Multidimensional Brownian Motion Research Articles

Pricing Contingent Claims in a Two-Interest-Rate Multi-Dimensional Jump-Diffusion Model via Market Completion

This paper investigates a financial market where asset prices follow a multi-dimensional Brownian motion process and a multi-dimensional Poisson process characterized by diverse credit and deposit rates where the credit rate is higher than the deposit rate. The focus extends to evaluating European options by establishing upper and lower hedging prices through a transition to a suitable auxiliary market. Introducing a lemma elucidates the same solution to the pricing problem in both markets under specific conditions. Additionally, we address the minimization of shortfall risk and determine no-arbitrage price bounds within the framework of incomplete markets. This study provides a comprehensive understanding of the challenges posed by the multi-dimensional jump-diffusion model and varying interest rates in financial markets.

On the singular values of complex matrix Brownian motion with a matrix drift

Let MatC(K,N) be the space of K×N complex matrices. Let Bt be Brownian motion on MatC(K,N) starting from the zero matrix and M∈MatC(K,N). We prove that, with K≥N, the N eigenvalues of Bt+tM∗Bt+tM form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman (Ann. Probab. 9 (1981) 573–582) for multidimensional Brownian motion with drift which corresponds to N=1. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe (Z. Wahrsch. Verw. Gebiete 31 (1975) 115–124) and studied by Pitman and Yor (In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980) (1981) 285–370 Springer), conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.

On a multidimensional Brownian motion with a membrane located on a given hyperplane

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [−1,1]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1/n, 2/n, …, [nt]/n (for fixed t>0) is proved under the assumption that n→∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane.

A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.

Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

A class of solvable multidimensional stopping problems in the presence of Knightian uncertainty

AbstractWe investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity-averse decision-maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst-case measure in terms of a family of explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may lead to stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.

A free boundary characterisation of the Root barrier for Markov processes

We study the existence, optimality, and construction of non-randomised stopping times that solve the Skorokhod embedding problem (SEP) for Markov processes which satisfy a duality assumption. These stopping times are hitting times of space-time subsets, so-called Root barriers. Our main result is, besides the existence and optimality, a potential-theoretic characterisation of this Root barrier as a free boundary. If the generator of the Markov process is sufficiently regular, this reduces to an obstacle PDE that has the Root barrier as free boundary and thereby generalises previous results from one-dimensional diffusions to Markov processes. However, our characterisation always applies and allows, at least in principle, to compute the Root barrier by dynamic programming, even when the well-posedness of the informally associated obstacle PDE is not clear. Finally, we demonstrate the flexibility of our method by replacing time by an additive functional in Root’s construction. Already for multi-dimensional Brownian motion this leads to new class of constructive solutions of (SEP).

On the approximation and simulation of iterated stochastic integrals and the corresponding Lévy areas in terms of a multidimensional Brownian motion

A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding Lévy areas driven by a multidimensional Brownian motion is proposed. The algorithm is based on a truncated Fourier series approach. However, the approximation of the remainder terms differs from the approach considered by Wiktorsson (2001). As the main advantage, the presented algorithm makes use of a diagonal covariance matrix for the approximation of one part of the remainder term and has a higher accuracy due to an exact approximation of the other part of the remainder. This results in a significant reduction of the computational cost compared to, e.g., the algorithm introduced by Wiktorsson. Convergence in -norm with for the approximations calculated with the new algorithm as well as for approximations calculated by the basic truncated Fourier series algorithm is proved and the efficiency of the new algorithm is analyzed.

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<frac{1}{2}. Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

Robust Classical-Impulse Stochastic Control Problems in an In finite Horizon

This paper establishes a general analytical framework for classical and impulse stochastic control problems in the presence of model uncertainty. We consider a set of dominated models, which are induced by the measures equivalent to that of a reference model. The state process under the reference model is a multidimensional Markov process with multidimensional Brownian motion, controlled by continuous and impulse control variates. We propose quasi-variational inequalities (QVI) associated with the value function of the control problem and prove a veri cation theorem for the solution to the QVI. With the relative entropy constraints and piecewise linear intervention penalty, we show that the QVI can be degenerated to the non-robust case and it can be solved via the solution to a free boundary problem. To illustrate the tractability of the proposed framework, we apply it to a linearquadratic setting, which covers a broad class of problems including robust mean-reverting inventory controls.

On the limiting law of the length of the longest common and increasing subsequences in random words with arbitrary distribution

Let (Xk)k≥1 and (Yk)k≥1 be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet Am:={1,…,m}, m≥2, having respective probability mass function p1X,…,pmX and p1Y,…,pmY. Let LCIn be the length of the longest common and weakly increasing subsequences in X1,...,Xn and Y1,...,Yn. Once properly centered and normalized, LCIn is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.

On the cumulative Parisian ruin of multi-dimensional Brownian motion risk models

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalisations of some of the results derived in Dbicki et al. [(2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and Their Applications 128, 4171–4206]. As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

Stability and asymptoticity of stochastic epidemic model with interim immune class and independent perturbations

In this paper, we formulate a stochastic epidemic model involved with interim-immune period. To make a better simulation of the transmission of the disease, multidimensional Brownian motion is taken to be the stochastic perturbation. By introducing a parameter R0, we choose a more general Lyapunov function to present sufficient conditions on stability and global asymptoticity of disease-free solutions.

Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion

Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

Tail asymptotics of the Brownian signature

The signature of a path \gamma is a sequence whose n-th term is the order-n iterated integrals of \gamma. It arises from solving multidimensional linear differential equations driven by \gamma. We are interested in relating the path properties of \gamma with its signature. If \gamma is C1, then an elegant formula of Hambly and Lyons relates the length of \gamma to the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion,with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion.

On explicit local solutions of Itô diffusions

Strong solutions of p -dimensional stochastic differential equations d X t = b ( X t , t ) d t + σ ( X t , t ) d W t , X s = x that can be represented locally in explicit simulation form X t = ϕ x , s ( ∫ s t V s , u d W u , t ) are considered. Here; W is a multidimensional Brownian motion; u → V s , u , ϕ x , s are continuous functions; and b , σ , ϕ x , s are locally continuously differentiable. The following three-way equivalence is established: 1) There exists such a representation from all starting points ( x , s ) , 2) V s , u , ϕ x , s satisfies a set differential equations, and 3) b , σ satisfy commutation relations. (For generality, the function V s , t is allowed to depend upon ϕ x , s via V s , t = U s , t ϕ x , s for some operators U s , t .) Moreover, construction theorems, based on a diffeomorphism between the solutions X and the strong solutions to a simpler Itô integral equation, with a possible deterministic component, are given. Finally, motivating examples are provided and its importance in simulation methods, including sequential Monte Carlo, financial risk assessment and path-dependent option pricing, is explained.

Portfolio Selection in Mean-Minimum Return Level-Expected Bounded First Passage Time Framework

This paper explores the selection of optimal portfolio by replacing the standard Mean-Variance model by Mean-Minimum Return Level (MRL) framework and adding one important dimension—expectation of bounded First Passage Time (FPT) towards the MRL. To measure how much a given portfolio is exposed to risk, the new model can capture both, the amount of the largest possible loss at a certain confidence level and time to such an event occurring. The novelty of this paper is the introduction of bounded first passage time towards MRL and taking its expectation into consideration as an additional factor in portfolio selection decision making. Assuming that the asset price dynamics follow multi-dimensional Geometric Brownian Motion with drift, we obtain a portfolio wealth process for multiple assets and we evaluate the lowest possible value to which it can drop by a high confidence level. Then we extend our examination of the optimal portfolio selection by ultimately obtaining the efficient surface of risky portfolios. As a result, the paper shows that the third dimension can make a significant difference while choosing the asset weights compared to classical models ignoring the portfolio return paths as long as they achieve a desired combination of risk and return.

An extension of the Clark–Haussmann formula and applications

ABSTRACTThis work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.

A weak version of path-dependent functional Itô calculus

We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.

Smoothness of self-intersection local time of multidimensional fractional Brownian motion

In this paper, we study the existence, Hölder continuity and fractional smoothness of the self-intersection local time with respect to fractional Brownian motion:where BH is a d-dimensional fractional Brownian motion with Hurst index . We also consider the case of intersection local time for two independent multidimensional fractional Brownian motions.