stochastic-calculus

Friends, I will come back with a detailed latex post and commented code later tonight. But very quickly for friends trying to play with the program, I have changed the order of variables and indexing and it is different in new output variable cnOptim that comes out of Newton optimization. Please let me explain here. In the expansion of nth power of Gaussian, if there are K correlated variables, t…

My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity that the stock is log-normally distributed at time t=T (I know how to derive it). The idea is to derive the B-S-M pricing formula as the expected value of the present value of max(S_t-K, 0) using t…

This paper formalises two memoryless arithmetic processes—the Prime Gap State Machine and the Collatz odd backbone—as exact Markov chains on the six vortex residues {1,2,4,5,7,8} (mod 9) defined by the Prime Lattice Coherence Theorem (PLCT). The chains arise directly from the PLCT’s zone structure and its core theorems. For the Collatz process, the transition matrix is derived analytically from t…

This study assessed the internal and external determinants affecting junior high school mathematics performance among public secondary school students in Dalaguete I District, Cebu, during the school year 2024–2025. Anchored on Bandura’s Social Cognitive Theory and Weiner’s Attribution Theory, the study examined how psychological, environmental, school-related, teacher-related, and family-related…

Although ethnomathematics studies have been widely conducted on various local cultures in Indonesia, research that specifically discusses the identification of mathematical values in Balikpapan community culture remains limited. This study aims to identify and describe the mathematical values contained in Balikpapan culture through an ethnomathematics study. This study employed a qualitative appr…

Traditional investigations into the Collatz $(3x+1)$ problem primarily target forward path trajectories or long-term statistical graph densities, which frequently behave like pseudo-random systems. This paper introduces an exact arithmetic invariant operating within the inverse Collatz map, demonstrating that the global value-sum ratio of distinct structural subtrees converges monotonically to th…

This paper establishes a conditionally rigorous closure theorem for the low-energy spatial spectral dimension dspace = 3 in Algebraic Quan tum Morphogenesis (AQM). Earlier AQM material proposed a dimensional emergence argument based on nested Dirac recursion and a free-convolution attractor. The present paper makes a rigorous correction to that argu ment: free probability is no longer used as the…

I am new to stochastic calculus. I would like to compute the closed-form solution for $$ \int_0^t \exp \left( \alpha s - \sigma W_s \right) \; {\rm d}s \tag{1}$$ $$ \int_0^t \exp \left( \alpha s - \sigma W_s \right) \; {\rm d} W_s \tag{2} $$ which I encountered when trying to solve the following stochastic differential equation (SDE) $$ dX_t = \theta(\mu - X_t)\; dt + \sigma X_t \; dW_t $$ How to…

Liu regression, originally developed in the context of classical (crisp) statistics as a biased estimation method to mitigate multicollinearity, has not been previously extended to fuzzy regression frameworks. In this study, we propose a novel fuzzy adaptation of Liu regression, implemented within an [Formula: see text]-cut-based estimation structure. This approach provides a systematic methodolo…

We report two randomized evaluations of educational interventions targeting children in low-income neighborhoods of Delhi, India. Four math games, played in social groups, associated early emerging and universally intuitive concepts of number and geometry with the mathematical language, symbols, and operations taught in Delhi government schools. The first experiment, conducted in 231 NGO-run pres…

Statistical modelling and machine learning offer a vast toolbox of inference methods with which to model the world, discover patterns and reach beyond the data to make predictions when the truth is not certain. This concise book provides a clear introduction to those tools and to the core ideas – probabilistic model, likelihood, prior, posterior, overfitting, underfitting, cross-validation – that…

This study addressed the absence of a comprehensive bibliometric analysis of mental arithmetic research in educational psychology by systematically mapping its evolutionary trajectory over 83 years (1942–2024). Using Scopus database searches, 1,749 unique deduplicated records were retrieved, with 626 "arithmetic skills" records serving as the core dataset for descriptive bibliometric analyses of …

Abstract The free positive multiplicative Brownian motion $$(h_t)_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> is the large N limit in non…