A pivotal concept in the framework is that of expectation gaps, representing the discrepancy between an individual’s aspirations or ideals and their perceived reality within specific life domains. These gaps are posited to significantly influence various aspects of well-being.

For each relevant life domain $i$ (e.g., mental health, stress, coping ability), we define the following components:

• Experienced Reality $E_{real,i}\left(t\right)$: This variable reflects the individual’s current state or perception within domain $i$. It is typically directly linked to the state variable of the corresponding domain. For example, for mental health, $E_{\text{real},\text{hm}}\left(t\right)=H_m\left(t\right)$, and for coping, $E_{\text{real},\text{c}}\left(t\right)=C\left(t\right)$. For stress, $E_{\text{real},\text{s}}\left(t\right)=\text{Stress}\left(t\right)$, representing the currently experienced stress level.

• Idealized Expectation $E_{ideal,i}\left(t\right)$: This represents the individual’s desired state, aspiration, or ideal level for domain $i$. Crucially, $E_{\text{ideal},i}\left(t\right)$ is not static but adapts over time based on the persistence and magnitude of the expectation gap $G_i\left(t\right)$. The adaptation rules are as follows:

1) $E_{\text{ideal},i}\left(t\right)$ generally remains constant from one time step to the next.

2) If $G_i\left(t\right)\approx 0$ (i.e., the current expectation gap is negligible, meaning reality meets or exceeds the ideal): $E_{\text{ideal},i}\left(t\right)$ may increase via a “Capped Random Increment”. This reflects that when expectations are met, aspirations might rise, albeit cautiously.

$E_{\text{ideal},i}\left(t+\text{Δ}t\right)=E_{\text{ideal},i}\left(t\right)+\text{min}\left(\text{Δ}{E}_{\text{max\_inc}},\text{max}\left(0,Z_i\right)\right)$ (1)

where $Z_i$ is a small, non-negative random increment (e.g., drawn from a normal distribution with a small positive mean and capped at zero from below), and $\text{Δ}{E}_{\text{max\_inc}}$ is a cap on this upward adjustment.

3) If $G_i\left(t\right)>0$ and persists for a duration $T_{threshold,i}$: $E_{\text{ideal},i}\left(t\right)$ may decrease through a “Historical Reduction”. This signifies that if reality consistently falls short of ideals for a prolonged period, the individual might adjust their ideals downwards to align them more closely with achievable reality.

$E_{\text{ideal},i}\left(t+\text{Δ}t\right)=E_{\text{ideal},i}\left(t\right)-{\eta }_i\cdot {\bar{G}}_{i,\text{hist}}\left(t\right)$ (2)

where ${\eta }_i$ is an adjustment rate, and ${\bar{G}}_{i,\text{hist}}\left(t\right)$ is an average of the gap $G_i\left(s\right)$ over a recent historical window (e.g., the past $T_{\text{threshold},i}$ time steps during which $G_i\left(s\right)>0$). If $G_i\left(t\right)>0$ but the persistence is less than $T_{\text{threshold},i}$, $E_{\text{ideal},i}\left(t\right)$ remains unchanged.

The specific interpretation of $E_{\text{ideal},i}\left(t\right)$ and the direction of these adjustments depend on the nature of the domain. For instance, for stress, $E_{\text{ideal},\text{s}}\left(t\right)$ might represent a desired low level of stress. If actual stress $E_{\text{real},\text{s}}\left(t\right)$ is consistently above this ideal (i.e., $G_s\left(t\right)=\text{max}\left(0,E_{\text{real},\text{s}}\left(t\right)-E_{\text{ideal},\text{s}}\left(t\right)\right)>0$), then $E_{\text{ideal},\text{s}}\left(t\right)$ might increase (i.e., the individual becomes accustomed to or accepts a higher baseline stress).

• Expectation Gap $G_i\left(t\right)$: The gap for domain $i$ is defined algebraically as the non-negative difference between the idealized expectation and the experienced reality. The precise formulation depends on whether a higher ideal is “good” or “bad” for the domain:

- For domains like mental health or coping ability, where higher is better:

$G_i\left(t\right)=\text{max}\left(0,E_{\text{ideal},i}\left(t\right)-E_{\text{real},i}\left(t\right)\right)$ (3)

A positive gap here means ideals are not being met.

- For domains like stress, where lower is better, and $E_{\text{ideal},i}\left(t\right)$ represents a desired low level:

$G_i\left(t\right)=\text{max}\left(0,E_{\text{real},i}\left(t\right)-E_{\text{ideal},i}\left(t\right)\right)$ (4)

A positive gap here means current reality (e.g., high stress) exceeds the desired low ideal.

These domain-specific gaps $G_i\left(t\right)$ then act as inputs to the SDEs of their respective domains (e.g., $G_{\text{hm}}\left(t\right)$ affects $H_m\left(t\right)$) and potentially cross-domain influences, typically exerting a negative influence on well-being when the gap signifies an unmet aspiration or an undesirable state. This adaptive mechanism for $E_{\text{ideal},i}\left(t\right)$ allows the model to capture phenomena like aspiration adjustment and the psychological impact of persistent discrepancies between desires and outcomes.

• Expectation Gap Dynamics: Ideal expectation levels ( $E_{\text{ideal}}$) for Mental Health ( $H_m$), Stress, and Coping ( $C$) were dynamically updated based on the persistent discrepancy (gap, $G$) between the ideal and realized states, following the rules defined in the update_expectation_gaps function (governed by parameters like eta_ideal_adj, T_thresh_days, delta_E_max_inc).

- Standard Gap Influence (NoFD Scenarios): In scenarios without fractional dynamics (NoFD), the current gap $G\left(t\right)$ directly influences the relevant domain’s SDE (e.g., $H_m\left(t\right)$ is affected by $G_{H_m}\left(t\right)$ through a coefficient like ${\alpha }_{G_m{H}_m}$).

- Fractional Derivative Gap Influence (FD Scenarios): For scenarios where fractional dynamics (FD) were enabled, an additional term incorporating a fractional derivative of the historical gap sequence was introduced into the drift component of the SDEs for $H_m\left(t\right)$, $\text{Stress}\left(t\right)$, and $C\left(t\right)$. This term aims to model memory effects, where the past trajectory of the gap (not just its current value) influences the domain’s evolution.

* Fractional Derivative Calculation: The fractional derivative of order $\alpha$ for a gap history $G_t,G_{t-\text{Δ}t},\cdots$ was approximated using the Grünwald-Letnikov definition:

$D_t^{\alpha }G\left(t\right)\approx \frac{1}{{\left(\text{Δ}t\right)}^{\alpha }}\underset{j=0}{\overset{M}{{\displaystyle \sum }}}{\left(-1\right)}^j\left(\begin{array}{l}\alpha \hfill \\ j\hfill \end{array}\right)G\left(t-j\text{Δ}t\right)$ (5)

where $\left(\begin{array}{l}\alpha \hfill \\ j\hfill \end{array}\right)=\frac{\text{Γ}\left(\alpha +1\right)}{\text{Γ}\left(j+1\right)\text{Γ}\left(\alpha -j+1\right)}$ are generalized binomial coefficients, $M$ is

the memory length (memory_steps_..._frac_gap), and $\text{Δ}t$ is the time step. In the implementation, this was calculated using the calculate_fractional_derivative_gl function which utilizes a deque to store the gap history.

* Parameterization of FD Effect: For each domain with FD enabled (e.g., $H_m$), two parameters controlled this effect:

• alpha_..._frac_gap_order: The order $\alpha$ of the fractional derivative (e.g., alpha_Hm_frac_gap_order).

• ‘zeta_..._frac_gap_effect’: A coefficient $\zeta$ determining the strength and direction of the fractional derivative’s influence on the domain’s drift (e.g., ‘zeta_Hm_frac_gap_effect’).

The term added to the drift was typically of the form $\zeta \cdot D_t^{\alpha }G\left(t\right)$.

* Rationale: This introduction of fractional derivatives allows the model to capture long-range memory and non-local effects in how individuals adapt to or are influenced by persistent discrepancies between their ideal and actual states in key psychological domains. For example, a prolonged history of a large gap in mental well-being, even if the current gap is small, might still exert a lingering influence on $H_m\left(t\right)$‘s tendency to change.

Income is modeled as a drift-dominated process, reflecting the fact that, on average, income tends to increase over time in a predictable manner but can still exhibit short-term fluctuations. We assume that income follows a Fractional Brownian motion (fBM), where the drift term represents the long-term growth rate of income and the diffusion term captures short-term volatility.

The model incorporates deterministic income (monthly salary), fractional random variations, and jump processes to account for unexpected expenditures and rare income events. The stochastic differential equation (SDE) for income is defined as:

$\text{d}I\left(t\right)={\theta }_I\left(t\right)\left({\mu }_I\left(t\right)-I\left(t\right)\right)\text{d}t+{\sigma }_I\text{d}{B}_H\left(t\right)-\text{d}{J}_{\text{exp}}\left(t\right)+\text{d}{J}_{\text{inc}}\left(t\right)+S_{\text{periodic}}\left(t\right)\text{d}t$ (6)

where:

• $\theta$: Mean-reversion rate, determining how quickly income is pulled back to its running average ( ${\mu }_I\left(t\right)$). Capturing the natural reluctance, simulating an attempt to “save” and return towards the running average.

• ${\mu }_I\left(t\right)$: Running average tracks the overall trend of the income process and is defined as ${\mu }_I=\frac{1}{t}{\displaystyle {\int }_0^{t}I\left(s\right)\text{d}s}$ or in discrete time, for the numerical solution, as ${\mu }_I\left(t\right)={\mu }_I\left(t-1\right)+\frac{1}{t}\left(I\left(t\right)-{\mu }_I\left(t-1\right)\right)$.

• ${\sigma }_I\text{d}{B}_H\left(t\right)$: Fractional Brownian motion term that introduces smooth random fluctuations with long-range dependence, where $B_H\left(t\right)$ is fractional Brownian motion with Hurst exponent $H>1/2$.

• $-\text{d}{J}_{\text{exp}}\left(t\right)$: Jump process for unexpected expenditures, modeled using a Poisson process with exponentially distributed jump sizes.

• $\text{d}{J}_{\text{inc}}\left(t\right)$: Jump process for unexpected income, modeled using a Poisson process with lognormally distributed jump sizes.

• ${\lambda }_{\text{exp}}$ and ${\lambda }_{\text{inc}}$ are the expected rates of jumps for expenditure and income(normally set quite low and with ${\lambda }_{\text{exp}}>{\lambda }_{\text{inc}}$).

• $S_{\text{periodic}}\left(t\right)$ is a periodic cash injection function, defined as $S\left(t\right)=\{\begin{array}{ll}S\left(t\right)\hfill & \text{if}t\equiv 0\mathrm{mod}30\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$

1) Mean-Reversion ( ${\theta }_I\left({\mu }_I-I\left(t\right)\right)\text{d}t$): - The Ornstein-Uhlenbeck (OU) process ensures that income fluctuates around its the running average ${\mu }_I\left(t\right)$. The mean-reversion rate ${\theta }_I$ controls how quickly deviations from the base income are corrected. Incorporating the mean-reversion term to reflect natural tendency to save or return to a previous financial “comfort zone” aligns well with real-world behavior.

2) Fractional Brownian Motion ( ${\sigma }_I\text{d}{B}_H\left(t\right)$): - We use fractional Brownian motion (fBm) with Hurst exponent $H>1/2$ to model persistent, smooth variations in income. Unlike standard Brownian motion, which has independent increments, fBm introduces memory effects where increments are positively correlated over time. This captures gradual, small-scale fluctuations in income that are realistic for most individuals.

3) Jump Process for Expenditures ( $-J_{\text{exp}}\left(t\right)$): - Expenditures are modeled as negative jumps in income, occurring randomly according to a Poisson process. The sizes of these jumps are sampled from an exponential distribution, reflecting rare but potentially significant costs such as unexpected bills or repairs. The jump rate for expenditures ( ${\lambda }_{\text{exp}}$) is set slightly higher than for income jumps, to account for the general tendency of unexpected costs to outweigh unexpected income.

4) Jump Process for Income ( $J_{\text{inc}}\left(t\right)$): - Positive jumps represent rare sources of unexpected income, such as bonuses, refunds, or windfalls. These jumps are modeled using a Poisson process with lognormally distributed jump sizes to account for the fact that large income events are less common than small ones. The rate of these income jumps ( ${\lambda }_{\text{inc}}$) is set lower than the rate for expenditure jumps.

The proposed model provides a flexible framework for simulating income dynamics over time. It can be adapted to fit specific scenarios by modifying the parameters or processes:

• Scaling for Different Income Levels: The base income ( ${\mu }_I$) can be adjusted to represent varying salary levels for different individuals or demographics.

• Customizing Jump Processes: The distributions for expenditure and income jumps (e.g., exponential, lognormal) can be replaced or re-parameterized to capture specific real-world patterns (e.g., heavy-tailed distributions for larger expenses).

• Adjusting Time Scales: The model is currently designed for daily simulations over one month but can easily be adapted for longer periods (e.g., annual income dynamics) by adjusting the time step ( $\text{d}t$) and total simulation time ( $T$).

• Including Seasonal Effects: Additional terms can be added to account for periodic changes in income or expenditure, such as holiday spending or tax refunds.

The quality and state of relationships, $R\left(t\right)$, are dynamic and influenced by several factors including income, mental health, and personality traits, as well as inherent randomness. The SDE is:

$\begin{array}{c}\text{d}R\left(t\right)={\theta }_R\left({\mu }_R-R\left(t\right)\right)\text{d}t+{\alpha }_{I_R}{f}_{I_R}\left(I\left(t\right)\right)\text{d}t+{\alpha }_{M_R}{H}_m\left(t\right)\text{d}t\\ +{\alpha }_{P_R}{P}_{\text{pers}}\text{d}t+{\sigma }_R\text{d}{B}_{H_R}\left(t\right)\end{array}$ (7)

where:

• ${\theta }_R\left({\mu }_R-R\left(t\right)\right)\text{d}t$ describes mean reversion to a baseline relationship quality ${\mu }_R$.

• ${\alpha }_{I_R}{f}_{I_R}\left(I\left(t\right)\right)\text{d}t$ represents the non-linear contribution from income. $f_{I_R}\left(I\left(t\right)\right)=-{\beta }_1{\left(I_{\text{low}}-I\left(t\right)\right)}^2+{\beta }_2{\left(I\left(t\right)-I_{\text{high}}\right)}^2$, indicating that income deviations from certain thresholds (both low and high) can impact relationships.

• ${\alpha }_{M_R}{H}_m\left(t\right)\text{d}t$ reflects the positive influence of mental health $H_m\left(t\right)$ on relationship quality.

• ${\alpha }_{P_R}{P}_{\text{pers}}\text{d}t$ is a contribution from personality traits $P_{\text{pers}}$ (e.g., a parameter representing introversion/extroversion influencing the tendency or quality of relationships).

• ${\sigma }_R\text{d}{B}_{H_R}\left(t\right)$ introduces smooth, memory-based random fluctuations in relationship dynamics.

This formulation emphasizes the interplay between material resources, mental well-being, and intrinsic personality in shaping an individual’s relational life.

Mental health, $H_m\left(t\right)$, is a core component of well-being, significantly influenced by expectation gaps, stress, and the quality of relationships. The SDE is simplified here by focusing on the impact of a general mental well-being gap and other key interactors:

$\begin{array}{c}\text{d}{H}_m\left(t\right)={\theta }_m\left({\mu }_m-H_m\left(t\right)\right)\text{d}t+{\sigma }_m\text{d}{B}_{H_{Hm}}\left(t\right)-a_{G_m}{G}_{\text{mental}}\left(t\right)\text{d}t\\ -a_{S_m}\text{Stress}\left(t\right)\text{d}t+a_{R_m}R\left(t\right)\text{d}t\end{array}$ (8)

where:

• ${\theta }_m\left({\mu }_m-H_m\left(t\right)\right)\text{d}t$ is the mean-reversion to a baseline mental health level ${\mu }_m$.

• ${\sigma }_m\text{d}{B}_{H_{Hm}}\left(t\right)$ captures random fluctuations in mental health using fBm.

• $a_{G_m}{G}_{\text{mental}}\left(t\right)\text{d}t$ represents the negative impact of expectation gaps specific to mental well-being or overall life satisfaction, $G_{\text{mental}}\left(t\right)$ (derived as per Section 0). $a_{G_m}$ is the sensitivity.

• $a_{S_m}\text{Stress}\left(t\right)\text{d}t$ models the detrimental effect of current stress levels $\text{Stress}\left(t\right)$ on mental health.

• $a_{R_m}R\left(t\right)\text{d}t$ reflects the supportive role of positive relationships $R\left(t\right)$ in bolstering mental health.

This model positions mental health as a dynamic state, constantly buffered or eroded by life’s expectations, stressors, and social supports.

Physical health, $H_p\left(t\right)$, evolves based on baseline health, lifestyle choices, random fluctuations, and susceptibility to external shocks. Its SDE is:

$\text{d}{H}_p\left(t\right)={\theta }_p\left({\mu }_p-H_p\left(t\right)\right)\text{d}t+L_{\text{lifestyle}}\left(H_p\left(t\right)\right)\text{d}t+{\sigma }_p\text{d}{B}_{H_{Hp}}\left(t\right)+\text{d}{J}_{\text{health\_shocks}}\left(t\right)$ (9)

where:

• ${\theta }_p\left({\mu }_p-H_p\left(t\right)\right)\text{d}t$ represents mean reversion to a baseline physical health ${\mu }_p$, influenced by factors like age and genetics.

• $L_{\text{lifestyle}}\left(H_p\left(t\right)\right)\text{d}t$ encapsulates the deterministic inputs from lifestyle factors, which are themselves influenced by the current health state $H_p\left(t\right)$: $L_{\text{lifestyle}}\left(H_p\left(t\right)\right)={\beta }_{E}E\left(H_p\left(t\right)\right)+{\beta }_{D}D\left(H_p\left(t\right)\right)+{\beta }_S{S}_{\text{sleep}}\left(H_p\left(t\right)\right)$. Each term $E(\cdot ),D(\cdot ),S_{\text{sleep}}(\cdot )$ (Exercise, Diet, Sleep) is a sigmoidal function: e.g., $E\left(H_p\left(t\right)\right)=E_0/\left(1+\text{exp}\left(-{\alpha }_E\cdot \left(x\left(H_p\left(t\right)\right)-x_0\right)\right)\right)$, where $x\left(H_p\left(t\right)\right)=x_0+{\alpha }_H\cdot \left(H_p\left(t\right)-H_{\text{min}}\right)$ reflects that the effectiveness or engagement in these activities depends on current health.

• ${\sigma }_p\text{d}{B}_{H_{Hp}}\left(t\right)$ models smooth random variations in health.

• $\text{d}{J}_{\text{health\_shocks}}\left(t\right)$ is a jump process representing sudden adverse health events like illness or injury, typically modeled as negative jumps.

This SDE captures the feedback loop where health status influences health behaviors, which in turn affect health outcomes, alongside random and acute health events.

Stress, $\text{Stress}\left(t\right)$, is modeled as a mean-reverting process influenced by expectation gaps and the effectiveness of coping strategies. Higher stress levels are detrimental to overall well-being. The SDE is:

$\text{dStress}\left(t\right)={\theta }_S\left(S_{\text{min}}-\text{Stress}\left(t\right)\right)\text{d}t+a_{G_S}{G}_{\text{stress}}\left(t\right)\text{d}t-a_{C_S}C\left(t\right)\text{d}t+{\sigma }_S\text{d}{B}_{H_{\text{Stress}}}\left(t\right)$ (10)

where:

• ${\theta }_S\left(S_{\text{min}}-\text{Stress}\left(t\right)\right)\text{d}t$ represents mean reversion towards a baseline or minimum achievable stress level $S_{\text{min}}$.

• $a_{G_S}{G}_{\text{stress}}\left(t\right)\text{d}t$ models the increase in stress due to expectation gaps $G_{\text{stress}}\left(t\right)$ related to perceived manageability of life demands or achievement of stress-related goals. $a_{G_S}$ is the sensitivity.

• $a_{C_S}C\left(t\right)\text{d}t$ signifies that effective coping strategies $C\left(t\right)$ reduce stress levels.

• ${\sigma }_S\text{d}{B}_{H_{\text{Stress}}}\left(t\right)$ introduces smooth random fluctuations to the stress level via fBm.

This formulation captures stress as a dynamic balance between environmental/internal pressures (via gaps) and the individual’s capacity to manage them (via coping).

Coping strategies, $C\left(t\right)$, are adaptive mechanisms individuals use to manage stress and are influenced by perceived stress, expectation gaps related to coping efficacy, relationships, and available resources like income. The SDE for coping is:

$\begin{array}{c}\text{d}C\left(t\right)={\theta }_C\left({\mu }_C-C\left(t\right)\right)\text{d}t+\frac{{\alpha }_{S_{\text{coping}}}}{1+{\beta }_{S_{\text{coping}}}\text{Stress}\left(t\right)}\text{d}t-a_{G_C}{G}_{\text{coping}}\left(t\right)\text{d}t\\ +a_{R_C}R\left(t\right)\text{d}t+a_{I_C}I\left(t\right)\text{d}t+{\sigma }_C\text{d}{B}_{H_C}\left(t\right)\end{array}$ (11)

where:

• ${\theta }_C\left({\mu }_C-C\left(t\right)\right)\text{d}t$ describes mean reversion to a baseline level of coping strategies ${\mu }_C$.

• $\frac{{\alpha }_{S_{\text{coping}}}}{1+{\beta }_{S_{\text{coping}}}\text{Stress}\left(t\right)}\text{d}t$ is the contribution from stress. Higher stress reduces coping ability (or that coping effectiveness is inversely related to stress levels). ${\alpha }_{S_{\text{coping}}}$ and ${\beta }_{S_{\text{coping}}}$ are parameters modulating this non-linear effect.

• $a_{G_C}{G}_{\text{coping}}\left(t\right)\text{d}t$ represents the negative impact of expectation gaps $G_{\text{coping}}\left(t\right)$ concerning the perceived effectiveness of one’s coping mechanisms.

• $a_{R_C}R\left(t\right)\text{d}t$ indicates that supportive relationships $R\left(t\right)$ improve coping strategies.

• $a_{I_C}I\left(t\right)\text{d}t$ signifies that income $I\left(t\right)$ enhances coping by providing resources.

• ${\sigma }_C\text{d}{B}_{H_C}\left(t\right)$ models smooth random variations in the deployment or effectiveness of coping strategies.

This model highlights that coping is not static but dynamically adapts based on demands (stress, gaps) and available supports (relationships, income).

Voluntary actions, $V\left(t\right)$, encompass self-driven activities that contribute to personal well-being, such as hobbies, learning, or community engagement. These are influenced by income, physical and mental health, and available free time. The SDE is:

$\begin{array}{c}\text{d}V\left(t\right)={\theta }_V\left({\mu }_V-V\left(t\right)\right)\text{d}t+{\alpha }_{I_V}{f}_{I_V}\left(I\left(t\right)\right)\text{d}t+{\alpha }_{P_{h}V}{H}_p\left(t\right)\text{d}t\\ +{\alpha }_{M_{h}V}{H}_m\left(t\right)\text{d}t+{\alpha }_{T_V}{T}_{\text{free}}\left(t\right)\text{d}t+{\sigma }_V\text{d}{B}_{H_V}\left(t\right)\end{array}$ (12)

where:

• ${\theta }_V\left({\mu }_V-V\left(t\right)\right)\text{d}t$ represents mean reversion to a baseline level of voluntary actions ${\mu }_V$.

• ${\alpha }_{I_V}{f}_{I_V}\left(I\left(t\right)\right)\text{d}t$ is the contribution from income, where $f_{I_V}\left(I\left(t\right)\right)=I\left(t\right)/\left(1+{\beta }_{I_V}I\left(t\right)\right)$ models diminishing returns of income on voluntary actions.

• ${\alpha }_{P_{h}V}{H}_p\left(t\right)\text{d}t$ signifies that better physical health $H_p\left(t\right)$ enables more voluntary actions.

• ${\alpha }_{M_{h}V}{H}_m\left(t\right)\text{d}t$ indicates that better mental health $H_m\left(t\right)$ also promotes engagement in voluntary actions.

• ${\alpha }_{T_V}{T}_{\text{free}}\left(t\right)\text{d}t$ is the contribution from available free time $T_{\text{free}}\left(t\right)$, which itself can be modeled as $T_{\text{free}}\left(t\right)=T_{\text{max}}-{\gamma }_{\text{workload}}W\left(t\right)$, where $W\left(t\right)$ is workload (potentially related to income generation efforts).

• ${\sigma }_V\text{d}{B}_{H_V}\left(t\right)$ introduces smooth random fluctuations in the level or intensity of voluntary actions.

This model captures the idea that engaging in fulfilling voluntary activities depends on having the necessary resources, health, and time.

The simulation of this happiness model presents several layers of complexity:

1) Coupled System of SDEs: The model consists of eight primary stochastic processes ( $I\left(t\right)$, $H_m\left(t\right)$, $H_p\left(t\right)$, $\text{Stress}\left(t\right)$, $C\left(t\right)$, $R\left(t\right)$, $V\left(t\right)$, and $H\left(t\right)$), plus potentially several auxiliary processes for expectation gaps ( $E_{\text{ideal},i}\left(t\right)$ if modeled stochastically). These SDEs are tightly coupled, meaning the drift or volatility term of one SDE often depends on the current state of one or more other SDEs. This requires careful handling of the update order within each time step or the use of values from the previous time step for contemporaneous dependencies.

2) Fractional Brownian Motion (fBm): Most SDEs incorporate fBm ( $B_H\left(t\right)$ with $H>1/2$) for their stochastic components. Unlike standard Brownian motion, fBm has correlated increments (long-range dependence). Generating fBm paths or their increments is more computationally intensive than generating standard Wiener process increments. It requires methods like the Davies-Harte algorithm (for exact generation of a full path then differencing) or Hosking’s method (for sequential generation of increments), each with its own computational trade-offs.

3) Jump Processes: The income SDE includes compound Poisson processes for expenditures and unexpected income, and the physical health SDE includes jumps for health shocks. Simulating these requires generating Poisson arrival times and then sampling jump sizes from their respective distributions (exponential, lognormal).

4) Non-Linearities: Many SDEs feature non-linear terms in their drift components. For instance, the income effect on relationships ( $f_{I_R}\left(I\left(t\right)\right)$), the effect of stress on coping, the lifestyle contributions to physical health ( $L_{\text{lifestyle}}\left(H_p\left(t\right)\right)$ involving sigmoidal functions), and the logarithmic income effect on the happiness target ( $f_{I_H}\left(I\left(t\right)\right)$). These non-linearities are crucial for realistic dynamics but can make analytical solutions intractable, necessitating numerical simulation.

5) Expectation Gap Dynamics: While simplified in this sub-paper (Section 3), the calculation of $G_i\left(t\right)$ still involves updating $E_{\text{ideal},i}\left(t\right)$ (possibly stochastically) and $E_{\text{real},i}\left(t\right)$, then applying the $\text{max}\left(0,\cdot \right)$ function, before $G_i\left(t\right)$ feeds into other SDEs.

6) Parameter Space: The model involves a large number of parameters (mean-reversion rates, baseline levels, volatilities, sensitivities between domains, Hurst exponents, jump parameters, etc.). Managing, calibrating (even illustratively), and conducting sensitivity analyses across this space is a significant undertaking.

Despite these complexities, numerical simulation provides a viable path to explore the model’s behavior.

Given the nature of the SDEs, including those with non-additive noise (if fBm is considered generally) and path-dependent components, the Euler-Maruyama scheme is a common and relatively straightforward method for numerical approximation [18] . For a generic SDE of the form:

$\text{d}X\left(t\right)=a\left(X\left(t\right),t\right)\text{d}t+b\left(X\left(t\right),t\right)\text{d}W\left(t\right)$ (15)

where $a\left(X\left(t\right),t\right)$ is the drift term and $b\left(X\left(t\right),t\right)$ is the diffusion term, and $\text{d}W\left(t\right)$ is the increment of the stochastic process (e.g., fBm or standard BM for auxiliary processes), the discrete-time update rule is:

$X\left(t_{k+1}\right)=X\left(t_k\right)+a\left(X\left(t_k\right),t_k\right)\text{Δ}t+b\left(X\left(t_k\right),t_k\right)\text{Δ}{W}_k$ (16)

where $\text{Δ}t=t_{k+1}-t_k$ is the time step, and $\text{Δ}{W}_k=W\left(t_{k+1}\right)-W\left(t_k\right)$ is the increment of the stochastic process over that time step.

For the SDEs involving fractional Brownian motion, $\text{Δ}{W}_k$ becomes $\text{Δ}{B}_{H,k}=B_H\left(t_{k+1}\right)-B_H\left(t_k\right)$. These increments are not independent and must be generated accordingly. For jump processes, at each time step $\text{Δ}t$, the probability of one or more jumps occurring is determined (e.g., $1-{\text{e}}^{-\lambda \text{Δ}t}\approx \lambda \text{Δ}t$ for small $\lambda \text{Δ}t$). If a jump occurs, its size is sampled from the specified distribution and added to the state variable.

The simulation for this paper is configured as follows:

• Time Horizon ( $T_{final}$): The total duration for which the trajectories are simulated (e.g., representing several years in daily or weekly steps). For instance, $T_{\text{final}}=1095$ days (3 years).

• Time Step ( $\Delta t$): A sufficiently small time step is chosen to ensure numerical stability and accuracy of the Euler-Maruyama scheme. For example, $\text{Δ}t=0.1$ (if units are days, this allows for sub-daily resolution) or $\text{Δ}t=1$ day. The choice depends on the characteristic time scales of the SDEs.

• Initial Conditions: Each state variable ( $I\left(0\right),H_m\left(0\right),H_p\left(0\right),\text{Stress}\left(0\right),C\left(0\right),R\left(0\right),V\left(0\right),H\left(0\right)$, and $E_{\text{ideal},i}\left(0\right)$ for all relevant gaps) must be initialized. These can be set to their respective baseline levels ( ${\mu }_i$) or other chosen starting values to represent a particular individual profile.

• Parameter Values: A consistent set of illustrative parameter values is used for all SDEs. These include:

- Mean-reversion rates ( ${\theta }_i$).

- Baseline/target levels ( ${\mu }_i,S_{\text{min}}$).

- Volatilities ( ${\sigma }_i$).

- Hurst exponents ( $H_i$) for each fBm process (typically $0.5<H_i<1$).

- Interaction coefficients ( ${\alpha }_{ij},a_{G_i},{\lambda }_j$, etc.).

- Jump process parameters (rates ${\lambda }_{\text{jump}}$, distribution parameters for jump sizes).

- Parameters for expectation gap dynamics ( ${\kappa }_{\text{ideal},i},{\sigma }_{\text{ideal},i}$).

A table of key parameters used in the illustrative simulations will be provided in Section 8. It is acknowledged that these parameters are chosen for demonstration and are not empirically calibrated in this study.

• Number of Trajectories: While this sub-paper may focus on presenting a few illustrative individual trajectories, for statistical analysis or ensemble averages (not the primary focus here), multiple trajectories ( $N_{\text{ensemble}}$) would be generated.

The simulations are implemented in Python 3.x, leveraging several scientific computing libraries:

• NumPy: For numerical operations, array management, and random number generation (for standard normal deviates needed for fBm and jump processes).

• SciPy: Potentially for statistical distributions (e.g., lognormal, exponential for jump sizes) and other scientific functions.

• fbm package (or similar): A dedicated Python library such as ‘fbm’ (by Christopher Flynn and collaborators) or ‘stochastic’ is used for generating fractional Brownian motion paths or increments with a specified Hurst exponent. Alternatively, custom implementations of algorithms like Hosking’s method can be used.

• Matplotlib/Seaborn: For visualizing the simulated trajectories.

The simulation proceeds iteratively: at each time step $t_k$, the values of all state variables $X\left(t_k\right)$ are used to calculate the drifts. Stochastic increments $\text{Δ}{B}_{H,k}$ and jump occurrences are generated. Then, all state variables are updated to $X\left(t_{k+1}\right)$ using the Euler-Maruyama scheme. Dependencies between SDEs are handled by using the values from $t_k$ to compute all drifts and updates for $t_{k+1}$.

• Psychological Theories: Contemporary psychology views subjective well-being (SWB) not as a monolithic entity, but as a multifaceted construct encompassing life satisfaction, positive affect, and the absence of negative affect [2] . More specific theories further delineate key dimensions:

- Ryff’s Model of Psychological Well-Being (PWB) [19] [20] identifies six core dimensions: self-acceptance, personal growth, purpose in life, environmental mastery, autonomy, and positive relations with others.

- Self-Determination Theory (SDT) [21] posits that well-being arises from the satisfaction of three basic psychological needs: autonomy (feeling volitional), competence (feeling effective), and relatedness (feeling connected).

- Seligman’s PERMA Model [22] conceptualizes flourishing through five pillars: Positive emotion, Engagement, Relationships, Meaning, and Accomplishment.

• Model Consistency: The model reflects this multi-component view by defining overall happiness $H\left(t\right)$ (Equation 11) as dynamically influenced by the states of several key life domains: income (resource for mastery/accomplishment), mental and physical health (foundational), stress (detriment), coping strategies (mastery), relationships (relatedness, positive relations), and voluntary actions (engagement, purpose, autonomy, growth). Each domain (Section 3) is modeled with its own SDE, acknowledging its unique dynamics while also being interconnected. For example, Relationship Dynamics $R\left(t\right)$ directly map to “positive relations” and “relatedness”, while Voluntary Action Dynamics $V\left(t\right)$ can encompass “engagement”, “purpose”, and opportunities for “personal growth”.

• Psychological Theories:

- Set-Point Theory [1] [23] suggests that individuals have a genetically influenced baseline level of happiness. Life events may cause temporary deviations, but individuals tend to revert to this set-point.

- Hedonic Adaptation [8] [9] describes the psychological process by which individuals’ affective responses to positive or negative events diminish over time. We adapt to new circumstances, and their initial emotional impact fades.

• Model Consistency: The model incorporates these concepts through:

- Mean Reversion: Each life domain $X\left(t\right)$ (e.g., $H_m\left(t\right)$, $R\left(t\right)$) includes a mean-reversion term (e.g., ${\theta }_m\left({\mu }_m-H_m\left(t\right)\right)\text{d}t$ in Equation 6), pulling the domain towards a baseline level $\mu$.

- Dynamic Happiness Target: The overall happiness $H\left(t\right)$ mean-reverts towards a dynamic target level ${\mu }_{H_{\text{target}}}\left(t\right)$ (Equation 11), which itself is a function of the current states of the life domains (Equation 12). This offers a nuanced view of the set-point: while there’s a tendency towards an equilibrium, this equilibrium is not static but adapts to sustained changes in life circumstances, aligning with findings that set-points can, to some extent, be shifted [24] .

- Income Adaptation: The logarithmic function for income’s contribution to ${\mu }_{H_{\text{target}}}\left(t\right)$ ( $f_{IH}\left(I\left(t\right)\right)=\text{ln}\left(1+I\left(t\right)/I_{\text{scale}}\right)$) inherently models diminishing returns, a form of adaptation where each additional unit of income provides less additional happiness.

• Psychological Theories:

- Discrepancy Theories of Happiness (e.g., [25] [26] – Self-Discrepancy Theory) posit that satisfaction and emotional states are influenced by the perceived gap between one’s current state and a standard of comparison (e.g., ideal self, ought self, aspirations, past self, others’ states).

- Aspiration Level Theory suggests that individuals adjust their aspirations based on past successes and failures.

- Goal Pursuit & Well-Being: The process of setting and striving for goals, and the feedback from goal attainment (or non-attainment), significantly impacts well-being [27] .

• Model Consistency: The “Expectation Gaps” $G_i\left(t\right)$ (subsection on page 5, Equations 3 and 4) are central to the model and directly operationalize discrepancy theories.

- $G_i\left(t\right)$ is defined as the difference between an “Idealized Expectation” $E_{\text{ideal},i}\left(t\right)$ and “Experienced Reality” $E_{\text{real},i}\left(t\right)$ for relevant domains.

- These gaps directly impact domain dynamics (e.g., $G_{\text{mental}}\left(t\right)$ negatively affects $H_m\left(t\right)$ in Equation 6; $G_{\text{stress}}\left(t\right)$ increases $\text{Stress}\left(t\right)$ in Equation 8).

- The adaptive nature of $E_{\text{ideal},i}\left(t\right)$ (Equations 1 and 2), where ideals may rise cautiously with success (“Capped Random Increment”) or lower after prolonged unmet expectations (“Historical Reduction”), mirrors psychological processes of aspiration adjustment and coping with dissonance [28] .

• Psychological Theories:

- Transactional Model of Stress and Coping [15] : Stress is viewed as an outcome of an appraisal process where an individual evaluates environmental demands (stressors) against their perceived resources to manage them. Coping strategies are the cognitive and behavioral efforts to manage these demands.

- Conservation of Resources (COR) Theory [29] [30] : Stress occurs when there is a threat of resource loss, actual resource loss, or a failure to gain resources following significant resource investment. Resources can be objects, conditions, personal characteristics, or energies.

• Model Consistency: The SDEs for $\text{Stress}\left(t\right)$ (Equation 8) and Coping Strategy Dynamics $C\left(t\right)$ (Equation 9) embody these principles:

- $\text{Stress}\left(t\right)$ increases due to expectation gaps $G_{\text{stress}}\left(t\right)$ (perceived unmanageability of demands or failure to achieve stress-related goals) and is reduced by effective coping $C\left(t\right)$.

- Coping $C\left(t\right)$ is influenced by factors like income $I\left(t\right)$ (material resource), relationships $R\left(t\right)$ (social support resource), and is also affected negatively by high stress levels (reflecting how overwhelming stress can impair coping efficacy, as seen in the denominator of the stress term in Equation 9).

- The interplay between stress, coping, and other domains like mental health ( ${\alpha }_{sm}\text{Stress}\left(t\right)\text{d}t$ in Equation 6) reflects the systemic impact of stress and the resources available to manage it.

• Psychological Theories:

- Self-Determination Theory (SDT) [21] emphasizes that activities pursued out of intrinsic motivation (for inherent satisfaction) or well-internalized extrinsic motivation contribute more to well-being than controlled or amotivated behaviors.

- Flow Theory [31] highlights the state of optimal experience achieved when individuals are fully immersed in challenging activities that match their skill level, often leading to feelings of enjoyment and accomplishment.

- Purpose in Life: Having a sense of purpose or meaning is consistently linked to higher well-being [19] [32] .

• Model Consistency: The “Voluntary Action Dynamics” $V\left(t\right)$ (Equation 10) represents engagement in self-driven activities like hobbies, learning, or community involvement.

- $V\left(t\right)$ contributes positively to the happiness target ${\mu }_{H_{\text{target}}}\left(t\right)$ ( ${\lambda }_{V}V\left(t\right)$ in Equation 12).

- The ability to engage in $V\left(t\right)$ is supported by enabling factors such as physical health $H_p\left(t\right)$, mental health $H_m\left(t\right)$, income $I\left(t\right)$ (providing resources and opportunity), and available free time $T_{\text{free}}\left(t\right)$, consistent with the idea that basic needs and resources are necessary to pursue higher-order growth activities.

The proposed income model, while stylized, incorporates several features that align well with established economic theories and observed financial behaviors:

1) Mean Reversion and Adaptive Expectations ( ${\theta }_I\left(t\right)\left({\mu }_I\left(t\right)-I\left(t\right)\right)dt$ with ${\mu }_I\left(t\right)$ as a running average)

- Consumption Smoothing & Habit Formation: The mean-reversion mechanism, where income (or more accurately, liquid funds represented by $I\left(t\right)$) tends to revert to a running average ${\mu }_I\left(t\right)$, is consistent with individuals attempting to smooth their consumption. When $I\left(t\right)$ is significantly above ${\mu }_I\left(t\right)$, individuals may increase spending or transfer funds to less liquid savings, effectively pulling $I\left(t\right)$ down. Conversely, if $I\left(t\right)$ falls below ${\mu }_I\left(t\right)$, they might curtail discretionary spending or draw on short-term credit/buffers, pulling $I\left(t\right)$ up. This reflects a “financial comfort zone” or habit persistence.

- Adaptive Expectations: The use of a running average ${\mu }_I\left(t\right)$ as the reversion target implies that individuals adapt their perception of "normal" income based on past experience. This is a common assumption in models of expectation formation.

2) Drift-Dominated Process and Periodic Income ( $S_{periodic}\left(t\right)dt$)

- Lifecycle Income Profile: The model’s ability to be “drift-dominated” primarily through periodic salary injections ( $S_{\text{periodic}}\left(t\right)\text{d}t$) reflects the fundamental reality of most individuals’ income streams, which are characterized by regular, predictable payments (salaries, wages). Over the long term, these injections typically lead to an upward trend in cumulative income, consistent with career progression and general economic growth.

3) Stochastic Fluctuations (Fractional Brownian Motion ${\sigma }_{I}d{B}_H\left(t\right)$)

- Persistent Shocks & Autocorrelation: The use of Fractional Brownian Motion (fBm) with $H>1/2$ introduces persistence or long-range dependence in the income fluctuations. This acknowledges that real-world income streams are often not memoryless; periods of slightly higher or lower income (or spending patterns affecting net cash flow) can persist beyond what a standard Brownian motion would suggest. This can be due to unobserved factors, minor changes in earning capacity, or ingrained spending habits.

- Smoothness: fBm provides smoother paths than standard Brownian motion, which can be more realistic for daily fluctuations in available funds, barring discrete events.

4) Jump Processes for Shocks ( $-d{J}_{exp}\left(t\right)+d{J}_{inc}\left(t\right)$)

- Unexpected Events & Lumpy Expenditures/Income: Economic life is punctuated by unexpected, significant financial events. The Poisson jump processes directly model these:

* $\text{d}{J}_{\text{exp}}\left(t\right)$: Captures lumpy, unforeseen expenditures (e.g., medical emergencies, urgent repairs) which are a common feature of household finance. The exponential distribution for jump sizes is plausible, reflecting many small unexpected costs and fewer very large ones.

* $\text{d}{J}_{\text{inc}}\left(t\right)$: Models rare positive windfalls (e.g., bonuses, small inheritances, tax refunds). The lognormal distribution allows for a skewed distribution where large positive jumps are possible but less frequent, a common characteristic of such income events.

- Asymmetry of Shocks: The typical parameterization ( ${\lambda }_{\text{exp}}>{\lambda }_{\text{inc}}$) reflects the common experience that unexpected negative financial shocks tend to be more frequent than unexpected positive windfalls for most individuals.

In summary, the model combines a deterministic component (salary) with various stochastic elements (mean-reversion to an adaptive average, persistent smooth fluctuations, and discrete jumps) that mirror key aspects of economic theory concerning income dynamics, expectation formation, consumption behavior, and the impact of uncertainty and rare events on individual financial well-being.

• Psychological Theories: Human lives are not static; they are characterized by continuous change, unexpected events, and internal fluctuations. [1] explicitly described happiness as a “stochastic phenomenon”. Psychological states are rarely stable and are subject to numerous, often unobservable, influences.

• Model Consistency: The model’s foundation on stochastic differential equations (SDEs) intrinsically captures this dynamism.

- Fractional Brownian Motion (fBm): The use of $\text{d}{B}_H\left(t\right)$ terms in most SDEs, where $H$ represents a Hurst parameter typically > 0.5, models persistent randomness—where past fluctuations have a lingering influence on future ones, creating smoother, more correlated paths than standard Brownian motion. This can represent mood persistence or slowly evolving unobserved factors.

- Jump Processes ( $dJ\left(t\right)$): The inclusion of jump processes (e.g., $\text{d}{J}_{\text{health\_shocks}}\left(t\right)$ in Equation 7 for physical health) allows for the modeling of significant, abrupt life events (e.g., sudden illness, job loss, unexpected windfall), which are known to have immediate and sometimes lasting impacts on well-being.

In conclusion, the proposed stochastic model of interacting life domains and happiness trajectories integrates multiple key insights from psychological theories of well-being, adaptation, stress, motivation, and the inherent dynamism of human life. By translating these psychological constructs into a system of SDEs, the model offers a quantitative framework to explore the complex, emergent patterns of happiness over time, grounded in established psychological understanding.

The target happiness level, ${\mu }_{H_{\text{target}}}\left(t\right)$, is defined as a weighted sum of various life domains (Equation):

$\begin{array}{c}{\mu }_{H_{\text{target}}}\left(t\right)={\lambda }_{const}+{\lambda }_I{f}_{IH}\left(I\left(t\right)\right)+{\lambda }_{Hm}{H}_m\left(t\right)+{\lambda }_{Hp}{H}_p\left(t\right)\\ -{\lambda }_S\text{Stress}\left(t\right)+{\lambda }_{C}C\left(t\right)+{\lambda }_{R}R\left(t\right)+{\lambda }_{V}V\left(t\right)\end{array}$ (17)

The core idea is to treat the weights $\lambda =\left({\lambda }_I,{\lambda }_{Hm},{\lambda }_{Hp},{\lambda }_S,{\lambda }_C,{\lambda }_R,{\lambda }_V\right)$ as learnable parameters. We aim to find a ${\lambda }^{\text{*}}$ that maximizes the expected long-term average happiness, $E\left[\bar{H}\right]$, where $\bar{H}=\frac{1}{T_{\text{final}}}{\displaystyle {\int }_0^{T_{\text{final}}}H\left(t\right)\text{d}t}$.

Since $H\left(t\right)$ is a stochastic process driven by the SDE system, its evolution depends on $\lambda$ through ${\mu }_{H_{\text{target}}}\left(t\right)$. The inherent randomness and path-dependency make direct analytical optimization intractable, motivating the use of simulation-based stochastic optimization.

The endeavor to identify and prioritize factors that enhance human well-being and fulfillment is a central theme across psychology, particularly within positive psychology and intervention research [33] [34] . While individuals do not typically engage in formal mathematical optimization of “happiness coefficients”, their choices and efforts to improve their lives often reflect an implicit process of prioritizing certain life domains or activities believed to yield greater well-being.

Our proposed optimization of the $\lambda$ coefficients, which weight the contribution of various life domains to the overall happiness target ${\mu }_{H_{\text{target}}}\left(t\right)$, is conceptually aligned with these broader efforts in several ways:

1) Focus on Key Life Domains: The model’s structure, based on distinct yet interconnected life domains (e.g., health, relationships, income, purposeful engagement), mirrors multi-component theories of well-being [19] [21] [2] . These theories inherently suggest that nurturing these domains is crucial for flourishing.

2) Implicit Weighting in Interventions: Psychological literature abounds with evidence for interventions targeting specific domains—such as fostering social connections [35] , cultivating gratitude, or engaging in meaningful activities [32] —as effective means to boost happiness. This implicitly assigns a significant “weight” or leverage to these factors. Our optimization seeks to make such weightings explicit within the model’s framework.

3) Resource Allocation and Goal Pursuit: Individuals continuously make decisions about allocating their finite resources (time, energy, attention) towards goals they believe will lead to a more satisfying life [36] . The optimization of $\lambda$ coefficients can be viewed as a computational analogue to an idealized process of learning which areas of life (domains) offer the greatest return in terms of sustained well-being, given the system’s dynamics.

4) Personalization and Context-Dependence: The strategy of running the optimization under different initial conditions (e.g., varying levels of income or relationship quality) resonates with the “person-activity fit” model in positive psychology [37] . This model posits that the effectiveness of well-being strategies can vary depending on individual characteristics and circumstances. Our approach allows for exploring how the “optimal” (model-derived) focus might shift based on an individual’s starting point or simulated life context.

Thus, while the methodology of using Stochastic Gradient Descent with simplex projection to find optimal $\lambda$ coefficients is a formal computational abstraction, the underlying goal—to understand how different life factors contribute to and can be leveraged for enhanced well-being—is deeply consonant with established psychological inquiry. The optimization provides a quantitative, model-based lens to explore questions of balance, prioritization, and the pursuit of a fulfilling life.

We employ a simulation-based Stochastic Gradient Descent (SGD) approach to find the vector of sensitivity coefficients $\lambda =\left({\lambda }_I,{\lambda }_{Hm},{\lambda }_{Hp},{\lambda }_S,{\lambda }_C,{\lambda }_R,{\lambda }_V\right)$ that maximizes the expected long-term average happiness, $J\left(\lambda \right)=E\left[\bar{H}\left(\lambda \right)\right]$. The coefficients ${\lambda }_i$ (for $i\in \left\{I,Hm,Hp,C,R,V\right\}$) represent the positive contributions of their respective domains, while ${\lambda }_S$ represents the magnitude of the negative impact of stress. To ensure these coefficients are interpretable as relative positive contributions and to maintain stability, we impose constraints:

1) Non-negativity: ${\lambda }_i\ge 0$ for all $i$.

2) Sum-to-one for positive contributors: ${\displaystyle {\sum }_{i\in \left\{I,Hm,Hp,C,R,V\right\}}{\lambda }_i}=1$.

The coefficient ${\lambda }_S$ (magnitude of stress impact) is optimized as a non-negative value but is not included in the sum-to-one constraint of the positive contributors. The constant term ${\lambda }_{const}$ is typically kept fixed or optimized separately.

The SGD algorithm iteratively updates the parameters $\lambda$ in the direction of an estimated gradient of $J\left(\lambda \right)$:

${{\lambda }^{\prime }}_{k+1}={\lambda }_k+\eta {\nabla }_{\lambda }\bar{H}\left({\lambda }_k;{\omega }_k\right)$ (18)

where ${\lambda }_k$ is the vector of weights at iteration $k$, $\eta$ is the learning rate, and ${\nabla }_{\lambda }\bar{H}\left({\lambda }_k;{\omega }_k\right)$ is an estimate of the gradient of the average happiness $\bar{H}$ obtained from one or more simulation runs (using stochastic path(s) ${\omega }_k$) with the current parameters ${\lambda }_k$.

The gradient ${\nabla }_{\lambda }\bar{H}$ is estimated using a finite difference method. For each component ${\lambda }_j$ in $\lambda$:

$\frac{\partial \bar{H}}{\partial {\lambda }_j}\approx \frac{\bar{H}\left({\lambda }_k\text{with}{\lambda }_j+\delta ;{{\omega }^{\prime }}_k\right)-\bar{H}\left({\lambda }_k;{{\omega }^{″}}_k\right)}{\delta }$ (19)

where $\delta$ is a small perturbation, and ${{\omega }^{\prime }}_k,{{\omega }^{″}}_k$ represent (potentially different) stochastic paths for the perturbed and baseline simulations. To improve gradient stability, $\bar{H}$ at each point can be averaged over multiple simulation runs.

To enforce the constraints, the updated vector ${{\lambda }^{\prime }}_{k+1}$ from Equation 18 is then projected onto the feasible set. Specifically:

1) The sub-vector ${{\lambda }^{\prime }}_{pos,k+1}$ corresponding to the positively contributing domains ( ${\lambda }_I,{\lambda }_{Hm},{\lambda }_{Hp},{\lambda }_C,{\lambda }_R,{\lambda }_V$) is projected onto the standard probability simplex (denoted $\text{Δ}$). This operation, ${\text{Proj}}_{\text{Δ}}(\cdot )$, ensures these components are non-negative and sum to 1.

${\lambda }_{pos,k+1}={\text{Proj}}_{\text{Δ}}\left({{\lambda }^{\prime }}_{pos,k+1}\right)$ (20)

2) The updated stress magnitude coefficient, ${{\lambda }^{\prime }}_{S,k+1}$, is ensured to be non-negative:

${\lambda }_{S,k+1}=\text{max}\left(0,{{\lambda }^{\prime }}_{S,k+1}\right)$ (21)

The vector ${\lambda }_{k+1}$ is then formed by combining ${\lambda }_{pos,k+1}$ and ${\lambda }_{S,k+1}$. The projection onto the simplex is a standard operation in optimization, often achieved efficiently using sorting-based algorithms (see Appendix for details). This iterative process of gradient estimation, update, and projection is repeated until convergence or for a fixed number of iterations. The average overall happiness $\bar{H}$ using the optimized ${\lambda }^{*}$ is then evaluated by averaging over multiple independent simulation runs.

The resulting optimized ${\lambda }^{*}$ vector provides insights into the relative importance of different life domains for maximizing long-term happiness within the confines of the model and the specific simulation setup (initial conditions, other parameters).

• Identifying Key Drivers: Domains with larger (positive for positive influences like $H_m,R,V,C,I$; negative for $\text{Stress}$) magnitudes in ${\lambda }^{*}$ would be identified by the model as more critical levers for achieving higher average happiness.

• Sensitivity to Initial Conditions/Scenarios: The optimization can be run starting from different initial states for the life domains (e.g., low income, poor mental health vs. high income, good mental health) or under different persistent environmental conditions (e.g., high baseline societal stress).

- For an individual starting with low income, the optimized ${\lambda }_I^{\text{*}}$ might be relatively higher, suggesting that, from that starting point, focusing on improving income (as modeled) yields significant happiness gains.

- For an individual with chronic health issues, ${\lambda }_{Hp}^{\text{*}}$ might be less prominent if improvements are difficult to achieve within the model, and perhaps ${\lambda }_C^{\text{*}}$ (coping) or ${\lambda }_{Hm}^{\text{*}}$ (mental health resilience) might become more important.

• Guidance for a “Well-Balanced Life”: The optimized ${\lambda }^{*}$ does not prescribe direct actions but can highlight areas of focus. If, for instance, ${\lambda }_R^{\text{*}}$ (relationships) consistently emerges as very high across various scenarios, it reinforces the psychological importance of social connections. A “well-balanced” set of ${\lambda }_i$ values (where no single ${\lambda }_i$ is zero, and several have significant magnitudes) would suggest that attention to multiple life domains is crucial for optimal well-being, aligning with holistic views of a fulfilling life.

• What to “Pay Attention To”: The optimization can reveal non-obvious interactions. For example, if investing in $V\left(t\right)$ (Voluntary Actions) strongly boosts $H_m\left(t\right)$ (Mental Health), which in turn has a high ${\lambda }_{Hm}^{\text{*}}$, then the indirect path through $V\left(t\right)$ might be more important than its direct ${\lambda }_V^{\text{*}}$ might suggest. The SGD process implicitly captures these systemic effects.

It is crucial to interpret the results of such an optimization with caution:

• Model Dependence: The “optimal” ${\lambda }^{*}$ is entirely dependent on the model’s structure, assumptions, and the chosen parameter values (other than $\lambda$). It reflects what is optimal within the simulated world, not necessarily in reality.

• Simplification of Choice: Real-life decisions are far more complex than adjusting abstract weights. Individuals don’t directly “choose” their ${\lambda }_i$; they make choices that affect the state of their life domains. This optimization is a high-level abstraction.

• Ethical Considerations: The notion of “optimizing happiness” can be misconstrued. The goal here is analytical insight, not a reductionist recipe for living.

• Computational Cost: Estimating gradients through multiple simulations per SGD step can be extremely computationally expensive.

• Local Optima: SGD can get stuck in local optima, especially in complex, non-convex landscapes.

• Definition of “Average Happiness”: Maximizing average happiness might neglect other important aspects like minimizing volatility or avoiding extremely low happiness states.

Despite these limitations, performing this optimization can provide valuable insights into the model’s behavior and generate hypotheses about the relative importance of different life factors under various simulated circumstances, prompting reflection on pathways to a more fulfilling life as conceptualized by the model. The resulting “optimized average happiness” serves as a benchmark achievable within the model’s framework given a flexible weighting scheme.