Wednesday, December 24, 2025

x̄ - > Limits and Stochastic Theorem in Finance

Understanding Limits: The Art of Approaching

What a Limit Really Means

In calculus, we write

\lim_{x \to a} f(x) = L

to say something subtle but profound: as x moves arbitrarily close to a (from either side), the values of f(x) move arbitrarily close to L.

Why This Matters

The value at the point may differ.
Arrival is irrelevant; approach is everything.

Limits as the Foundation of Calculus

Derivatives

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Integrals

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x

Worked Visual Examples

A Hole in the Graph

f(x)=\frac{x^2-1}{x-1}

One-Sided Limits

g(x)=\frac{|x|}{x}

From Deterministic to Stochastic

So far, limits have described how smooth functions behave as we zoom in: derivatives as velocity, integrals as accumulated area. In modern finance, limits also describe how random price movements emerge from many tiny shocks.

A simple model of a stock price $S(t)$ assumes its relative change over a very short time step $\Delta t$ has two parts:

\frac{\Delta S}{S} \approx \mu\,\Delta t + \sigma\,\Delta W,

where $\mu$ is the drift, $\sigma$ is the volatility, and $\Delta W$ is a small random shock coming from a Brownian motion $W(t)$.

Brownian Motion as a Limit

Brownian motion $W(t)$ itself can be defined as the limit of a scaled symmetric random walk: many tiny up/down moves of size $\pm\sqrt{\Delta t}$, each occurring with probability $1/2$.

W(t) = \lim_{n \to \infty} \sum_{k=1}^{\lfloor t / \Delta t_n \rfloor} \sqrt{\Delta t_n}\,X_k,

where $X_k$ are independent random variables taking values $+1$ or $-1$. This is a probabilistic analogue of how a Riemann sum converges to an integral as the mesh size goes to zero.

Stochastic Integrals as Random Limits

In calculus, the definite integral is the limit of Riemann sums. In stochastic calculus, integrals with respect to Brownian motion are defined in a similar way, but now the limit takes place in a probabilistic sense.

\int_0^T H(t)\,dW(t) = \lim_{n \to \infty} \sum_{k=0}^{n-1} H(t_k)\,\bigl(W(t_{k+1}) - W(t_k)\bigr),

where $(t_k)$ is a partition of $[0,T]$ whose mesh size tends to zero. This is the Itô integral, the central object used to model continuous‑time trading strategies and hedging in mathematical finance.

Geometric Brownian Motion for Prices

Putting these ideas together, a standard model for a stock price is the stochastic differential equation

dS_t = \mu S_t\,dt + \sigma S_t\,dW_t,

whose solution is the geometric Brownian motion

S_t = S_0 \exp\!\Bigl( \bigl(\mu - \tfrac{1}{2}\sigma^2\bigr)t + \sigma W_t \Bigr).

Here, the deterministic limit ideas from calculus and the random limits from probability meet: prices evolve as the exponential of a drift term plus a limit of many small random shocks, encoded by $W_t$.

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