level by level as $A_1,A_2,\dots$. They are independent and identically distributed standard normal random variables, so the squares have mean one. The strong law of large numbers impliesConsequently,A convergent sequence in a Hilbert space must be bounded in norm. Hence this random Haar wavelet series diverges in $\mathcal{H}$ almost surely.

Enumerate the orthonormal basis consisting of $H_0$ and the Haar wavelets $H_{n,k}$ as $(e_j{)}_{j\ge 1}$, and enumerate the corresponding independent standard normal random variables as $(A_j{)}_{j\ge 1}$. Set $c_j=⟨h,e_j{⟩}_{\mathcal{H}}$. The Parseval identity for a Hilbertian basis givesThe partial sums $T_m={\sum }_{j=1}^m{c}_j{A}_j$ form a martingale for $\sigma (A_1,\dots ,A_m)$, since the next summand is independent of the past and has mean zero. Moreover,The martingale convergence theorem for a martingale bounded in L2 space proves convergence almost surely and convergence in L2. Equivalently one may apply the Kolmogorov convergence theorem to the summable variances $c_j^2$.

A finite linear combination of independent normal random variables is normal. Its characteristic function is $\mathbb{E}{e}^{iu{T}_m}=\exp (-u^2{\sum }_{j\le m}{c}_j^2/2)$. Passing to the limit identifiesFor $h=0$ this is the constant zero random variable. Each fixed $h$ has its own permissible null set; simultaneous pathwise convergence for all $h\in \mathcal{H}$ is unnecessary.

For any $h_1,\dots ,h_m\in \mathcal{H}$ and real $u_1,\dots ,u_m$, linearity of the finite partial sums and convergence in L2 giveBy part (ii), this random variable has the normal distributionA finite family is Jointly Gaussian precisely when every real linear combination has a possibly degenerate normal distribution. Thus every finite subfamily of $(Wh{)}_{h\in \mathcal{H}}$ is Jointly Gaussian, which is the meaning of joint Gaussianity for this collection.

Use the same finite truncation $T_m(h)={\sum }_{j\le m}⟨h,e_j⟩A_j$ for $f$ and $g$. Since $\mathbb{E}[A_i{A}_j]={\delta }_{ij}$ by independence, zero means and unit variances,Here ${\delta }_{ij}$ is the Kronecker delta. The Cauchy-Schwarz inequality and convergence in L2 justify passing to the limit in the product's expectation. The Parseval identity for a Hilbertian basis now givesBecause each $Wh$ has mean zero, this is also its covariance identity. Thus $W$ is an isonormal Gaussian process over $\mathcal{H}$.

Write $e_t=1_{[0,t]}$ as an element of L2 space. Parts (iii) and (iv) show that $B_t=W{e}_t$ is a centered Gaussian process with covariance functionIn particular $B_t-B_s\sim N(0,|t-s|)$. The fourth moment of a centered normal random variable of variance $v$ is $3v^2$, soThe Kolmogorov continuity theorem, with moment exponent $4$ and time exponent $2>1$, provides a continuous modification $\hat{B}$. At each finite set of times this modification of a stochastic process has the same joint law as $B$.

For $0\le t_0<t_1<\cdots <t_m\le 1$, the successive increments are Jointly Gaussian. The covariance of two increments on disjoint intervals equals the inner product of their disjoint interval indicators and is zero. Zero cross-covariances of Jointly Gaussian variables imply independence, and each increment has variance $t_j-t_{j-1}$. Finally $e_0=0$ in L2 space, so ${\hat{B}}_0=0$ almost surely. ThereforeIndependence from the whole past in its natural Brownian filtration follows by extending finite-dimensional independence to the generated sigma-algebra.

For any real orthonormal basis $(G_i)$, the argument of (b)(ii) gives almost sure convergence and convergence in L2 of $\tilde{W}h$. Finite linear combinations have normal distributions, and the Parseval identity for a Hilbertian basis givesThus $W$ and $\tilde{W}$ are centered Gaussian processes with the same covariance function. More explicitly, their finite-dimensional characteristic functions are bothThis proves the basis invariance of an isonormal Gaussian process: equality of all finite-dimensional distributions, and hence equality in law of the indexed collections.

For the specified cosine orthonormal basis, put ${\lambda }_i=\pi (i-\frac{1}{2})$. Its coefficient against an interval indicator isConsequently the Brownian half-integer sine expansion, a Karhunen–Loève expansion, isFor every fixed $t$, the sum converges almost surely and in L2 space because the coefficient squares sum to $\Vert 1_{[0,t]}{\Vert }^2=t$. Its continuous modification is a standard Brownian motion by (b)(v). The expansion also holds in $L^2(\Omega \times [0,1])$: the expectation of the squared tail is the integral of the coefficient-square tail, which tends to zero by the dominated convergence theorem, with bound $t$.

To realize this expansion for the original $B$, choose $b_i=W{G}_i$. These are Jointly Gaussian with covariance matrix equal to the identity, hence are independent standard normal random variables. The finite orthogonal projections of $1_{[0,t]}$ onto $G_1,\dots ,G_N$ converge in $\mathcal{H}$, and the identity $\mathbb{E}|Wh{|}^2=\Vert h{\Vert }^2$ transfers that convergence to the displayed expansion for $B_t$. With an independently chosen sequence $(b_i)$ the resulting Brownian motion has the same law; a pathwise identification uses the coupling just given.