The Multiplication Rules in Stochastic Calculus in layman’s terms…

In the realm of quantitative finance and stochastic calculus, certain multiplication rules are foundational in the modeling and analysis of random processes, such as the Brownian motion. These rules, though succinct, encapsulate profound implications that enable the derivation of complex models integral to financial engineering, option pricing, and risk management.

I. The dW^2 = dt Rule

This rule originates from the quadratic variation of Brownian motion. When we evaluate the square of the Brownian motion's increments over a brief time interval Δt, it equals the time interval itself. We express this mathematically by considering a sum of the squares of increments of the Wiener process W_t over a partition of an interval:

Sum (W_(t_(i+1)) - W_(t_i))^2 ≈ Sum Δt_i = Δt

As the partition becomes finer, this approximation turns into an equality in the limit, leading to the rule dW^2 = dt. This mathematical behavior underscores the roughness of the Brownian motion’s path. This roughness means that there is no point where a tangent can be drawn, illustrating the motion’s non-differentiability at any given point in time. Consequently, Brownian motion is nowhere differentiable with probability one, a characteristic integral to the modeling of random and unpredictable movements in various fields, including finance.

II. The Zero Rules: dt^2 = 0 and dtdW = 0

These rules stem from the infinitesimal nature of the terms involved. For dt^2 = 0, it’s understood as the second-order term that becomes infinitesimally small and is thus neglected in differential equations.

For dtdW = 0, the rationale lies in the nature of dW, which represents an increment of Brownian motion over an infinitesimal time interval dt. Since Brownian motion has a mean of zero and variance of dt, the product of dt and dW tends to zero faster than dt itself as dt approaches 0.

The statement "dtdW = 0" is a result of the mathematical properties of Brownian motion and the infinitesimal increments involved in stochastic calculus.

Brownian motion is characterized by its randomness, where each increment is random and independent of others. Its mean is zero, indicating that upward and downward movements are equally likely over an infinitesimal time increment.

The variance of the Brownian motion over an infinitesimal time interval dt is equal to dt. It signifies the dispersion or spread of the possible values of the Brownian motion, and in this case, it scales linearly with time.

When you multiply the infinitesimal time increment dt by the infinitesimal increment of the Brownian motion dW, you're multiplying two small quantities. Since dW has a mean of zero and is random, on average, the product dtdW becomes much smaller than the individual increments dt or dW.

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