Stochastic Calculus Notes 07 | SDEs, Feynman–Kac and Stopping Times

21 Oct 2026 • Ziyuan Zhang

Overview

The last part zooms out. Stock prices, interest rates and volatilities are all modelled as solutions of stochastic differential equations. We solve the Ornstein–Uhlenbeck SDE explicitly, meet the CIR process, and see why every SDE solution is Markov. Then Feynman–Kac closes the loop of the whole course: a conditional expectation of a payoff and a PDE are the same object — Black–Scholes is just one instance. We finish with dividend-paying stocks, stopping times and the optional sampling theorem.

1. Stochastic differential equations

Fix \((\Omega,\mathcal F,\mathbb F,\mathbb P)\) with a Brownian motion \(W\) and \(\mathbb F=\mathbb F^W\). An SDE is an equation of the form

\[dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dW_t,\qquad X_0=x,\]

where \(\mu:[0,\infty)\times\mathbb R\to\mathbb R\) is the drift function and \(\sigma:[0,\infty)\times\mathbb R\to\mathbb R\) the diffusion (dispersion) function. Unlike an Itô process, the coefficients depend on the unknown \(X\) itself.

A solution is a process \(X\) such that for all \(t\ge0\)

\[\mathbb P\Big[X_t=x+\int_0^t\mu(u,X_u)\,du+\int_0^t\sigma(u,X_u)\,dW_u\Big]=1,\]

with all integrals well defined.

GBM is an SDE. With \(\mu(t,x)=\mu x\) and \(\sigma(t,x)=\sigma x\): \(dX_t=\mu X_tdt+\sigma X_tdW_t\), solved by \(X_t=x\,e^{(\mu-\frac12\sigma^2)t+\sigma W_t}\) — the Black–Scholes stock.

Existence and uniqueness. Not every SDE has a solution, and few have closed-form ones. Under mild conditions one can prove existence and uniqueness; the key property is that \(\mu\) and \(\sigma\) are Lipschitz in space,

\[\vert f(x)-f(y)\vert\le K\vert x-y\vert,\]

just as for ordinary differential equations. In this course we always assume solutions exist.

2. The Ornstein–Uhlenbeck process

\[dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t,\qquad \kappa>0 .\]

\(\theta\): long-run mean; \(\kappa\): speed of mean reversion; \(\sigma\): volatility. When \(X\) is above \(\theta\) the drift pulls it down, and vice versa — a spring with noise.

Solving it. (1) Assume a solution exists and find its form; (2) check it. Multiply by the integrating factor \(e^{\kappa t}\) and apply Itô (product rule with a deterministic factor):

\[d(e^{\kappa t}X_t)=e^{\kappa t}\big(\kappa X_t\,dt+\kappa(\theta-X_t)\,dt+\sigma\,dW_t\big)=e^{\kappa t}\big(\kappa\theta\,dt+\sigma\,dW_t\big).\]

Integrate: \(e^{\kappa t}X_t=x+\theta(e^{\kappa t}-1)+\int_0^te^{\kappa u}\sigma\,dW_u\), so

\[X_t=\theta+e^{-\kappa t}\Big((x-\theta)+\int_0^te^{\kappa u}\sigma\,dW_u\Big).\]

Check: differentiating, \(dX_t=-\kappa e^{-\kappa t}\big((x-\theta)+\int_0^te^{\kappa u}\sigma dW_u\big)dt+\sigma\,dW_t=\kappa(\theta-X_t)dt+\sigma dW_t\). ✓

Distribution. The integrand \(e^{\kappa u}\sigma\) is deterministic, so by Part 04 the stochastic integral is Gaussian, with variance from the Itô isometry:

\[e^{-\kappa t}\int_0^te^{\kappa u}\sigma\,dW_u\sim N\big(0,\sigma_t^2\big),\qquad \sigma_t^2=\sigma^2e^{-2\kappa t}\int_0^te^{2\kappa u}\,du=\frac{\sigma^2}{2\kappa}\big(1-e^{-2\kappa t}\big).\]

Hence

\[X_t\sim N\Big(\theta+e^{-\kappa t}(x-\theta),\ \frac{\sigma^2}{2\kappa}\big(1-e^{-2\kappa t}\big)\Big)\ \xrightarrow{t\to\infty}\ N\Big(\theta,\ \frac{\sigma^2}{2\kappa}\Big).\]

The starting point is forgotten exponentially fast; \(\theta\) is the long-run mean and \(\sigma^2/(2\kappa)\) the long-run variance.

Time-dependent coefficients. With non-random \(\kappa(t),\theta(t),\sigma(t)\) (they may vary with \(t\) but not with \(\omega\)), \(dX_t=\kappa(t)\big(\theta(t)-X_t\big)dt+\sigma(t)dW_t\) is solved by

\[X_t=e^{-\int_0^t\kappa(u)du}\Big(x+\int_0^te^{\int_0^u\kappa(v)dv}\kappa(u)\theta(u)\,du+\int_0^te^{\int_0^u\kappa(v)dv}\sigma(u)\,dW_u\Big).\]

3. The Cox–Ingersoll–Ross process

\[dX_t=\kappa(\theta-X_t)\,dt+\xi\sqrt{X_t}\,dW_t,\qquad X_0=x>0 .\]

Same mean reversion as OU, but the instantaneous conditional variance is \(\xi^2X_t\,dt\): noise shrinks as \(X\) approaches \(0\). The solution must stay positive for \(\sqrt{X_t}\) to make sense; strict positivity holds if \(\kappa\theta\ge\frac12\xi^2\) (the Feller condition). There is no closed-form solution, but a solution exists and the distribution of \(X_t\) can be computed. It is a standard interest-rate and variance model.

4. The Markov property

Solutions of \(dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\) are Markov: to evaluate the future of the process at time \(t\), knowing \(X_t\) is enough — all other information in \(\mathcal F_t\) is irrelevant. In practice:

\[E[g(X_u)\mid\mathcal F_t]=E[g(X_u)\mid X_t],\qquad E\Big[e^{-\int_t^Tg(X_u)du}\,\Big\vert\,\mathcal F_t\Big]=E\Big[e^{-\int_t^Tg(X_u)du}\,\Big\vert\,X_t\Big].\]

Why: at time \(t\), the next step \(dX_t\) is determined by \(X_t\) (through \(\mu\) and \(\sigma\)) and by \(dW_t\), which is independent of \(\mathcal F_t\). Expanding \(E[g(X_t+dX_t)\mid\mathcal F_t]\) and \(E[g(X_t+dX_t)\mid X_t]\) gives the same answer, step after step.

5. Feynman–Kac

Let \(dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\), let \(g\) be a payoff and \(r(t,x)\) a discount rate. Then

\[f(t,x):=E\Big[e^{-\int_t^Tr(u,X_u)du}\,g(X_T)\,\Big\vert\,X_t=x\Big],\qquad t\le T,\ x\in\mathbb R,\]

corresponds one-to-one with the PDE

\[\begin{cases}0=\Big(f_t+\mu f_x+\tfrac12\sigma^2f_{xx}-rf\Big)(t,x),&t<T,\ x\in\mathbb R,\\[4pt] f(T,x)=g(x).\end{cases}\]

Without discounting, \(u(t,x)=E[g(X_T)\mid X_t=x]\) solves \(0=u_t+\mu u_x+\frac12\sigma^2u_{xx}\) with \(u(T,x)=g(x)\).

Why it is true (one line). By Markov, \(e^{-\int_0^tr\,du}f(t,X_t)=E\big[e^{-\int_0^Tr\,du}g(X_T)\mid\mathcal F_t\big]\) is a martingale. Apply Itô to it: the \(dt\) coefficient is \(e^{-\int_0^tr}\big(f_t+\mu f_x+\frac12\sigma^2f_{xx}-rf\big)\), and a martingale has no \(dt\) term.

Black–Scholes is a special case. Take \(X=S\) under \(\mathbb Q\), i.e. \(\mu(t,s)=rs\), \(\sigma(t,s)=\sigma s\), constant \(r\), and \(g(s)=(s-K)^+\). The PDE becomes exactly the Black–Scholes PDE of Part 05, and the expectation becomes the risk-neutral formula of Part 06. The two halves of the course — PDEs and expectations — are the same thing.

6. Dividend-paying stocks

If the stock pays a continuous dividend at rate \(a_t\), its price drops by the dividend:

\[\frac{dS_t}{S_t}=\mu_t\,dt+\sigma_t\,dW_t-a_t\,dt .\]

Wealth. Holding \(\Delta_t\) shares over \([t,t+dt]\), we also receive dividends \(\Delta_tS_ta_t\,dt\):

\[X_{t+dt}=\Delta_tS_{t+dt}+\Delta_tS_ta_t\,dt+(X_t-\Delta_tS_t)\frac{B_{t+dt}}{B_t},\] \[\begin{aligned} dX_t&=\Delta_t\,dS_t+\Delta_tS_ta_t\,dt+(X_t-\Delta_tS_t)r_t\,dt\\ &=\Delta_tS_t\big((\mu_t-a_t)dt+\sigma_tdW_t\big)+\Delta_tS_ta_t\,dt+(X_t-\Delta_tS_t)r_t\,dt\\ &=r_tX_t\,dt+\Delta_t\sigma_tS_t\big(dW_t+\theta_t\,dt\big),\qquad\theta_t=\frac{\mu_t-r_t}{\sigma_t}. \end{aligned}\]

The wealth dynamics are the same as without dividends (the dividend you receive offsets the price drop), but the stock dynamics change. So European option prices do change, because the stock price is different. Under \(\mathbb Q\),

\[dX_t=r_tX_t\,dt+\Delta_tS_t\sigma_t\,dW^{\mathbb Q}_t,\qquad V_t=E^{\mathbb Q}\Big[e^{-\int_t^Tr_udu}V_T\,\Big\vert\,\mathcal F_t\Big],\]

and under \(\mathbb Q\) the stock now drifts at \(r_t-a_t\). In practice, write the payoff as a function of \(S_T=S_t\cdot\frac{S_T}{S_t}\), freeze \(S_t=s\), and compute — exactly as in the Black–Scholes derivation.

7. Stopping times

A random time \(\tau\) is a stopping time if

\[\{\omega:\tau(\omega)\le t\}\in\mathcal F_t\quad\text{for all }t,\]

i.e. at every time you can tell from current information whether you have already stopped. Example: a non-random time \(\tau\equiv t_0\), for which \(\{\tau\le t\}\) is \(\Omega\) if \(t_0\le t\) and \(\varnothing\) otherwise.

Hitting times. \(\tau(\omega)=\inf\{t:X_t(\omega)=\lambda\}\), the first time \(X\) hits level \(\lambda\). For adapted \(X\) with continuous paths, hitting times are stopping times. (By contrast, “the last time \(X\) hits \(\lambda\) before \(T\)” is not: you need the future to know it.)

Stopped values and stopped processes. \(X_\tau\) is the random variable \(\omega\mapsto X_{\tau(\omega)}(\omega)\) — e.g. \(X_1(\omega)\) if \(\tau(\omega)=1\), \(X_3(\omega)\) if \(\tau(\omega)=3\). The stopped process \(X^\tau\) is

\[X^\tau_t(\omega):=X_{t\wedge\tau(\omega)}(\omega):\]

it follows \(X\) until \(\tau\) and then stays flat.

8. The optional sampling theorem

Theorem (OST). Let \(M\) be a martingale and \(\alpha\le\tau\) two bounded stopping times. Then

\[E[M_\tau\mid\mathcal F_\alpha]=M_\alpha .\]

Corollaries for a bounded stopping time \(\tau\):

the stopped process \(t\mapsto M_{t\wedge\tau}\) is a martingale;
\(E[M_\tau]=M_0\);
if \(\tau\le T\), the price of a claim at the random time \(\tau\) is \(V_\tau=E^{\mathbb Q}\big[e^{-r(T-\tau)}V_T\mid\mathcal F_\tau\big]\) — the pricing formula of Part 06 with \(t\) replaced by \(\tau\).

Intuition: a fair game stays fair even if you decide when to look using a rule that does not peek into the future. Boundedness matters: “play until you are ahead” is a stopping time, but an unbounded one, and with it \(E[M_\tau]>M_0\) is possible.

Series contents

01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
02 · Random Walks, Quadratic Variation and Brownian Motion
03 · Stochastic Integration
04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
05 · Black–Scholes: Replication, the PDE and the Greeks
06 · Risk-Neutral Pricing, Change of Measure and Girsanov
07 · SDEs, Feynman–Kac and Stopping Times (this post)

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This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.