# Evolutionary Optimization of Control Strategies for Non-Stationary Immersion Environments

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Observation Model and Problem Statement

_{k}= x

_{k}+ v

_{k}, k = 1, …, n

_{k}, k = 1, …, n is the system component used in the process of making management decisions (i.e., open, close or retain current position), and v

_{k}, k = 1, …, n is the noise.

_{k}in (1) is modeled as an output signal of a nonlinear system observed in the conditions of non-stationary and non-Gaussian interference v

_{k}(dynamic chaos model) [5,6,7,8,9,10,11,12,13]. Lyapunov functions [25] and identification methods based on higher-order spectra [26] are used in order to substantiate such problem statements. There is a large area of research on the direct reconstruction of stochastic differential equations [27,28] for the model (1). Other points of view are based on nonlinear transformations of the y

_{k}process, and, for example, on investigating fractal properties of the process trajectory [29,30,31].

_{k}, k = 1, …, n is usually used to isolate the system component x

_{k}, k = 1, …, n from (1) in real time. For this purpose, we utilize an exponential filter [32]:

_{k}= αy

_{k}+ (1 − α)y

_{(k−1)}= x

_{(k−1)}+ α(y

_{k}− x

_{(k−1)}), k = 2, …, n

_{k}, k = 1, …, n is modeled as an oscillatory non-periodic process with a large number of local trends. This description indicates the possibility of interpreting this process as an implementation of the dynamic chaos model [5,6,7,8,9,10,11,12,13]. Its second distinctive feature is the noise component v

_{k}, k = 1, …, n being interpreted as a non-stationary random process described by an approximate Gaussian model with fluctuating parameters. In particular, correlations and spectral characteristics of this process change significantly over time [23,24].

_{k}= x

_{k}+ v

_{k}, k = 1, …, n be a sequence of observations corresponding to a given time interval of asset management T = nΔt, where Δt is the selected interval between time counts. During the specified time, M operations are carried out in the trading process, each being determined by their start and finish (k

_{open,}k

_{close})

_{j}, j = 1, …, M.

_{j}, j = 1, …, M to obtain maximum profit G(S):

_{open,}k

_{close})

_{j}, j = 1, …, M, and, in some cases, the lot size. The sum of the operation results at the k-th step G

_{k}(S) becoming smaller than the trader’s available deposit G

_{0}means the management process resulted in complete loss.

#### 2.2. Channel Asset Management Strategies

_{k}= x

_{k}± B, k = 1, …, n [33]. Variations inside the channel |y

_{k}− x

_{k}| = |δy

_{k}| ≤ B, k = 1, …, n, are fluctuations that do not contain an obvious trend, in which case, the process can be referred to as a sideways trend or a flat. Channel width B can be selected depending on various considerations. It usually lies in the range from s

_{y}to 3s

_{y}, where s

_{y}is the estimate of the standard deviation (SD) δy

_{k}, k = 1, …, n. In general, the choice of channel width depends on the nature of the data and the specificity of the selected management strategy. In some cases, the channel width may be some variable value B

_{k}= B

_{k}(y

_{k}), k = 1, …, n.

_{k}, k = 1, …, n that breaks out of the channel is interpreted as the emergence of a trend in some management strategies. In the case of managing assets according to the trend direction, such events give rise to a recommendation to open a position in accordance with the sign of the channel boundary. Due to strong variability, a trend is often considered to be present when the system component x

_{k}, k = 1, …, n formed by the exponential filter (2) with a given level of smoothing quits the channel. The values of the model parameters α, B, TP, SL are optional. Their selection depends on the knowledge and intuition of the trader, and they fully determine the management effectiveness. But it is often the case that intuition and other abilities of a human person appear to be ineffective in trading. Therefore, there is a need for strictly formalized and mathematically sound solutions.

_{k}> x

_{k}+ B or Open Dn at y

_{k}< x

_{k}− B. Otherwise, a position will be opened at each step outside the channel. In this regard, the more often used rules are based on determining the time of crossing the channel boundary (y

_{k−1}≤ x

_{k−1}+ B) & (y

_{k}> x

_{k}+ B) or (y

_{k−1}≥ x

_{k−1}− B) & (y

_{k}< x

_{k}− B), k = 1, …, n.

_{close}> y

_{open}+ TP or = y

_{close}< y

_{open}− SL levels are reached (at Open Up) or when y

_{close}< y

_{open}− TP or y

_{close}> y

_{open}+ SL (at Open Dn).

_{0}= {0.01, 10, 10, 10, 10}. The values of the iteration step, respectively, were Step = {0.01, 1, 1, 1, 1}.

#### 2.3. Features of Evolutionary Optimization for Chaotic Immersion Environments

#### 2.4. Algorithm of Evolutionary Optimization of the Management Model

_{A}= {S

_{A}

_{1}, …, S

_{ANa}} with N

_{a}elements

_{,}each of which is defined by its structure R (the decision-making rule) and a set of corresponding numerical parameters a, i.e., S = {R, a}. The effectiveness of a strategy Eff(s) is assessed via applying it to the time series of observations Y(t), which together form an experimental retrospective dataset. We introduce two nonlinear operators.

- The operator of variability and multiplication of strategies:

_{b}> 1 is the multiplication coefficient of strategies. The union of ancestor strategies and descendant strategies is a generation of size N

_{g}= N

_{a}+ N

_{d}= N

_{a}(1 + k

_{b}):

- 2.
- A selection operator that selects the “surviving” strategies from the generation ${S}_{G}=\left\{{S}_{{G}_{1}},\dots ,{S}_{G{N}_{g}}\right\}$ that become the ancestors of the next generation:

_{g}of the generation being formed. It can be assumed that the convergence rate will be higher if the multiplication coefficient k

_{b}is made variable so that the number of descendant strategies N

_{d}depends on the effectiveness of parent strategies, i.e., N

_{d}= k(Eff(S

_{a})), k > 1. In other words, a more effective ancestor can produce more offspring. However, this statement requires additional verification.

#### 2.5. Computational Aspects of the Evolutionary Optimization Algorithm

_{G}. Furthermore, each of the first-generation strategies undergoes a testing by being applied to a set of retrospective observations {Y(t), Y(t − Y)}, where T is the size of the validation dataset. The created strategies are ranked by their effectiveness Eff(S

_{i}), i = 0, …, N

_{g}and a specified number N

_{a}of “surviving” strategies that are allowed for further “reproduction” (modification) are selected. The selected strategies are the parents of a new set of modified descendant strategies and together with them form the second generation.

- Small single changes. In the strategy undergoing modification, relatively small changes are made to only one parameter (gene) selected by a random draw. The size of the change range depends on the parameter. This one-time change does not usually exceed 10% of the original value. The choice of the parameter is carried out by a random draw, similar to how it happens in the Monte Carlo method.
- Small group changes. They are carried out similarly to the previous case, but are made to several gene parameters at once instead. Their amount and their numbers are selected via a random draw.
- Strong single mutation or parametric mutation. The gene number is selected via a random draw. Usually, the number of mutations in a generation is small, and the probability of their occurrence does not exceed 2–3%. The size of the change field also depends on the parameter, usually a single mutation can reach 30–50% of the original value.
- Strong group changes are similar to the previous case, but are made immediately in several parameters, as in case 2.
- Structural (nonparametric) mutations. The parent strategy with some relatively small probability (usually less than 0.01) may undergo nonparametric mutation. In this case, the number of genes in the original genome may change, or, in a more radical case, the management strategy itself may be completely modified. The most rational way in this case consists of randomly choosing a management strategy from an a priori created knowledge base.

## 3. Results

## 4. Discussion

- the study of potential characteristics of self-adjusting asset management systems for various sets of dynamic properties of observation intervals of chaotic processes;
- the development of a knowledge base of management strategies and its application for implementing structural mutations of the management model in the mechanism of variability of the evolutionary optimization algorithm;
- the development of randomized synthesis of management strategies using multi-expert data analysis [51];
- the use of composite algorithms combining the capabilities of robustification and adaptation in management decision-making.
- the effectiveness of the application of evolutionary optimization in markets and periods that differ in the degree of market efficiency within the Efficient Market Hypothesis (EMH) [52]. It is supposed that the greatest profit can be made in a highly inefficient market. At the same time various exchange markets all have multifractal structural properties with different levels in the sample and sub-samples that cause inefficiency with different levels in these foreign exchange markets [30]. Another work reveals that the efficiency of the cryptocurrency markets varies over time, which is consistent with adaptive market hypothesis (AMH) [53]. The question about the level of current market inefficiency which is acceptable for self-adjusting asset management systems needs to be investigated.
- the use of external add-ons that carry information exogenous to technical analysis on expected trends of the considered financial instrument and market mood in general.
- The outlined issues constitute the subject of our further research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Performance of the CSF strategy for example in Figure 1.

**Figure 5.**Performance of applying the CSF strategy with parameters optimal for the first day to the next nine days of management.

**Figure 6.**Dynamics of quotations during the indicated 10 days, corresponding to the performance of the CSF strategy in Figure 5.

**Figure 8.**An example implementation of the CSF strategy with parameters obtained via evolutionary optimization.

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**MDPI and ACS Style**

Musaev, A.; Makshanov, A.; Grigoriev, D.
Evolutionary Optimization of Control Strategies for Non-Stationary Immersion Environments. *Mathematics* **2022**, *10*, 1797.
https://doi.org/10.3390/math10111797

**AMA Style**

Musaev A, Makshanov A, Grigoriev D.
Evolutionary Optimization of Control Strategies for Non-Stationary Immersion Environments. *Mathematics*. 2022; 10(11):1797.
https://doi.org/10.3390/math10111797

**Chicago/Turabian Style**

Musaev, Alexander, Andrey Makshanov, and Dmitry Grigoriev.
2022. "Evolutionary Optimization of Control Strategies for Non-Stationary Immersion Environments" *Mathematics* 10, no. 11: 1797.
https://doi.org/10.3390/math10111797