Brownian Filtration Research Articles

On the value of a time-inconsistent mean-field zero-sum Dynkin game

We study a mean-field zero-sum Dynkin game (MF-ZSDG) with time-inconsistent performance functionals adapted to the Brownian filtration. Despite the time-inconsistency of the MF-ZSDG, we show that it admits a value and that the pair of first times the value process hits the upper and lower obstacles, respectively, is a saddle point for the game. We solve the problem by approximating the associated lower and upper value processes with a sequence of value processes of interacting time-consistent zero-sum Dynkin games for which the saddle point of each of the value processes is the pair of first times each of those value processes hits the associated upper and lower obstacles, respectively. Under mild assumptions, we show that this sequence of saddle points converges in probability to the pair of first hitting times of the value process of the upper and lower obstacles, respectively, and that the limit is a saddle point for the time-inconsistent MF-ZSDG.

Sandwiched filtrations

Abstract In this paper, we prove that a filtration sandwiched (in the sense of immersion) between two Brownian filtrations, is a weak Brownian filtration.

Time-inconsistent mean-field optimal stopping: A limit approach

We provide a characterization of an optimal stopping time for a class of finite horizon time-inconsistent optimal stopping problems (OSPs) of mean-field type, adapted to the Brownian filtration, including those related to mean-field diffusion processes and recursive utility functions. Despite the time-inconsistency of the OSP, we show that it is optimal to stop when the value-process hits the reward process for the first time, as is the case for the standard time-consistent OSP. We solve the problem by approximating the corresponding value-process with a sequence of Snell envelopes of processes, for which a sequence of optimal stopping times is constituted of the hitting times of each of the reward processes by the associated value-process. Then, under mild assumptions, we show that this sequence of hitting times converges in probability to the hitting time for the mean-field OSP and that the limit is optimal.

Probabilistic representation of viscosity solutions to quasi-variational inequalities with non-local drivers

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon $ [0,T\wedge\tau] $, where $ \tau $ is a given random jump time for the underlying state process and $ T $ a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon $ [0,T\wedge\tau] $. By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $ [0,T] $. Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.

On the near-viability property of controlled mean-field flows

We aim at studying the property of controlled stochastic flows with mean-field dynamics to comply with some (closed) state restrictions. This property, known as (near)-viability, is tackled via (quasi-)tangency methods. Law restrictions and mixed state-law restrictions are considered as the interplay between the two classes. As an auxiliary result used in this process, Theorem 1.2, whose importance exceeds the present framework, dissociates, through a class of elementary controls, the contribution of the Brownian filtration and that of the initial $ \sigma $-field. Explicit conditions for the coefficient functions are provided in the invariance context. Moreover, specific applications to comparison in the convex order illustrate the theoretical results.

Power Mixture Forward Performance Processes

We consider the forward investment problem in market models where the stock prices are continuous semimartingales adapted to a Brownian filtration. We construct a broad class of forward performance processes with initial conditions of power mixture type, $u(x) = \int_{\mathbb{I}} \frac{x^{1-\gamma}}{1-\gamma }\nu(\mathrm{d} \gamma)$. We proceed to define and fully characterize two-power mixture forward performance processes with constant risk aversion coefficients in the interval $(0,1)$, and derive properties of two-power mixture forward performance processes when the risk aversion coefficients are continuous stochastic processes. Finally, we discuss the problem of managing an investment pool of two investors, whose respective preferences evolve as power forward performance processes.

McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations

We study a McKean–Vlasov optimal control problem with common noise in order to establish the corresponding limit theory as well as the equivalence between different formulations, including strong, weak, and relaxed formulations. In contrast to the strong formulation, in which the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an (H)-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classic optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean–Vlasov control problem, that is, proving that it consists in the limit of a large population control problem with common noise.

Infinite Horizon Stochastic Impulse Control with Delay and Random Coefficients

We study a class of infinite horizon impulse control problems with execution delay when the dynamics of the system is described by a general stochastic process adapted to the Brownian filtration. The problem is solved by means of probabilistic tools relying on the notion of Snell envelope and infinite horizon reflected backward stochastic differential equations. This allows us to establish the existence of an optimal strategy over all admissible strategies.

Projections of martingales in enlargements of Brownian filtrations under Jacod’s equivalence hypothesis

We consider the initial and progressive enlargements of a Brownian filtration with a random time, that is, a strictly positive random variable. We assume Jacod’s equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the Brownian filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged Brownian filtration, we derive explicit expressions for the components which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the Brownian filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the Brownian filtration.

On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the “Optimized Certainty Equivalent” risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.

Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

The aim of this work is to prove an existence and uniqueness result of Kato–Fujita type for the Navier–Stokes equations, in vorticity form, in 2D and 3D, perturbed by a gradient-type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by Barbu and Rockner (J Differ Equ 263:5395–5411, 2017) that treats the stochastic 3D Navier–Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in Barbu and Rockner (2017), but the transformation is different and adapted to our gradient-type noise. Then, global unique existence results are proved for the transformed equation, while for the original stochastic Navier–Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.

Martingale representation for degenerate diffusions

Let (W,H,μ) be the classical Wiener space on IRd. Assume that X=(Xt) is a diffusion process satisfying the stochastic differential equation dXt=σ(t,X)dBt+b(t,X)dt, where σ:[0,1]×C([0,1],IRn)→IRn⊗IRd, b:[0,1]×C([0,1],IRn)→IRn, B is an IRd-valued Brownian motion. We suppose that the weak uniqueness of this equation holds for any initial condition. We prove that any square integrable martingale M w.r.t. to the filtration (Ft(X),t∈[0,1]) can be represented asMt=E[M0]+∫0tPs(X)αs(X).dBs where α(X) is an IRd-valued process adapted to (Ft(X),t∈[0,1]), satisfying E∫0t(a(Xs)αs(X),αs(X))ds<∞, a=σ⋆σ and Ps(X) denotes a measurable version of the orthogonal projection from IRd to σ(Xs)⋆(IRn). In particular, for any h∈H, we have(0.0)E[ρ(δh)|F1(X)]=exp⁡(∫01(Ps(X)h˙s,dBs)−12∫01|Ps(X)h˙s|2ds), where ρ(δh)=exp⁡(∫01(h˙s,dBs)−12|H|H2). In the case the process X is adapted to the Brownian filtration, this result gives a new development as an infinite series of the L2-functionals of the degenerate diffusions. We also give an adequate notion of “innovation process” associated to a degenerate diffusion which corresponds to the strong solution when the Brownian motion is replaced by an adapted perturbation of identity. This latter result gives the solution of the causal Monge–Ampère equation.

Concentration of dynamic risk measures in a Brownian filtration

Motivated by liquidity risk in mathematical finance, Lacker (2015) introduced concentration inequalities for risk measures, i.e. upper bounds on the liquidity risk profile of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type inequalities in terms of the generator of the BSDE, which in the present case can grow arbitrarily fast.

Optimal position targeting via decoupling fields

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward–backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a monotonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.

Immersion of strongly Brownian filtrations with honest time avoiding stopping times

In this paper, we give a partial answer to the following question: if $\mathbb{F}\hookrightarrow\mathbb{G}\hookrightarrow\mathbb{H}$ (where the symbol ($\hookrightarrow$) indicates the immersion property), $\mathbb{F}$ and $\mathbb{H}$ are two strongly Brownian filtrations, is $\mathbb{G}$ also a strongly Brownian filtration ?\\We prove that $\mathbb{G}$ is weakly Brownian in the case of progressive enlargement of $\mathbb{F}$ with an honest time $\tau$ that avoids all stopping times.

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type [Formula: see text]-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty is studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

A note on the long rate in factor models of the term structure

AbstractIn this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two‐dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three‐dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.

On One Integral Representation of Functionals of Brownian Motion

In this paper we propose a method for finding an integrand in the Clark stochastic integral representation of quadratic integrable Brownian functional $F$ with Malliavin differentiable conditional mathematical expectation ${E}\,[F\,|\,\Im_t^B],$ $t<T$, where $(\Im_t^B)$ is a natural Brownian filtration. The proposed method allows us to determine the explicit form of the integrand in the case when $F$ has no Malliavin derivative. We give several illustrations as examples.

Mean-Variance Hedging Under Multiple Defaults Risk

We solve a mean-variance hedging problem in an incomplete market where multiple defaults can occur. For this purpose, we use a default-density modeling approach. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of the default times is modelled using a conditional density hypothesis. We prove the quadratic form of each value process between consecutive default times and recursively solve systems of coupled quadratic backward stochastic differential equations (BSDEs). We demonstrate the existence of these solutions using BSDE techniques. Then, using a verification theorem, we prove that the solutions of each subcontrol problem are related to the solution of our global mean-variance hedging problem. As a byproduct, we obtain an explicit formula for the optimal trading strategy. Finally, we illustrate our results for certain specific cases and for a multiple defaults case in particular.