Kinetic and macroscopic models for active particles exploring complex environments with an internal navigation control system

A large number of biological systems — from bacteria to sheep — can be described as ensembles of self-propelled agents (active particles) with a complex internal dynamic that controls the agent’s behavior: resting, moving slow, moving fast, feeding, etc. In this study, we assume that such a complex internal dynamic can be described by a Markov chain, which controls the moving direction, speed, and internal state of the agent. We refer to this Markov chain as the Navigation Control System (NCS). Furthermore, we model that agents sense the environment by considering that the transition rates of the NCS depend on local (scalar) measurements of the environment such as e.g. chemical concentrations, light intensity, or temperature. Here, we investigate under which conditions the (asymptotic) behavior of the agents can be reduced to an effective convection–diffusion equation for the density of the agents, providing effective expressions for the drift and diffusion terms. We apply the developed generic framework to a series of specific examples to show that in order to obtain a drift term three necessary conditions should be fulfilled: (i) the NCS should possess two or more internal states, (ii) the NCS transition rates should depend on the agent’s position, and (iii) transition rates should be asymmetric. In addition, we indicate that the sign of the drift term — i.e. whether agents develop a positive or negative chemotactic response — can be changed by modifying the asymmetry of the NCS or by swapping the speed associated to the internal states. The developed theoretical framework paves the way to model a large variety of biological systems and provides a solid proof that chemotactic responses can be developed, counterintuitively, by agents that cannot measure gradients and lack memory as to store past measurements of the environment.

Bootstrap unit-root test for random walk with drift: The bsrwalkdrift command

In this article, we introduce the command bsrwalkdrift, which is primarily intended to perform a bootstrap unit-root test under the null hypothesis of random walk with drift. The method implemented in this command is considerably more precise than the corresponding case of the conventional augmented Dickey–Fuller test, which can be inaccurate when the true value of the drift term is small relative to the standard deviation of the innovations. The command also has an option to account for deterministic linear trend and another option to perform bootstrap unit-root tests under the null hypothesis of random walk without drift.

A financial market with singular drift and no arbitrage

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay $$\theta > 0$$ θ > 0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as $$\theta > 0$$ θ > 0 . This implies that there is no arbitrage in the market in that case. However, when $$\theta $$ θ goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.