The stochastic exponential Z t = exp{M t − M 0 − (1/2) M, M t } of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where M t = t 0 b(Y u ) dW u and Y is a one-dimensional diffusion driven by a Brownian motion W . Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y . As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.

In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this paper, we consider a limit order book model that allows for time-dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a tradingdependent spread that increases when market orders are matched against the order book. In this model, no price manipulation occurs and the optimal strategy is of the wait region/buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption, we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation, there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form.

Abstract. We obtain a deterministic characterisation of the no free lunch with vanishing risk, the no generalised arbitrage and the no relative arbitrage conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.

In this paper we describe the pathwise behaviour of the integral functional t 0 f (Yu) du for any t ∈ [0, ζ], where ζ is (a possibly infinite) exit time of a one-dimensional diffusion process Y from its state space, f is a nonnegative Borel measurable function and the coefficients of the SDE solved by Y are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion. Convergence of integral functionalsThe proof of the results consists of two steps. The first step, which uses only basic properties of diffusion processes and their local times, reduces the original problem to a question of the convergence of an integral functional of Brownian motion. The answer to this question is given in Lemma 4.1, which is not new. This result or its generalizations appeared in [8, p. 225-226], [1, Lem. 1.4.1], [3, Lem. 2], [2, Prop. A1.8], [11, Prop. 3.2]. In the second step, we present two short proofs of Lemma 4.1: (i) Williams' theorem (see [18, Ch. VII, Cor. 4.6]) and the result on integral functionals of Bessel processes from [4] are applied; (ii) a direct approach based on the first Ray-Knight theorem (see [18, Ch. XI, Th. 2.2]) is followed. The second proof uses the same ideas as the proofs in the references mentioned above (with a single technical point worked out differently -see the discussion following the statement of Lemma 4.1). The two proofs are presented for their simplicity and in order to make the paper self-contained.In [11] the convergence of the integral functionals of the form t 0 f (X s ) ds is investigated, where f is a nonnegative Borel function and X a strong Markov continuous local

In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form Ex τ 0 e −λs f (Xs) ds, λ ≥ 0 for one-dimensional diffusions X. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem.