Optimal portfolio choice with path dependent labor income: the infinite horizon case

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agentreceives labor income which adjusts to financial market shocks in a path dependent way. Thispath-dependency is the novelty of the model, and leads to an infinite dimensional stochasticoptimal control problem. We solve the problem completely, and find explicitly the optimalcontrols in feedback form. This is possible because we are able to find an explicit solutionto the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if stateconstraints are present. To the best of our knowledge, this is the first infinite dimensionalgeneralization of Merton’s optimal portfolio problem for which explicit solutions can be found.The explicit solution allows us to study the properties of optimal strategies and discuss theirfinancial implications

Dynasore, a Dynamin Inhibitor, Inhibits Trypanosoma cruzi Entry into Peritoneal Macrophages

BACKGROUND: Trypanosoma cruzi is an intracellular parasite that, like some other intracellular pathogens, targets specific proteins of the host cell vesicular transport machinery, leading to a modulation of host cell processes that results in the generation of unique phagosomes. In mammalian cells, several molecules have been identified that selectively regulate the formation of endocytic transport vesicles and the fusion of such vesicles with appropriate acceptor membranes. Among these, the GTPase dynamin plays an important role in clathrin-mediated endocytosis, and it was recently found that dynamin can participate in a phagocytic process. METHODOLOGY/PRINCIPAL FINDINGS: We used a compound called dynasore that has the ability to block the GTPase activity of dynamin. Dynasore acts as a potent inhibitor of endocytic pathways by blocking coated vesicle formation within seconds of its addition. Here, we investigated whether dynamin is involved in the entry process of T. cruzi in phagocytic and non-phagocytic cells by using dynasore. In this aim, peritoneal macrophages and LLC-MK2 cells were treated with increasing concentrations of dynasore before interaction with trypomastigotes, amastigotes or epimastigotes. We observed that, in both cell lines, the parasite internalization was drastically diminished (by greater than 90% in LLC-MK2 cells and 70% in peritoneal macrophages) when we used 100 microM dynasore. The T. cruzi adhesion index, however, was unaffected in either cell line. Analyzing these interactions by scanning electron microscopy and comparing peritoneal macrophages to LLC-MK2 cells revealed differences in the stage at which cell entry was blocked. In LLC-MK2 cells, this blockade is observed earlier than it is in peritoneal macrophages. In LLC-MK2 cells, the parasites were only associated with cellular microvilli, whereas in peritoneal macrophages, trypomastigotes were not completely engulfed by a host cell plasma membrane. CONCLUSIONS/SIGNIFICANCE: Taken together our results demonstrate that dynamin is an essential molecule necessary for cell invasion and specifically parasitophorous vacuole formation by host cells during interaction with Trypanosoma cruzi

Viscosity solutions of path-dependent pdes with randomized time

none2We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. With the new definition, we prove the two important results, until now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/supersolutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.noneRen Z.; Rosestolato M.Ren, Z.; Rosestolato, M

Partial regularity of viscosity solutions for a class of Kolmogorov equations arising from mathematical finance

We study value functions which are viscosity solutions of certain Kolmogorov equations. Using PDE techniques we prove that they are C1+α regular on special finite dimensional subspaces. The problem has origins in hedging derivatives of risky assets in mathematical finance

C0-sequentially equicontinuous semigroups

We present and apply a theory of one-parameter C0-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of C0-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille-Yosida theorem. Then we particularize the theory in some functional spaces and identify two locally convex topologies that allow us to gather-under a unified framework-various notions of C0-semigroups introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations (SDEs)

Robustness for path-dependent volatility models

In this paper, we consider a generalisation of the Hobson-Rogers model proposed by Foschi and Pascucci (Decis Eocon Finance 31(1):1-20, 2008) for financial markets where the evolution of the prices of the assets depends not only on the current value but also on past values. Using differentiability of stochastic processes with respect to the initial condition, we analyse the robustness of such a model with respect to the so-called offset function, which generally depends on the entire past of the risky asset and is thus not fully observable. In doing this, we extend previous results of Blaka Hallulli and Vargiolu (2007) to contingent claims, which are globally Lipschitz with respect to the price of the underlying asset, and we improve the dependence of the necessary observation window on the maturity of the contingent claim, which now becomes of linear type, while in Blaka Hallulli and Vargiolu (2007), it was quadratic. Finally, in this framework, we give a characterisation of the stationarity assumption used in Blaka Hallulli and Vargiolu (2007), and prove that this model is stationary if and only if it is reduced to the original Hobson-Rogers model. We conclude by calibrating the model to the prices of two indexes using two different volatility shapes. © 2012 Springer-Verlag

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution a` la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space

Optimal control of Path-dependent Mckean–Vlasov SDEs in infinite-dimension

We study the optimal control of path-dependent McKean–Vlasov equations valued in Hilbert spaces motivated by non-Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions (Lions (Audio Conference, 2006–2012)), and prove a related functional Itô formula in the spirit of Dupire ((2009), Functional Itô Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS) and Wu and Zhang (Ann. Appl. Probab. 30 (2020) 936–986). The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control

The use of confocal laser scanning microscopy to analyze the process of parasitic protozoon-host cell interaction

In this communication we review the results obtained with the confocal laser scanning microscope to characterize the interaction of epimastigote and trypomastigote forms of Trypanosoma cruzi and tachyzoites of Toxoplasma gondii with host cells. Early events of the interaction process were studied by the simultaneous localization of sites of protein phosphorylation, revealed by immunocytochemistry, and sites of actin assembly, revealed by the use of labeled phaloidin. The results obtained show that proteins localized in the interaction sites are phosphorylated. The process of formation of the parasitophorous vacuole was monitored by labeling the host cell surface with fluorescent probes for lipids (PKH26), proteins (DTAF) and sialic acid (FITC-thiosemicarbazide) before interaction with the parasites. Evidence was obtained indicating transfer of components of the host cell surface to the parasite surface in the beginning of the interaction process. We also analyzed the distribution of cytoskeletal structures (microtubules and microfilaments visualized with specific antibodies), mitochondria (visualized with rhodamine 123), the Golgi complex (visualized with C6-NBD-ceramide) and the endoplasmic reticulum (visualized with anti-reticulin antibodies and DIOC6) during the evolution of intracellular parasitism. The results obtained show that some, but not all, structures change their position during evolution of the intracellular parasitism