Independent Brownian Motions Research Articles

GENERALIZED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY TWO MUTUALLY INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

This paper deals with a class of generalized backward stochastic differential equations driven by two mutually independent fractional Brownian motions (FGBSDEs in short). The existence and uniqueness of solutions for FGBSDE as well as a comparison theorem are obtained.

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

Stochastic maximum principle for partially observed optimal control problem of McKean–Vlasov FBSDEs with Teugels martingales

Abstract In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov’s theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering.

The set of intersections of several independent Brownian motions on Carnot group

In this paper the existence of intersections for functions of several Brownian motions on the Carnot group is studied. A condition is presented for the existence of such intersections with Probability 1, which is in the form of the asymptotics of a measure on a specific family of small balls. The measure is arbitrary but can be chosen as a surface measure on the manifold related to intersections, and the balls are constructed using the distances related to the processes. Additionally, if the same condition holds in a weaker form, it is shown that there is a Hausdorff measure, such that the value of this Hausdorff measure on the set of intersection points is finite with Probability 1.

Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations

Abstract Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

Optimal Investment with a Noisy Signal of Future Stock Prices

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock’s price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, but it even allows for a closed-form solution. We describe this solution and the resulting problem value for this stochastic control problem with partial observation by solving its convex-analytic dual problem.

Different topological solution structures in a two-dimensional controlled ruin problem depending on the optimization criterion

We consider two insurance companies with wealth processes given by independent Brownian motions with controllable non-negative drift. The drift rates sum up to 1. The companies aim at finding a strategy for the drift rates to maximize the probability that at least one of them survives forever. We prove that the strategy, where the company with higher wealth obtains the maximal drift rate, is optimal. Our result differs considerably from the numerical result of McKean and Shepp in [H.P. McKean and L.A. Shepp, The advantage of capitalism vs. socialism depends on the criterion, J. Math. Sci. 139(3) (2006), pp. 6589–6594]. Furthermore, we numerically obtain candidates for the optimal strategy if the common aim of the two companies is to maximize a convex combination of the probability that both firms survive and the probability that exactly one firm survives forever. Our numerical results indicate that in general it is optimal to assign the maximal drift rate to one company but whether it is the company with higher or less wealth depends on the size of both wealth processes.

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

Erratum to “Itô’s formula for noncommutative $C^2$ functions of free Itô processes”

The results in [Doc. Math. 27 (2022), 1447–1507] are stated for free Itô processes driven by n -tuples of freely independent semicircular Brownian motions. They should be stated instead for free Itô processes driven by n-dimensional semicircular Brownian motions. We explain and correct this error.

On the joint survival probability of two collaborating firms

AbstractWe consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.

On the Sum of Gaussian Martingale and an Independent Fractional Brownian Motion

On the Sum of Gaussian Martingale and an Independent Fractional Brownian Motion

Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion

The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent.

Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble

We consider the symmetric tridiagonal matrix-valued process associated with Gaussian beta ensemble (GβE) by putting independent Brownian motions and Bessel processes on the diagonal entries and upper (lower)-diagonal ones, respectively. Then, we derive the stochastic differential equations that the eigenvalue processes satisfy, and we show that eigenvalues of their (indexed) principal minor sub-matrices appear in the stochastic differential equations. By the Cauchy’s interlacing argument for eigenvalues, we can characterize the sufficient condition that the eigenvalue processes never collide with each other almost surely, by the dimensions of the Bessel processes.

Current fluctuations in stochastically resetting particle systems.

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for Q<Q_{crit}(t), while it remains time dependent for Q>Q_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

Tail asymptotics for the delay in a Brownian fork-join queue

We study the tail behavior of maxi≤Nsups>0Wi(s)+WA(s)−βs as N→∞, with (Wi,i≤N) i.i.d. Brownian motions and WA an independent Brownian motion. This random variable can be seen as the maximum of N mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around σ22βlogN. Here, we analyze the rare event that this random variable reaches the value (σ22β+a)logN, with a>0. It turns out that its probability behaves roughly as a power law with N, where the exponent depends on a. However, there are three regimes, around a critical point a⋆; namely, 0<a<a⋆, a=a⋆, and a>a⋆. The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the N suprema, with a nontrivial transition at a=a⋆.

A sharp Lp-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE)(0.1)du=(aij(ω,t)uxixj+f)dt+(σik(ω,t)uxi+gk)dwtk,u(0,x)=u0, where {wtk:k=1,2,⋯} is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω,t) and can be unbounded and fully degenerate, that is, coefficients aij, σik merely satisfy(0.2)(αij(ω,t))d×d:=(aij(ω,t)−12∑k=1∞σik(ω,t)σjk(ω,t))≥0. In this article, we prove that there exists a unique solution u to (0.1), and(0.3)‖uxx‖Hpγ(τ,δ)≤N(d,p)(‖u0‖Bpγ+2(1−1/p)+‖f‖Hpγ(τ,δ1−p)+‖gx‖Hpγ(τ,|σ|pδ1−p,l2)p+‖gx‖Hpγ(τ,δ1−p/2,l2)), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of αij(ω,t), Hpγ(τ,δ) is a weighted stochastic Sobolev space, and Bpγ+2(1−1/p) is a stochastic Besov space.

Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

A dual risk model with additive and proportional gains: ruin probability and dividends

AbstractWe consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a&gt;0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

A stationary model of non-intersecting directed polymers

We consider the partition function of non-intersecting continuous directed polymers of length t in dimension , in a white noise environment, starting from positions and terminating at positions . When , it is well known that for fixed x, the field solves the Kardar–Parisi–Zhang equation and admits the Brownian motion as a stationary measure. In particular, as t goes to infinity, converges to the exponential of a Brownian motion B(y). In this article, we show an analogue of this result for any . We show that converges as t goes to infinity to an explicit functional of independent Brownian motions. This functional admits a simple description as the partition sum for non-intersecting semi-discrete polymers on lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formula in the case of two polymers. To obtain these results, we show that the stationary measure of the O’Connell–Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric Robinson–Schensted–Knuth correspondence found in Corwin-O’Connell-Seppäläinen-Zygouras (2014 Duke Math. J. 163 513–63).

Approximation and error analysis of forward–backward SDEs driven by general Lévy processes using shot noise series representations

We consider the simulation of a system of decoupled forward–backward stochastic differential equations (FBSDEs) driven by a pure jump Lévy process L and an independent Brownian motion B. We allow the Lévy process L to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise series representation method by [26] to approximate the driving Lévy process L. We compute the Lp error, p ≥ 2, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the Lp error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme.