In this paper, we extend the Merton model of investment in discrete time to the cases when there is a finite number of investors and the market is with frictions represented by convex penalty functions defined for each investor. In the main result of this paper, we proved the existence of optimal strategy of investment by using a new approach based on the formulation of an equivalent general equilibrium economy model.

Innovation has emerged as a critical driver of economic growth in the digital era, revolutionizing industries and reshaping economies worldwide. As traditional funding avenues continue to be dominated by risk-averse lenders and investors, entrepreneurs are increasingly turning to alternative funding mechanisms to fuel their innovative ideas and ventures. This article delves into the evolving landscape of financing innovation in the digital era, with a primary focus on the role of alternative funding mechanisms. By drawing on a comprehensive review of existing literature and empirical evidence, we aim to provide a conceptual framework that elucidates the interplay between alternative finance platforms, innovation ecosystems, and entrepreneurial activities. Moreover, this article examines the unique characteristics and challenges associated with financing innovation through crowdfunding, venture capital, and angel investing. The insights from this research contribute to a better understanding of how these alternative funding mechanisms can effectively support and catalyze innovation across industries and economies.

The research is a deep study of the CBOE Volatility Index (VIX), which represent the market sentiments of the S & P 500 Index (SPX) over a period of 33 years. It evaluates how the two indexes perform in the stock market crash and how reliable VIX is in predicting market sentiments. The research distributes the market rally in two forms, and one is a trust rally, when investors are happy to see the market in a stable form, and investors' sentiments are positive towards the stock market. Second is a fear rally, where investors' sentiments are negative, and their outlook is bearish for various reasons. It observes the pattern of rallies, like the longest trust or fear rally, in which year and its duration. The research analyzes data to reach a meaningful conclusion and various inferences that can be drawn from the study of historical data.

Pakistan is under developing country and it has an unpredictable market nature of shareholder-investors observe the company’s performance. This research could help to companies in understanding financial behavior, attitude and investors’ satisfaction in stock trade. Financial behavior is comparatively new subject in Pakistan therefore; this study has examined the financial behavior and attitude of investors. The behavioral finance that has been attempted to understand the positive experiences influences investors’ financial behavior. This study has find out that investor satisfaction is strongest in influence of positive financial behavior of investor and trader in stock trading; positive experience and brokers suggestions are strengthens the investment decision of investors and increases behavior loyalty to prefer over competitor. The main purpose of research to determine the effect of financial behavior on investors’ attitude and behavioral loyalty and investors’ satisfaction to preference over competitor. The research framework links with experiences in stock trade for positive (negative) experiences, attitude and financial behavior is developed. The research framework is measured data from sample of Karachi and Karachi Stock Exchange; the data is analyzed in smart PLS based on PLS-SEM. This study focused on trading experience with company’s active investors and traders in banking industry in Pakistan. The future research could be research in other sectors with inter-related issue of investors and traders (brokers) in stock trade. This is the first study in this research area; this study will be determine the experiences with positive (negative) financial behavior, attitude, satisfaction and behavioral loyalty of investors and traders in stock trade. Therefore, adding in this area of study which will help understanding the investors and traders attitude, preference and financial behavior in financial market.