# Strong law of large numbers - Continuous martingales - Reference request

Let $$(\Omega, \mathcal F P)$$ be a probability space with a filtration $$\mathcal F_t$$ and $$M = (M_t,t\geq 0)$$ be a real-valued continuous $$\mathcal F_t$$-martingale such that its quadratic variation satisfies $$\langle M\rangle_\infty = \infty$$ a.s. Then, the strong law of large numbers for martingales yields $$\lim_{t\to\infty} \frac{M_t}{\langle M\rangle_t}=0,$$ a.s. (see e.g. Lipster & Shirayev Theory of Martingales, Corollary 1.1, p. 144).

I'm looking for a reference for this result, but in the case of vector-valued continuous martingales. In particular, let $$M$$ be a vector-valued $$\mathcal F_t$$-martingale (with values in $$\mathbb R^d$$). In this case, the quadratic variation $$\langle M \rangle$$ is defined as the unique $$Q = (Q_t, t \geq 0)$$ with values in $$\mathbb R^{d\times d}$$ such that $$M_t \otimes M_t - Q_t$$ is an $$\mathcal F_t$$-martingale (see e.g. Pavliotis & Stuart, Multiscale Methods: Averaging and Homogenization, Definition 3.18, p. 46).

What can we say in this case? For example, if $$\langle M \rangle_t$$ is a non-singular (thus in this case symmetric positive definite) matrix a.s. for all $$t \geq 0$$ and such that $$\|\langle M\rangle_\infty\| = \infty$$ a.s., can we say that $$\lim_{t\to\infty} \langle M \rangle_t^{-1} M_t = 0, \qquad \qquad (1)$$ a.s. in $$\mathbb R^d$$?

Example

I give a simple example of a martingale I am interested in. Let $$W$$ be a one-dimensional Wiener process with natural filtration $$\mathcal F_t$$ and let $$\beta = (\beta_t, t\geq 0)$$ be a $$\mathcal F_t$$-adapted stochastic process with values in $$\mathbb R^d$$ and components $$\beta_t = (\beta_t^{1}, \ldots, \beta_t^{d})^\top$$. Then consider the $$\mathcal F_t$$-martingale $$M_t = \int_0^t \beta_s \, \mathrm{d} W_s = \begin{pmatrix}\int_0^t \beta_s^1 \, \mathrm{d} W_s, \ldots, \int_0^t \beta_s^d \, \mathrm{d} W_s \end{pmatrix}^\top.$$ The quadratic variation of $$M$$ satisfies (by Itô isometry) $$\langle M \rangle_t=\int_0^t \beta_s \beta_s^\top \, \mathrm{d} s.$$ Can we conclude something in the line of $$(1)$$?

The simplest method to solve your problem is use above mentioned result to each component of $$M_t=(M^{(1)}_t,\cdots, M^{(d)}_t)$$, that is $$\begin{equation*} \{\langle M^{(i)} \rangle_\infty=\infty, 1\le i\le d \} \subset \{\lim_{t\to\infty} D_t^{-1} M_t=0\}, \qquad \text{a.s.} \end{equation*}$$ where $$D_t=\mathrm{diag}(\langle M^{(1)} \rangle_t ,\cdots, \langle M^{(d)} \rangle_t )$$ is a diagonal matrix.
Let $$M = \{M_t\}_{t\geq 0}$$ be a real-valued continuous local martingale vanishing at $$t = 0$$. Then $$\lim_{t \to \infty} \langle M\rangle_t = \infty \quad \text{a.s.} \implies \lim_{t \to \infty} M_t / \langle M\rangle_t = 0 \quad \text{a.s.}$$ and also $$\limsup_{t \to \infty} \langle M\rangle_t / t < \infty \quad \text{a.s.} \implies \lim_{t \to \infty} M_t / t = 0 \quad \text{a.s.}$$