This page follows the same structure as the derivatives FAQ: each question has a Short Answer, a concrete Example, and a Detailed Explanation with formulas, edge cases, and practical caveats.
1) Brownian Motion / Wiener Process
Short Answer
A continuous-time stochastic process with stationary independent Gaussian increments, continuous paths, zero mean and variance proportional to time: for $W_t, W_0 = 0$, $W_t - W_s \sim \mathcal{N}(0, t-s)$.
Example
If $W_t$ is a Brownian motion, then over 1 day ($t=1/252$), the increment $W_{t+1/252}-W_t$ is approximately $\mathcal{N}(0, 1/252)$.
Detailed Explanation
- Definitions: almost surely continuous paths, independent increments, Gaussian increments, and $W_0=0$. Scaling: $W_{ct}$ has law $\sqrt{c} W_t$.
- Construction: limit of random walk (Donsker’s theorem) or via Karhunen–Loève expansion.
- Caveats: Brownian paths are nowhere differentiable and have infinite variation on any interval; quadratic variation over $[0,T]$ equals $T$.
2) Quadratic Variation
Short Answer
Quadratic variation $[W]_T$ of Brownian motion over $[0,T]$ equals $T$; for a continuous semimartingale $X$, pathwise limit of sum of squared increments.
Example
Partition $[0,T]$ into $n$ equal subintervals; sum $(W_{t_{i+1}}-W_{t_i})^2 \to T$ as $n\to\infty$.
Detailed Explanation
- Importance: justifies Itô calculus; Itô integral uses quadratic variation term.
- For finite-variation processes the quadratic variation is 0; for diffusions it’s positive.
- In practice, realized variance estimators estimate quadratic variation from high-frequency returns, with microstructure noise caveats.
3) Itô’s Lemma
Short Answer
Stochastic chain rule: for $X_t$ an Itô process and $f(t,x)$ twice differentiable, \(df(t,X_t)=\left(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\right)dt+\sigma f_x dW_t.\)
Example
If $X_t = S_t$ follows $dS = \mu S dt + \sigma S dW$, and $f(S)=\ln S$, then
\[d\ln S=(\mu-\tfrac12\sigma^2)dt+\sigma dW.\]Detailed Explanation
- Derivation via Taylor expansion keeping up to second-order $dW^2=dt$ term and dropping higher-order terms.
- Contrasts with Stratonovich calculus where the chain rule looks like ordinary calculus; conversion: $dX\circ dW = dX dW + \tfrac12 d\langle X,W\rangle$.
- Use cases: derive Black–Scholes dynamics for log-price, change variables, compute SDEs of functions of processes.
4) Itô vs Stratonovich
Short Answer
Itô integral uses nonanticipative integrands (martingale-friendly) and has extra drift (1/2) term in chain rule; Stratonovich obeys ordinary calculus and is convenient for change of variables in physics.
Example
For SDE $dX=a(X)dt+b(X)dW$, the Stratonovich form $dX = A dt + b(X) \circ dW$ relates via $A=a+\tfrac12 b b’$.
Detailed Explanation
- The conversion formula: $a_{Itô}=a_{Strat}+\tfrac12 b b’$ (where derivative is wrt state variable).
- When modeling from physical considerations (limits of smooth noise), Stratonovich is natural; for probabilistic pricing and martingale measures, Itô is standard.
- Numerics: Euler–Maruyama approximates Itô; the midpoint (Stratonovich) scheme converges to Stratonovich.
5) Stochastic Integrals (Itô integral)
Short Answer
Integral $\int_0^T H_s dW_s$ defined as mean-square limit of nonanticipative Riemann sums; martingale with variance $\mathbb{E}[\int_0^T H_s^2 ds]$. Example
If $H_s = 1{s\le\tau}$ (stopping time), $\int_0^T H_s dW_s = W{T\wedge\tau}$.
Detailed Explanation
- Construction: approximate by simple predictable processes; isometry: $\mathbb{E}[(\int_0^T H dW)^2]=\mathbb{E}[\int_0^T H^2 ds]$.
- Conditions: H must be predictable and square-integrable. Pathwise interpretations exist (Föllmer) but require different frameworks.
- Use in finance: martingale representation theorem says local martingales can be written as stochastic integrals under completeness assumptions.
6) SDE Existence & Uniqueness
Short Answer
Lipschitz drift and diffusion in the state variable guarantee strong existence and uniqueness for SDE solutions; weaker conditions yield weak solutions.
\[df(t,X_t)=\left(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\right)dt+\sigma f_x\,dW_t.\]Example
For $dX = \mu(X) dt + \sigma(X) dW$ with Lipschitz $\mu,\sigma$, there exists a unique strong solution. For geometric Brownian motion $dS = \mu S dt + \sigma S dW$, solution is $S_t = S_0 \exp((\mu - \tfrac12 \sigma^2)t + \sigma W_t)$. For $f(S)=\ln S$,
\(d\ln S=(\mu-\tfrac12\sigma^2)\,dt+\sigma\,dW.\) Detailed Explanation
- Picard iterations give existence/uniqueness under global Lipschitz and linear growth. For CIR (square-root) process, Feller conditions ensure positivity and uniqueness despite non-Lipschitz at 0.
- Strong vs weak solutions: strong = adapted to given Brownian motion, weak = existence of probability space and Brownian motion; Girsanov change-of-measure can convert between laws.
7) Girsanov’s Theorem
Short Answer
Change of measure that shifts Brownian drift: under Radon–Nikodym density $Z_T = \exp(\int_0^T \theta_s dW_s - \frac12\int_0^T \theta_s^2 ds)$, process $W^Q_t = W_t - \int_0^t \theta_s ds$ is a Brownian motion under Q.
Example
Risk-neutral pricing: change from real-world measure P with drift $\mu$ to $Q$ with drift $r-q$ by choosing $\theta=(\mu-(r-q))/\sigma$ for geometric Brownian motion.
Detailed Explanation
- Novikov/Kazamaki conditions ensure Z is a martingale. If they fail, the candidate density may be a strict local martingale, causing measure-change problems.
- Applications: pricing, risk premia identification, and filtering/measure transforms in stochastic control.
8) Martingales & Local Martingales
Short Answer
| Martingale: $E[X_t | F_s]=X_s$ for $t\ge s$; local martingale: martingale property holds up to stopping times. Not every local martingale is a martingale. |
Example
Exponential local martingale $M_t=\exp(\sigma W_t - 0.5\sigma^2 t)$ is a martingale; but some strict local martingales (e.g., certain Bessel processes) have $E[M_t]<M_0$.
Detailed Explanation
- Distinctions matter: pricing via a candidate density that is only a local martingale can produce bubbles and asset price anomalies.
- Uniform integrability is a common sufficient condition to promote local martingales to martingales.
9) Risk-Neutral Measure & Martingale Representation
Short Answer
Under the risk-neutral measure $Q$, discounted asset prices are martingales; martingale representation says every square-integrable martingale is an Ito integral w.r.t. the driving Brownian motion (in a complete Brownian filtration).
Example
In Black–Scholes, discounted stock price $S_t e^{-rt}$ is a martingale under $Q$ and any derivative price process is a conditional expectation of its payoff.
Detailed Explanation
- Martingale representation underpins replication: the integrand gives the hedging strategy (delta is the integrand for option replication).
- In incomplete markets (extra sources of risk), the representation may require multiple Brownian motions or incomplete spanning.
10) Fokker–Planck / Kolmogorov Forward Equation
Short Answer
Describes time evolution of transition density $p(t,x)$ of an SDE: $\partial_t p = -\partial_x(\mu p) + \tfrac12 \partial_{xx}(\sigma^2 p).
Example
For GBM in log-space, the log-price density evolves as a normal distribution with drift $(\mu-\tfrac12\sigma^2)$ and variance $\sigma^2 t$.
\[Z_T=\exp\Bigl(\int_0^T\theta_s\,dW_s-\tfrac12\int_0^T\theta_s^2\,ds\Bigr).\]Detailed Explanation
- It’s the PDE dual to the backward Kolmogorov (pricing) PDE. Useful for density forecasting, likelihood-based calibration, and filtering.
- Numerical challenges: boundary conditions and absorbing/reflecting barriers must be specified carefully.
11) Backward Kolmogorov / Feynman–Kac
Short Answer
Feynman–Kac links linear parabolic PDEs to expectations of SDEs: solution $u(x,t)=\mathbb{E}^{x,t}[e^{-\int_t^T r(s,X_s) ds} g(X_T)]$.
Example
Black–Scholes PDE solution for option price is expectation of discounted payoff under risk-neutral GBM.
Detailed Explanation
- Use: derive pricing PDEs and represent solutions as risk-neutral expectations; forms the theoretical basis for Monte Carlo pricing.
- Requires sufficient regularity of coefficients; for payoffs with low regularity, weak solutions or viscosity solutions may be used.
12) Numerical Schemes: Euler–Maruyama & Milstein
Short Answer
Euler–Maruyama: basic time-discretization for SDEs (strong order 0.5); Milstein adds derivative term to achieve strong order 1 when diffusion is state-dependent.
Example
Euler step for $dX=\mu(X)dt+\sigma(X)dW$: $X_{n+1}=X_n+\mu(X_n)\Delta t+\sigma(X_n)\Delta W$.
Detailed Explanation
- Order: strong vs weak convergence. For option pricing (expectations) weak order is relevant; for path-dependent Greeks, strong order matters.
- For SDEs with non-Lipschitz coefficients (e.g., CIR), implicit schemes or specialized positivity-preserving methods are preferred.
13) Stopping Times & Optional Sampling
Short Answer
A stopping time is a random time $T$ where ${T\le t}$ is $\mathcal{F}_t$-measurable; optional sampling theorem gives conditions under which $\mathbb{E}[M_T]=\mathbb{E}[M_0]$ for a martingale $M$.
Example
Hitting time of a barrier for Brownian motion is a stopping time; optional sampling applied to bounded stopped martingales preserves expectation.
Detailed Explanation
- OST conditions: integrability and boundedness or uniformly integrable martingales; naive application can fail leading to paradoxes.
- Use in finance: pricing barrier options and proving optional decomposition for American claims.
14) Local Time
Short Answer
Local time $L_t^a$ measures how much time a semimartingale spends at level $a$; appears in Tanaka’s formula.
Example
| For Brownian motion, $L_t^0$ is the density of occupation near 0 and appears when decomposing $ | W_t | $. |
Detailed Explanation
-
Tanaka: $ X_t-a = X_0-a + \int_0^t \text{sgn}(X_s-a) dX_s + L_t^a$. - Local time is nondecreasing in $t$ and increases only when $X_t=a$. Numerically, occupation densities approximate local time.
15) Tanaka’s Formula
Short Answer
A generalisation of Ito’s lemma for convex (nonsmooth) functions like absolute value, including local time term.
Example
| For $f(x)= | x | $, Tanaka gives decomposition with local time at 0. |
Detailed Explanation
- A continuous-time stochastic process with stationary independent Gaussian increments, continuous paths, zero mean and variance proportional to time: for $W_t$, $W_0=0$, and $W_t-W_s\sim\mathcal{N}(0,t-s)$.
- Useful for reflecting boundaries and deriving properties of hitting times. Shows how non-differentiable payoffs introduce local-time terms in SDE transforms.
16) Martingale Problems (Stroock–Varadhan)
Short Answer
\[X_{n+1}=X_n+\mu(X_n)\Delta t+\sigma(X_n)\Delta W.\]Characterise diffusion processes by their generator: existence/uniqueness of solutions to martingale problem equivalent to weak uniqueness of SDE.
Example
Given generator $L=(1/2)\sigma^2(x)\partial_{xx}+\mu(x)\partial_x$, solving martingale problem constructs process with those local characteristics.
Detailed Explanation
Quadratic variation $[W]_T$ of Brownian motion over $[0,T]$ equals $T$; for a continuous semimartingale $X$ it’s the pathwise limit of sums of squared increments.
17) Jump Processes & Lévy Itô Decomposition
Short Answer
General semimartingales include jump terms; Lévy–Itô decomposition represents a Lévy process as drift + Brownian part + compensated Poisson integral for jumps.
Example
Poisson-driven jump-diffusion: $dX = \mu dt + \sigma dW + J dN_t$ with jump sizes $J$ and intensity of $N_t$.
Detailed Explanation
- Pricing: jumps break completeness; risk-neutral measure selection and option pricing need jump compensator adjustments.
- Numerics: simulate jumps with thinning or exact jump-time simulation; small-jump approximations (diffusion limits) sometimes used.
18) Change of Numeraire
Short Answer
Change measure using a positive tradable as numeraire; asset prices expressed in units of numeraire become martingales under new measure.
Example
Forward measure: use a T-bond as numeraire to price forward-start or bond options; simplifies caplet pricing.
Detailed Explanation
- Radon–Nikodym derivative involves the ratio of numeraires discounted; martingale property shifts accordingly and drift terms adjust.
19) Malliavin Calculus (brief)
Short Answer
A probabilistic calculus of variations on Wiener space; provides integration-by-parts to compute Greeks with lower variance estimators.
Example
Use Malliavin weights to produce unbiased delta estimators for path-dependent payoffs in Monte Carlo. Itô integral uses nonanticipative integrands (martingale-friendly) and has an extra drift \(\tfrac12\) term in the chain rule; Stratonovich obeys ordinary calculus and is convenient for physical limits. Detailed Explanation
- Advanced topic: requires smoothness on Wiener space; practical use in Monte Carlo Greeks (likelihood ratio vs pathwise vs Malliavin). Introduces Skorokhod integral (anticipating integrand).
20) Practical Caveats for Quant Work
Short Answer
Continuous-time theory is an idealisation; market data is discrete, has microstructure noise, jumps, and model risk. Be careful with calibration and hedging assumptions.
Example
Discrete hedging yields P&L different from continuous-time replication; realized volatility estimation is biased by microstructure noise unless corrected.
Detailed Explanation The integral \(\int_0^T H_s\,dW_s\) is defined as a mean-square limit of nonanticipative Riemann sums; it’s a martingale with variance \(\mathbb{E}[\int_0^T H_s^2\,ds]\).
- Robust hedging: model misspecification (stochastic vol, jumps) makes perfect replication impossible; consider robust bounds, super-hedging, or calibration-consistent hedges.
- Numerics and data: choose discretization, account for bid–ask, incorporate jump-robust estimators, and stress-test hedges under scenario moves.
References & Further Reading
- Karatzas & Shreve — Brownian Motion and Stochastic Calculus
- ksendal — Stochastic Differential Equations
- Shreve — Stochastic Calculus for Finance I & II
- Revuz & Yor — Continuous Martingales and Brownian Motion
Comments