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This is a review paper, concerning some extensions of the celebrated Merton's mutual fund theorem in infinite-dimensional financial models, in particular, the so-called Large Financial Markets (where a sequence of assets is taken into account) and Bond Markets Models (where there is a continuum of assets). In order to obtain these results, an infinite-dimensional stochastic integration theory is essential: the paper illustrates briefly a new theory introduced to this extent by M. De Donno and the author. [ABSTRACT FROM AUTHOR]

The current industry standard is to optimize loan portfolios with respect to variance. In this paper we show that optimization with respect to expected shortfall and expected regret is fairly easy to implement. [ABSTRACT FROM AUTHOR]

We consider, in a Hilbert space U, a class of Gaussian processes defined by a linear filter with a cylindrical Wiener process as input process. This noise is used as an additive perturbation to a family of fractional order (in time) partial differential equations. We give conditions such that the stochastic convolution process is well defined, both in finite time horizon and in an infinite interval. An important example of noise that is contained in the paper is the fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper we analyse the robustness of the Hobson-Rogers model with respect to the offset function, which depends on the whole past of the risky asset and is thus not fully observable. We prove that, if the offset function is the realisation of a stationary process, then the error in pricing a derivative asset decreases exponentially with respect to the observation window. We present sufficient conditions on the volatility in order to characterise the invariant density and three examples. [ABSTRACT FROM AUTHOR]

Suppose that the trajectory of a particle x(t; x, k) is a solution of the Newton equation $$ \ddot x\left( {t;x{\text{,}}k} \right) = \delta ^{1/2} F\left( {x\left( {t;x{\text{,}}k} \right),\dot x\left( {t;x{\text{,}}k} \right)} \right),{\mathbf{ }}x\left( {\text{0}} \right) = x,\dot x\left( 0 \right) = k $$ , where F(x, k) is a spatially homogeneous random force field defined over a certain probability space (Ω,Σ, ℙ). It has been proved by Kesten and Papanicolaou in [2] that if d ≥ 3 and F(x, k) is sufficiently regular, nondegenerate and mixing in the spatial variable, then the process $$ \left( {\delta ^{1/2} x\left( {\delta ^{ - 1} t;x{\text{,}}k} \right),\ddot x\left( {\delta ^{ - 1} t;x,k} \right)} \right),t \geqslant 0 $$ , converges weakly to a hypoelliptic diffusion. In this paper we prove power-like bounds on the convergence rate for one-dimensional marginals of the process. [ABSTRACT FROM AUTHOR]

We consider a stochastic equation in ℝn with time-dependent coefficients assuming that it has a unique solution and denote by Ps,t, s < t the corresponding transition semigroup. Then we consider a family of measures (νt)t∈ℝ such that $$ \smallint _{\mathbb{R}^d } P_{s,t} \varphi \left( x \right)\nu _s \left( {dx} \right) = \smallint _{\mathbb{R}^d } \varphi \left( x \right)\nu _t \left( {dx} \right),s \leqslant t $$ , for all continuous and bounded functions ϕ. The family (νt)t∈ℝ is called an evolution system of measures indexed by ℝ. It plays the role of a probability invariant measure for autonomous systems. In this paper we generalize the Krylov-Bogoliubov criterion to prove the existence of an evolution system of measures. Moreover, we study some properties of the corresponding Kolmogorov operator proving in particular that it is dissipative with respect to the measure ν(dt, dx) = νt(dx)dt. [ABSTRACT FROM AUTHOR]

The aim of this paper is to approximate the solution of a stochastic differential equation driven by fractional Brownian motion (with Hurst index greater than 1/2 ) using a series expansion for the noise. We prove that the solution of the approximating equations converge in probability to the solution of the given equation. We illustrate the approximation through an example from mathematical finance. [ABSTRACT FROM AUTHOR]

We present nonlinear (hypercontractive)Markov semigroups, which are constructed as solutions of infinite-dimensional (semilinear) Cauchy problems, and provide smoothness and ergodicity results. We also discuss a nonlinear path space functional as well as certain nonlinear transition phenomena. [ABSTRACT FROM AUTHOR]

We introduce a boundedness condition on the Malliavin derivative of a random variable to study sub-Gaussian and other non-Gaussian properties of functionals of random fields, with particular attention to the estimation of suprema. We relate the boundedness of the nth Malliavin derivative to a new class of "sub-nth-Gaussian chaos" processes. An expected supremum estimation, extending the Dudley theorem, is proved for such processes. Sub-nth-Gaussian chaos concentration inequalities for the supremum are obtained, using Malliavin derivative conditions; for n = 1, this generalizes the Borell-Sudakov inequality to a class of sub-Gaussian processes, with a particularly simple and efficient proof; for n = 2 a natural extension to sub-2nd-Gaussian chaos processes is established; for n ≥ 3 a slightly less efficient Malliavin derivative condition is needed. [ABSTRACT FROM AUTHOR]

The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set, and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one-dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes. [ABSTRACT FROM AUTHOR]

Using heat kernel estimates obtained in [18] and the Feynman-Kac formula, we investigate finite-time blow-up and stability of semilinear partial differential equations of the form ∂wt/∂t (x) = Δwt(x) − V (x)wt(x) + vt(x)G(wt(x)), w0(x) ≥ 0, x ∈ ℝd, where v and G are positive measurable functions subject to certain growth conditions, and V is a positive bounded potential. We recover the results of [19] and [14] by probabilistic arguments and in the quadratic decay case V (x) ∼+∞ a(1 + x2)−1, a > 0, we find two critical exponents β*(a), β*(a) with 0 < β*(a) ≤ β*(a) < 2/d, such that any nontrivial positive solution blows up in finite time if 0 < β < β*(a), whereas if β*(a) < β, then nontrivial positive global solutions may exist. [ABSTRACT FROM AUTHOR]

We consider the problem of maximizing expected utility from terminal wealth for a power utility of the risk-averse type assuming that the dynamics of the risky assets are affected by hidden "economic factors" that evolve as a finite-state Markov process. For this partially observable stochastic control problem we determine a corresponding complete observation problem that turns out to be of the risk sensitive type and for which the Dynamic programming approach leads to a nonlinear PDE that, via a suitable transformation, can be made linear. By means of a probabilistic representation we obtain a unique viscosity solution to the latter PDE that induces a unique viscosity solution to the former. This probabilistic representation allows us to obtain, on the one hand, regularity results, on the other hand, a computational approach based on Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

We present a version of a No Free Lunch and Hedging Theorem for security markets under transaction costs for continuous processes. We show that the (RNFL) condition, which requires that the absence of free lunches is preserved under a smaller bid-ask spread, is equivalent to the existence of a uniformly strictly consistent price system. We also characterize the super-replication price of bounded contingent claims as the supremum of expected values under all uniformly consistent price systems. [ABSTRACT FROM AUTHOR]

Biagini (2004) and Biagini-Frittelli (2005) faced the utility maximization problem in incomplete markets when the price process of financial assets is described by general semimartingales that are not necessarily locally bounded. They introduced a class of well-controlled admissible strategies in this (very) risky context and then they solved the maximization problem with an (L∞, ba)-duality technique. In this note we almost stick to their setup and we show that their dual result can be obtained via an Orlicz spaces duality, naturally associated with the utility function considered. This new formulation gives additional insight into the nature of the loss control in the good trading strategies. [ABSTRACT FROM AUTHOR]

Dynamic guaranteed pricing and management of a portfolio under transaction constraints is actually a problem straightforwardly set in terms of guaranteed capture basin of a time-dependent target that is viable in a time-dependent environment under (stochastic or tychastic) uncertain systems. The knowledge of the properties of "capture basin" of targets viable in evolving environments under an uncertain evolutionary system can be used for obtaining the corresponding properties for portfolios. They yield at each time both the evaluation of the capital and the transaction rule. They can be computed by viability algorithms and software providing the valuation of optimal portfolio and the management of their evolution. The capital function, which is actually the value function of a differential game, is the solution to a free boundary problem for nonlinear partial differential equations with discontinuous coefficients. This survey provides several examples. [ABSTRACT FROM AUTHOR]

We can view a Brownian sheet as a sequence of interacting Brownian motions or slices. Here we present a number of results about the slices of the sheet. A common feature of our results is that they exhibit phase transition. In addition, a number of open problems are presented. [ABSTRACT FROM AUTHOR]

The stability of a non-homogeneous measure-valued evolution equation is studied using a variational approach. We apply our results in particular to stochastic optimization algorithms and to the pathwise filter equation. In the latter example the variational approach leads to a new interpretation of the rate of stability. [ABSTRACT FROM AUTHOR]

We relate ergodicity, monotonicity and attractors of a random dynamical system (rds). Our first result states that an rds which is both monotone and ergodic has a weak random attractor which consists of a single point. Then we show that ergodicity alone is insufficient for the existence of a weak random attractor. In particular we present an rds in ℝd, d ≥ 2 namely an isotropic Brownian flow with drift, whose single-point motion is an ergodic diffusion process and which does not have a weak attractor. It seems that this is the first example of this kind in the literature. [ABSTRACT FROM AUTHOR]

We consider a geometric rough path associated with a fractional Brownian motion with Hurst parameter H ∈] 1/4, 1/2 [. We give an approximation result in a modulus type distance, up to the second order, by means of a sequence of rough paths lying above elements of the reproducing kernel Hilbert space. [ABSTRACT FROM AUTHOR]

Classical contact geometry is an odd-dimensional analogue of symplectic geometry. We show that a natural probabilistic deformation of contact geometry, compatible with the very irregular trajectories of diffusion processes, allows one to construct the stochastic version of a number of basic geometrical tools, like, for example, Liouville measure. Moreover, it provides an unified framework to understand the origin of explicit relations (cf. "quadrature") between diffusion processes, useful in many fields. Various applications are given, including one in stochastic finance. [ABSTRACT FROM AUTHOR]

We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme (Kurtz and Protter [39]) in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus. [ABSTRACT FROM AUTHOR]

We give a probabilistic proof of the classical long-time behaviour of the heat kernel on a compact manifold by using Bismut's integration-by-parts formula. [ABSTRACT FROM AUTHOR]

We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. We look for tractable large population approximations. By combining various scalings on population size, birth and death rates, mutation rate, mutation step, or time, a single microscopic model is shown to lead to contrasting macroscopic limits, of different nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. In the limit of rare mutations, we show that a possible approximation is a jump process, justifying rigorously the so-called trait substitution sequence. We thus unify different points of view concerning mutation-selection evolutionary models. [ABSTRACT FROM AUTHOR]

A 2-dimensional dissipative Euler equation, subject to a random perturbation is considered. Using compactness arguments, existence of martingale stationary solutions are proved. [ABSTRACT FROM AUTHOR]

We give a Bakry-Emery type criterion for self-interacting diffusions on a compact manifold. [ABSTRACT FROM AUTHOR]