# Transmission of Trading Orders through Communication Line with Relativistic Delay

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. M/M/G Transmission Theory

_{I}, (I = “Idle”), and the probability, p

_{r}, that the trading system reacts during the time between t and t + dt:

_{I}can be expressed through p(t,x) as follows:

_{2}is expressed through the response function of the trading system B(x):

_{1}(t, x) (e.g., “buy”) and a negative p

_{2}(t, x) (e.g., “sell”) signals propagate simultaneously along with the network. Henceforth, we replace Equation (6) with the following system of equations. Technically, this is the system of equations for the signed measures (Cohn 1997), but in what follows, we shall consider probability densities’ continuous functions of their argument and interpret derivatives in the sense of distributions:

_{2}(t, y

_{0}). This problem is investigated elsewhere.

_{12}= β

_{21}= β

_{2}(s). This means, quite intuitively, that the transmission coefficients for the buy and sell orders are the same. In the system of Equation (9), w*

_{1,2}(τ) are functions that are chosen in such a way that the zeroes of numerator and denominator coincide in the region Re(s) > 0 for Re(τ) > 0. Beneš (1957) showed the existence and uniqueness of this choice for Equation (6), but we suggest the existence—but not necessarily uniqueness―as a hypothesis for the system of Equation (9).

_{1,2}are the contours involved in the inverse Laplace transform (Jeanblanc et al. 2003; Davies 2002). The reaction of the system on an arbitrary trading signal $\overrightarrow{\mathsf{\Phi}}=\left(\begin{array}{c}{\phi}_{1}\left(x\right)\\ {\phi}_{2}\left(x\right)\end{array}\right)$ can then be computed as the convolution of the matrix defined by Equation (12) as follows:

## 4. Representation of the Propagating Signal

^{©}plot. Intuitively, it corresponds to the situation when the signal is completely absorbed at x = 0 and then reemitted by a secondary source in both forward and backward directions (Figure 2).

## 5. Discussion

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Author’s estimate, using geodesic distances. |

2 | Note that Equation (5) is for free propagation, and it does not contain Ito terms (Jeanblanc et al. 2003). The Ito equation would appear if one considers a slowly varying response in Equation (6), where the last term can be expanded in a Taylor series, as well. |

**Figure 1.**(

**A**) Relativistic light cone. Only part x > 0 is drawn for clarity. (

**B**) Schematic shape of the random signal propagating across the line. The upper axis shows “Buy” and “Sell” orders separately, and the lower axis shows the sum of the orders. The vertical axis may correspond to the number of shares traded, or to the aggregate dollar volume, depending on the organization of the trading venue and the type of order. (

**C**) “Topology” of the trading network. Exchange is located at point A, and the imbalances are absorbed (cleared) at point B. Traders are distributed along the line AB, according to the Poisson law.

**Figure 2.**Space/time propagation of the signal in the trading system. The signal propagates from the lower left corner of the frame to the upper right corner parallel to the diagonal. The purple line indicates $\tau =x/c-0.5a$ (the signal is in the future with respect to an observer at the origin), the green line is $\tau =x/c-0.25a$ (signal approaches the origin), and the yellow line is $\tau =x/c+0.25a$ (the signal passes the origin). One might consider a as a crude measure of the reaction time of the trading system. Values for the parameters of the Equations (9), (11), and (12) used for all figures are λ = 1.5a, ${\tilde{\beta}}_{11}={\tilde{\beta}}_{22}=0.6a$, and ${\beta}_{12}={\beta}_{21}=0.3a$ and were chosen for the best visual appearance of plots. Hyperbolic structure in the left quadrant indicates the space/time region where the signal is indistinguishable from the noise.

**Figure 3.**The shape of the pulse in arbitrary units (a response to the delta function) in the cases (

**A**) $\tau =x/c-0.5a$, (

**B**) $\tau =x/c-0.25a$, and (

**C**) $\tau =x/c+0.25a$. The spatial coordinate is measured in the units c·a (see Equation (5)).

**Figure 4.**Autocorrelation of the square of the Green function kernel $\mathrm{I}\left(\mathsf{\tau}\right)={{\displaystyle \int}}_{-10}^{10}{K}^{2}(x,\frac{x}{v}-\tau )dx$ (see Equation (14) in the text), as a function of time delay and v = 0.75c. Limits of integration are arbitrary as a truncation to approximate [−∞, +∞] with the necessary accuracy.

**Figure 5.**Kullback–Leibler distance (Equation (15)) between Green functions kernels as a function of time delay and v = 0.75c. A small negative tail is spurious and is related to numerical approximations.

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**MDPI and ACS Style**

Lerner, P.B.
Transmission of Trading Orders through Communication Line with Relativistic Delay. *Int. J. Financial Stud.* **2021**, *9*, 12.
https://doi.org/10.3390/ijfs9010012

**AMA Style**

Lerner PB.
Transmission of Trading Orders through Communication Line with Relativistic Delay. *International Journal of Financial Studies*. 2021; 9(1):12.
https://doi.org/10.3390/ijfs9010012

**Chicago/Turabian Style**

Lerner, Peter B.
2021. "Transmission of Trading Orders through Communication Line with Relativistic Delay" *International Journal of Financial Studies* 9, no. 1: 12.
https://doi.org/10.3390/ijfs9010012