Multidimensional Martingale Research Articles

Fundamental Theorem of Asset Pricing under fixed and proportional costs in multi-asset setting and finite probability space

The Fundamental Theorem of Asset Pricing is extended to a market model over a finite probability space with many assets that can be exchanged into one another under combined fixed and proportional transaction costs. The absence of arbitrage in this setting is shown to be equivalent to the existence of a family of absolutely continuous single-step probability measures and a multi-dimensional martingale with respect to such a family.

Optimal control of martingales in a radially symmetric environment

We study a stochastic control problem for continuous multidimensional martingales with fixed quadratic variation. In a radially symmetric environment, we are able to find an explicit solution to the control problem and find an optimal strategy. We show that it is optimal to switch between two strategies, depending only on the radius of the controlled process. The optimal strategies correspond to purely radial and purely tangential motion. It is notable that the value function exhibits smooth fit even when switching to tangential motion, where the radius of the optimal process is deterministic. Under sufficient regularity on the cost function, we prove optimality via viscosity solutions of a Hamilton–Jacobi–Bellman equation. We extend the results to cost functions that may become infinite at the origin. Extra care is required to solve the control problem in this case, since it is not clear how to define the optimal strategy with deterministic radius at the origin. Our results generalise some problems recently considered in Stochastic Portfolio Theory and Martingale Optimal Transport.

Introducing smooth amnesia to the memory of the Elephant Random Walk

This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach

This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization.

The continuous [formula omitted] inventory model with dual sourcing and emergency orders

This paper examines an inventory system of a retailer who uses a continuous-review base-stock policy to manage inventory. We consider two types of demands: Markov-modulated fluid demands with linear rates and batch-size demands that arrive according to a Markov arrival process (MAP). In addition, we consider two types of supply: (i) a regular supply that follows an (S,s) policy with stochastic lead times under a lost-sales assumption and (ii) an additional emergency supply that brings the inventory up to level 0<Se≤S when stockout. We assume that such an emergency supply has a negligible lead time but incurs an extra cost. Hence, we analyze four models that are each adequate for a different demand-supply pattern. Our cost structure includes variable and fixed costs for each type of order, together with holding and lost-sales costs. By integrating a multidimensional martingale technique and hitting-time transforms for fluid models and MAPs, we derive explicit formulas for the cost components that aim to minimize the expected discounted total cost. A numerical study provides insights into the best parameters and how they depend on different factors. For example, we show that the MAP-type arrivals yield higher control values; increasing the lead time (average and variance) and the lost cost increases these values, and eventually leads to the (S,s=S−1) policy. We further show that increasing the discount factor decreases s and the fluid models are less sensitive to the timing. By comparing the models, we provide managers with a framework for assessing when dual sourcing is more beneficial than single sourcing.

Asymptotic analysis of random walks on ice and graphite

The purpose of this paper is to investigate the asymptotic behavior of random walks on three-dimensional crystal structures. We focus our attention on the 1h structure of the ice and the 2h structure of graphite. We establish the strong law of large numbers and the asymptotic normality for both random walks on ice and graphite. All our analysis relies on asymptotic results for multi-dimensional martingales.

Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

&lt;p style='text-indent:20px;'&gt;The existence, uniqueness, and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are developed. The goal is to develop a general multi-asset framework encompassing a wide spectrum of non-linear financial models with jumps, including as particular cases, the setups studied by Peng and Xu [&lt;xref ref-type="bibr" rid="b27"&gt;27&lt;/xref&gt;, &lt;xref ref-type="bibr" rid="b28"&gt;28&lt;/xref&gt;] and Dumitrescu et al. [&lt;xref ref-type="bibr" rid="b7"&gt;7&lt;/xref&gt;] who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. &lt;/p&gt;

On the center of mass of the elephant random walk

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the center of mass of the elephant random walk. The asymptotic normality, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.

Irreducible convex paving for decomposition of multidimensional martingale transport plans

Martingale transport plans on the line are known from Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in $\mathbb{R}^{d}$, $d\ge1$. Our decomposition is a partition of $\mathbb{R}^{d}$ consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

On the Multi-dimensional Elephant Random Walk

The purpose of this paper is to investigate the asymptotic behavior of the multi-dimensional elephant random walk (MERW). It is a non-Markovian random walk which has a complete memory of its entire history. A wide range of literature is available on the one-dimensional ERW. Surprisingly, no references are available on the MERW. The goal of this paper is to fill the gap by extending the results on the one-dimensional ERW to the MERW. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the MERW. The asymptotic normality of the MERW, properly normalized, is also provided. In the superdiffusive regime, we prove the almost sure convergence as well as the mean square convergence of the MERW. All our analysis relies on asymptotic results for multi-dimensional martingales.

A threshold policy in a Markov-modulated production system with server vacation: the case of continuous and batch supplies

AbstractWe consider a Markov-modulated fluid flow production model under theD-policy, that is, as soon as the storage reaches level 0, the machine becomes idle until the total storage exceeds a predetermined thresholdD. Thus, the production process alternates between a busy and an idle machine. During the busy period, the storage decreases linearly due to continuous production and increases due to supply; during the idle period, no production is rendered by the machine and the storage level increases by only supply arrivals. We consider two types of model with different supply process patterns: continuous inflows with linear rates (fluid type), and batch inflows, where the supplies arrive according to a Markov additive process (MAP) and their sizes are independent and have phase-type distributions depending on the type of arrival (MAP type). Four types of cost are considered: a setup cost, a production cost, a penalty cost for an idle machine, and a storage cost. Using tools from multidimensional martingale and hitting time theory, we derive explicit formulae for these cost functionals in the discounted case. Numerical examples, a sensitivity analysis, and insights are provided.

Critical level policy for a production-inventory model with lost sales

We consider a storage process under the generalised order-up-to-level policy, based on a continuous-time Markov chain (CTMC). Specifically, the process starts at level S; whenever it drops to s, an order is sent, which is carried out after an exponential lead time. If during the lead time level S is reached, the order is cancelled, incurring some fee. This paper is written as an extension of Barron [2016. “An Fluid Inventory Model with Exponential Lead Times and Order Cancellations.” Stochastic Models 32 (2): 301–332]. While the latter paper considered a fluid inventory model with backlogging and focused on discounted analysis only, the case of lost sales was not solved. The present paper generalises the analysis to incorporate unsatisfied demand for the expected discounted costs and for the average costs per time unit. We consider four costs. There is a fixed nonzero ordering cost or a fee for each order cancellation, a purchase cost for each ordered item, a storage cost for the stock, and a penalty cost due to the unmet demand. Applying renewal theory, multi-dimensional martingales, and stopping time theory, we obtain explicit expressions of the cost components. Numerical study provides several guidelines on the optimal controls.

Moments for multidimensional Mandelbrot cascades

Abstract We consider the distributional equation Z =D ∑k=1NAkZ(k), where N is a random variable taking value in N0 = {0, 1, . . .}, A1, A2, . . . are p x p nonnegative random matrices, and Z, Z(1), Z(2), . . ., are independent and identically distributed random vectors in R+p with R+ = [0, ∞), which are independent of (N, A1, A2, . . .). Let {Yn} be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton–Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α &gt; 1, we show a sufficient condition for E|Y|α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee-t∙Z as |t| → ∞ and those of the tail probability P(y ∙ Z ≤ x) as x → 0 for given y = (y1, . . ., yp) ∈ R+p, and the existence of the harmonic moments of y ∙ Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the Lα convergence and the αth-moment of the Mandelbrot martingale {Yn} are also established.

An (s, k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.

BSDEs Driven by Multidimensional Martingales and Their Applications to Markets with Funding Costs

We establish some well-posedness and comparison results for BSDEs driven by one- and multidimensional martingales. On the one hand, our approach is largely motivated by results and methods developed by Carbone, Ferrario, and Santacroce in [Theory Probab. Appl., 52 (2008), pp. 304--314] and by El Karoui and Huang in [Backward Stochastic Differential Equations, Pitman Res. Notes in Math. Ser. 364, Longman, Harlow, 1997, pp. 27--36]. On the other hand, our results are also motivated by the recent developments in the theory of pricing of financial derivatives under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multidimensional martingale is established and applied to the pricing BSDEs studied by Bielecki and Rutkowski in [SIAM J. Financial Math., 6 (2015), pp. 594--655] and by Nie and Rutkowski in [Int. J. Theor. Appl. Finance, 18 (2015), 1550048], [Fair Bilateral Prices under Funding Costs and Exogenous Collateralization, Working paper, University of Sydney, Sydney, 2014], and [Finance Stoch., 20 (2016), pp. 855--900]. We establish the existence and uniqueness results for unilateral prices and demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have nonnegative initial endowments.

A make-to-stock production/inventory model with MAP arrivals and phase-type demands

We consider a make-to-stock production/inventory model in a random environment with finite storage capacity and restricted backlogging possibility. Our aim is to demonstrate that all cost quantities of interest can be derived in closed form under quite general assumptions on the demand arrival process and on the switches in the production rates. Specifically, the demands arrive according to a Markov additive process governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. The production process switches between predetermined rates which depend on the state of the environment and on the presence or absence of backlogs. Four types of costs are considered: the holding cost for the stock, the cost of lost production due to the finite storage capacity, the shortage cost for the backlogged demand and the cost due to unsatisfied demand. We obtain explicit formulas for these cost functionals in the discounted case and under the long-run average criterion. The derivations are based on optional sampling of a multi-dimensional martingale and on fluid flow techniques.

Randomization in the construction of multidimensional martingales

A general approach to the construction of multidimensional martingales is developed. The obtained results allow considerably extend the range of applicability of the martingale method to the theory of random fields, especially to the theory of limiting distributions of Gibbs random fields.

Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices

We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in Rd. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization.

The expected time to ruin in a risk process with constant barrier via martingales

Two risk models with a constant dividend barrier are considered. In the two models claims arrive according to a Poisson process. In the first model the claim size has a phase type distribution. In the second model the claim size is exponentially distributed, but the arrival rate, the mean claim size, and the premium rate are governed by a random environment, which changes according to a Markov process. Kella and Whitt [Kella, O., Whitt, W., 1992. Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probability 29, 396–403] martingale is applied in the first model. Asmussen and Kella [Asmussen, S., Kella, O., 2000. A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probability 32, 376–393] multi-dimensional martingale is applied in the second model. The expected time to ruin and the amount of dividends paid until ruin occurs are obtained for both models.

A Brownian motion with two reflecting barriers and Markov-modulated speed

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.