a stochastic calculus crash course for Financial Engineering. Building on Kolmogorov’s axiomatic probability theory, define continuous time stochastic process, for time $\mathcal T = [0,\infty)$ a sample path is essentially $$ \{X_t : t\ge 0\} $$Brownian Motion History anecdote: In 1828, British Botanist R. Brown observed the random movement exhibited by plant pollens suspended in water. In 1905, A. Einstein proposed the first quantitative model of Brownian motion in terms of a Gaussian process, which explains some principles of molecular movements. ...

reading write-up: How Food Affects the Mind and Body December 20, 2022, The Economist the trend The take-out delivery industry is booming, paralleling the general food industry’s growth in Shanghai. The United States has been facing serious obesity problem for decades, and now East Asia, particularly Shanghai, appears to be catching up. This trend is rooted in macroeconomic factors. The general decline in growth rate has increased stress levels, causing people to lower their consumption standards. In such times, easily accessible, unhealthy, yet delicious food becomes a cheap dopamine booster. The phrase “happiness is the same price as Red-Bottoms” holds true only if one can afford them. The economic downturn has led capital to venture into the general food industry, especially in Shanghai, where bakeries are flourishing. ...

The Timeless Tchaikovsky: best known, and much loved, with his epic work of ballets and symphonies. Among one of his most cherished ballet compositions is the Miniature Overture from The Nutcracker, a masterpiece of delicate orchestration and intricate musical storytelling. For a stellar rendition of this celebrated piece, listen to the 2010 performance conducted by Sir Simon Rattle and recorded by the Berliner Philharmoniker. Listen here on YouTube or on Apple Music which provides the Lossless fidelity. ...

Gather around for some bedtime tales from the beautiful land of the free market. Here are four classical antitrust-collusion economic stories. communication problems Once upon a time, in February 1982, the CEO of American Airlines, Robert Crandall, found price competition too fierce in the airline industry. He called Howard Putnam, the CEO of Braniff Airlines, to share his thoughts. Little did he know that Putnam was recording the conversation: Crandall: I think it’s dumb as hell for Christ’s sake, all right, to sit here and pound the **** out of each other and neither one of us making a ****ing dime. ...

The past one and a half month was wild. But it’s fun and I’ve pretty much enjoyed riding the wave. Whenever I take some time off I get confused –– I’m like, cause I’ll have to do something. I’ve come to realize that being in a healthy state (both physically and mentally) is crucial for productivity. Investing time in self-care sessions and routine exercise plans turns out to be unexpectedly rewarding. So, here’s an update of the about me homepage. Pretty much a reflection –– it’s interesting to see how ideas evolve and things unfold. ...

Definition Overture is a music instrumental introduction to a ballet or opera. During the early Romantic era, composers such as Beethoven and Mendelssohn composed overtures which were independent, self-existing, instrumental, programmatic works that foreshadowed genres such as the symphonic poem. In motion pictures, an overture is a piece of music setting the mood for the film before the opening credits start. Here’s a list of my favourite overtures. Ranked w.r.t. listening frequency: ...

My impression of Shostakovich used to be largely biased with his chromaticism–the squeakily sharp Symphony No. 1 and the rich underlying emotions embodied in Symphony No. 5, as Apple Music Classical introduced His Symphony No. 5 (1937) and subsequent works express a fury and despair that he could not voice in public, although these works were often leavened with moments of sarcasm, irony, and cheeky allusions to his own private life. ...

In 1889, when Tchaikovsky was 49 years old and at the height of his fame and powers, he composed his ballet The Sleeping Beauty–based on a romantic fairy tale. An elaborate waltz is performed and Princess Aurora arrives: No.6 Grande valse villageoise (a.k.a. The Garland Waltz) Undeniably, Tchaikovsky wrote the best waltz music. This waltz is just as popular as other Tchaikovsky classics, as well as the ballet suite–people hear it in Disney movie music tracks, Tom and Jerry, restaurants, and even elevators. The original version is pure perfection, absolute perfection: ...

Reading notes of Chapter 4 of ECONOMICS OF REGULATION AND ANTITRUST (2018), by Viscusi, Harrington Jr. and Sappinton. The theories of collusive and oligopoly pricing is powered by the mathematical tools of economic analysis and game theory. An oligopoly is an industry with a small number of sellers. The criterion of small is, “whether firms take into account their rivals’ actions when deciding on their own actions”. And we model this economic setting as a game–a list of oligopolies in the market as playes, whose actions includes setting prices or production quantities, and the actions of all players jointly determine the outcome for everyone. ...

Ariana Grande graced the 2024 Met Gala not merely as a guest but as a spellbinding performer under the grandeur of the Met’s storied arches. Clothed in a sheer green gown by Maison Margiela, designed by John Galliano, Grande radiated an ethereal charm. The gown, a diaphanous masterpiece, clung to her like morning dew, catching the light in a cascade of shimmering greens that seemed to paint her in the colors of a waking forest. ...