Stochastic Processes — Foundations¶

Job: Fix filtrations and stopping times; characterize martingales and Brownian motion; set up the Itô integral and Lipschitz SDEs; connect Markov semigroups to generators.

Core¶

Filtrations and adapted processes¶

A filtration $(\mathcal{F}_t)_{t\ge 0}$ is an increasing family of sub-$\sigma$-algebras of $\mathcal{F}$. Process $X$ is adapted if $X_t$ is $\mathcal{F}_t$-measurable. Progressive: $(s,\omega)\mapsto X_s(\omega)$ measurable on $[0,t]\times\Omega$ w.r.t. $\mathcal{B}([0,t])\otimes\mathcal{F}_t$. Usual conditions: right-continuous $(\mathcal{F}_{t+}=\mathcal{F}_t)$ and complete.

Stopping time $\tau:\Omega\to[0,\infty]$: $\{\tau\le t\}\in\mathcal{F}_t$ all $t$ (optional time: $\{\tau<t\}\in\mathcal{F}_t$). Stopped $\sigma$-algebra $\mathcal{F}_\tau=\{A\in\mathcal{F}:A\cap\{\tau\le t\}\in\mathcal{F}_t\}$. Debut of progressive sets is a stopping time under usual conditions.

Martingales¶

Integrable adapted $M$ is a martingale if $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$ a.s. ($s\le t$). Sub/super: $\ge$/$\le$. Local martingale: stopping times $\tau_n\uparrow\infty$ with $M^{\tau_n}$ a true martingale.

Object	Role
Cadlag martingale	right-continuous paths with left limits (standard modification)
Quadratic variation $\langle M\rangle$	unique increasing predictable process s.t. $M^2-\langle M\rangle$ local martingale
UI martingale	closable: $M_t=\mathbb{E}[M_\infty\mid\mathcal{F}_t]$

Brownian motion¶

Standard BM $W$: continuous paths, $W_0=0$, independent increments $W_t-W_s\sim N(0,t-s)$. Equivalent: continuous Gaussian process with $\mathbb{E}[W_s W_t]=s\wedge t$.

Lévy characterization. Continuous local martingale $M$ with $M_0=0$ and $\langle M\rangle_t=t$ is a BM (w.r.t. its filtration).

Quadratic variation: $\langle W\rangle_t=t$ in probability (along refining partitions). Paths a.s. nowhere differentiable; Hölder-$\gamma$ for all $\gamma<\tfrac12$, not $\tfrac12$.

Markov processes¶

Markov property: $\mathbb{E}[f(X_{t+s})\mid\mathcal{F}_t]=P_s f(X_t)$ a.s. for bounded measurable $f$, with transition semigroup $(P_t)$. Strong Markov: same at stopping times $\tau<\infty$.

Feller semigroup: $P_t:C_0(E)\to C_0(E)$, strongly continuous. Generator $$ Af=\lim_{t\downarrow 0}\frac{P_tf-f}{t}\quad\text{in }|\cdot|_\infty, $$ domain $D(A)\subset C_0$. For BM on $\mathbb{R}^d$: $A=\tfrac12\Delta$ on suitable domain. Hille–Yosida links generators of contraction semigroups to resolvents (FA language).

Infinitesimal generator of diffusion $dX=b\,dt+\sigma\,dW$: $$ Af=b\cdot\nabla f+\tfrac12\operatorname{tr}(\sigma\sigma^\top D^2 f) $$ on $C_c^2$ (under growth/Lipschitz hypotheses).

Itô integral (setup)¶

For simple progressive $H=\sum H_i 1_{(t_i,t_{i+1}]}$, $$ \int_0^t H_s\,dW_s:=\sum H_i\bigl(W_{t_{i+1}\wedge t}-W_{t_i\wedge t}\bigr). $$ Itô isometry: $\mathbb{E}\bigl[\bigl(\int_0^T H\,dW\bigr)^2\bigr]=\mathbb{E}\int_0^T H_s^2\,ds$ for $H\in L^2([0,T]\times\Omega)$. Extend by density to progressive $H$ with $\mathbb{E}\int H^2<\infty$; further localize to $$ H\in L^2_{\mathrm{loc}}\quad\bigl(\int_0^t H^2\,ds<\infty\text{ a.s. all }t\bigr). $$ Result: continuous local martingale; true martingale under $L^2$ integrability of $H$.

Itô formula (1D): $X_t=X_0+\int b+\int\sigma\,dW$, $f\in C^2$, $$ df(X)=f'(X)\,dX+\tfrac12 f''(X)\,d\langle X\rangle,\qquad d\langle X\rangle=\sigma^2\,dt. $$ Multidimensional / jump extensions: use full Hessian and jump measures (semimartingale calculus).

SDEs: Lipschitz existence¶

$$ dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dW_t,\qquad X_0=\xi. $$ Global Lipschitz + linear growth in $x$ (uniform in $t$ on compact time): unique strong solution; continuous paths; continuous dependence on $\xi$ in $L^2$. Proof idea: Picard iteration in Banach space of adapted continuous processes with sup-$L^2$ norm; contraction on small intervals, patch.

Weak existence under continuous coefficients + linear growth: tightness + martingale problem (Stroock–Varadhan). Pathwise uniqueness $\Rightarrow$ unique strong solution when weak existence holds (Yamada–Watanabe).

Martingale problem: law of $X$ such that $f(X_t)-\int_0^t Af(X_s)\,ds$ is a martingale for all $f\in C_c^2$. Well-posedness of the martingale problem $\Leftrightarrow$ uniqueness in law for the diffusion.

See it¶

Geometric Brownian motion $dS=\mu S\,dt+\sigma S\,dW$: $$ S_t=S_0\exp\bigl((\mu-\tfrac12\sigma^2)t+\sigma W_t\bigr) $$ by Itô on $\log S$. The $-\tfrac12\sigma^2$ drift correction is the canonical “Itô vs Stratonovich / ordinary calculus” signature. Under risk-neutral $\mu=r$, $e^{-rt}S_t$ is a martingale (with integrability).

Exit time of BM. $\tau_a=\inf\{t:W_t=a\}$; Laplace transform $\mathbb{E}_x[e^{-\lambda\tau_a}]=e^{-\sqrt{2\lambda}|a-x|}$. Optional stopping on $e^{\theta W_t-\theta^2 t/2}$ yields hitting distributions.

Check¶

Show that $W_t^2-t$ is a martingale (compute conditional expectation).

$\mathbb{E}[W_t^2-t\mid\mathcal{F}_s]=\mathbb{E}[(W_s+(W_t-W_s))^2-t\mid\mathcal{F}_s]=W_s^2+2W_s\cdot 0+(t-s)-t=W_s^2-s$ using independent increments and $\mathbb{E}[(W_t-W_s)^2]=t-s$.

Why can a continuous local martingale of finite variation be only constant?

FV continuous $\Rightarrow$ quadratic variation $0$; for local martingales $\langle M\rangle$ measures “martingale noise.” $M^2-\langle M\rangle$ local martingale with $\langle M\rangle=0$ forces $M$ constant (use localization + $L^2$ identity).

Pitfalls¶

Stratonovich $\circ$ vs Itô: Stratonovich obeys ordinary chain rule; conversion $b_{\mathrm{Itô}}=b_{\mathrm{Strat}}+\tfrac12\sigma\partial_x\sigma$ in 1D. Numerics and physics often mean Stratonovich.
Local martingale exponential $\mathcal{E}(M)=\exp(M-\tfrac12\langle M\rangle)$ need not be a true martingale (Novikov / Kazamaki conditions).
Feller generator domain is subtle: $C_c^2$ is a core for many diffusions, not the full domain; boundary conditions encode domain for reflecting/absorbing processes.
Weak uniqueness without strong: Tanaka $dX=\operatorname{sign}(X)\,dW$ has unique law but no strong solution.

Refs¶

Karatzas–Shreve; Revuz–Yor; Øksendal, Stochastic Differential Equations; Stroock–Varadhan (martingale problems); Ethier–Kurtz (Markov processes).

Next¶

Martingales · Index · Probability · SPDE · PDE · Functional analysis