The South African insurance sector is experiencing a positive growth as the nation is on high quality economic growth and development. However, there is little attention with regards to research on the growth analysis, hence the researchers aim to bridge the gap by analyzing the growth using a mathematically based approach. To verify the wide spread phenomenon behind insurance growth an extended Gompertz model (EGM) which is a member of the unified Richards family was used. The quantitative approach by means of functional limits, the cumulative distribution approach, initial value problem (IVP) and the qualitative derivative approach were used to fully analyze the model. We managed to derive a cumulative function was derived which can be used to estimate the number of insurance growth indicators. The maximum carrying capacity of an insurance industry was estimated using the IVP which in our case is time dependent hence does not concur with other Gompertz related works. Using both the qualitative and derivative approach, a growth function which produced the same pattern with the original Gompertz curve with K(t) as the asymptotically stable and non-constant growth limit were deduced. Hence we can conclude that the growth of insurance sectors does follow a sigmoid shape with non-constant maturity levels. Lastly, we performed a statistical analysis of the nexus between insurance sector growth and economic development using GDP and insurance indicators (net premiums) data. From the statistical analysis done the results showed a positive relationship between the two. This showed that, insurance sector indeed plays a significant role towards economic development and as such their growth patterns should be well attended.

Stock price volatility is considered the main matter of concern within the investment grounds. However, the diffusivity of these prices should as well be considered. As such, proper modelling should be done for investors to stay healthy-informed. This paper suggest to model stock price diffusions using the heat equation from physics. We hypothetically state that, our model captures and model the diffusion bubbles of stock prices with a better precision of reality. We compared our model with the standard geometric Brownian motion model which is the wide commonly used stochastic differential equation in asset valuation. Interestingly, the models proved to agree as evidenced by a bijective relation between the volatility coefficients of the Brownian motion model and the diffusion coefficients of our heat diffusion model as well as the corresponding drift components. Consequently, a short proof for the martingale of our model is done which happen to hold.

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In this paper we present time to time price dynamics associated with stock assets within stock markets. Our conjecture was that, stock prices are stochastic and time variant as such they do attain and possess different values from time to time. We then centrally aimed to model this old way phenomenon of stock price dynamics using a distinct model from the physics field so as to substantially expose the core idea of phynance on an open academic space. We used the two-form of Schrödinger wave Equation (SWE) to fully model our core study. We derived the time part and space (market) value functions for stock assets from the SWE. Meaning that, we managed to derive the time function measuring the time intervals taken by stock assets in the market space and the market value function which gives out the value of stocks without any time factor. Our results suggested that, stock price dynamics can well be modelled and presented using both time independent Schrödinger equation (TISE) and time dependent Schrödinger equation (TDSE) with traceable stock price and time changes. This supported our conjecture and our model proposition as stock prices are traditionally known to be stochastic in nature and normally they are non-stationary. As such we safely concluded that, physics indeed play important roles when modelling finance problems, hence phynance should be well credited in finance as it presents fruitful and powerful abstractions of the real time happenings in the financial markets.