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In this paper, we establish the almost sure convergence of two-timescalestochastic gradient descent algorithms in continuous time under general noiseand stability conditions, extending well known results in discrete time. Weanalyse algorithms with additive noise and those with non-additive noise. Inthe non-additive case, our analysis is carried out under the assumption thatthe noise is a continuous-time Markov process, controlled by the algorithmstates. The algorithms we consider can be applied to a broad class of bileveloptimisation problems. We study one such problem in detail, namely, the problemof joint online parameter estimation and optimal sensor placement for apartially observed diffusion process. We demonstrate how this can be formulatedas a bilevel optimisation problem, and propose a solution in the form of acontinuous-time, two-timescale, stochastic gradient descent algorithm.Furthermore, under suitable conditions on the latent signal, the filter, andthe filter derivatives, we establish almost sure convergence of the onlineparameter estimates and optimal sensor placements to the stationary points ofthe asymptotic log-likelihood and asymptotic filter covariance, respectively.We also provide numerical examples, illustrating the application of theproposed methodology to a partially observed Bene\v{s} equation, and apartially observed stochastic advection-diffusion equation.

Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement