M.3.6 Advanced Extensions

Basic Prerequisites: 05 · Martingales.

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1. Intuition & Practical Objective

These are the foundations of the foundations: the results that make “expectation” and “conditioning” well-defined and the theorems that let you move limits inside expectations. Radon–Nikodym turns one measure into another as a density - the mathematical license for the risk-neutral change of measure. The convergence theorems (Monotone, Fatou, Dominated) are why you may stop a martingale and let $t\to \infty$, pass a limit under $\int$, or claim $\frac{1}{n}\sum f(U_i)\to \mathbb{E}[f]$ almost surely (Glasserman §1.1). The Lebesgue/standard machine is how expectation itself is built on uncountable spaces.

The practical objective: understand the complete toolkit that (a) proves a conditional expectation exists (via Radon–Nikodym, Shreve I §9.5); (b) makes the change of measure $\mathbb{P}\to \tilde{\mathbb{P}}$ rigorous (Shreve II §1.6; Shreve I Ch 9); (c) justifies every Monte Carlo and every discrete limit - the theoretical reason “it’s just an average” is sound.

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2. Mathematical Ground Truth & Derivations

2.1 Radon–Nikodym & change of measure (Shreve II Thm 1.6.1; Shreve I Ch 9)

$\tilde{\mathbb{P}}$ is absolutely continuous w.r.t. $\mathbb{P}$ ($\tilde{\mathbb{P}}\ll \mathbb{P}$) if $\mathbb{P}(A)=0\Rightarrow \tilde{\mathbb{P}}(A)=0$. Then there is a nonnegative $Z$ with

$$\tilde{\mathbb{P}}(A)={\int }_{A}Zd\mathbb{P}\forall A\in \mathcal{F},Z=\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}.$$

Equivalence ($\tilde{\mathbb{P}}\sim \mathbb{P}$) iff $Z>0$ a.s.; then $1/Z$ is the RN derivative of $\mathbb{P}$ w.r.t. $\tilde{\mathbb{P}}$ and $\mathbb{E}Y=\tilde{\mathbb{E}}[Y/Z]$ (Shreve II Thm 1.6.1; Shreve I Ch 9). Existence of conditional expectation as a corollary (Shreve I §9.5): for $X\ge 0$ with $\int Xd\mathbb{Q}=1$, define ${\mathbb{P}}^{\prime }(A)={\int }_{A}Xd\mathbb{Q}$ on a sub-$\sigma$-algebra $\mathcal{G}$; the RN derivative of ${\mathbb{P}}^{\prime }$ w.r.t. $\mathbb{Q}$ restricted to $\mathcal{G}$ is $\mathcal{G}$-measurable and has the partial-averaging property, so it is $\mathbb{E}[X\mid \mathcal{G}]$.

On the finite market (Shreve I Ex 9.1), $Z(\omega )=\tilde{\mathbb{P}}(\omega )/\mathbb{P}(\omega )$ gives $Z(HH)=\frac{9}{4},Z(HT)=Z(TH)=\frac{9}{8},Z(TT)=\frac{9}{16}$ (verified), and $Z_k=\mathbb{E}[Z\mid {\mathcal{F}}_k]$ is a $\mathbb{P}$-martingale; the state-price density is ${\zeta }_k=(1+r{)}^{-k}{Z}_k$ with ${\zeta }_j{V}_j$ a $\mathbb{P}$-martingale and European value $V_0=\mathbb{E}[{\zeta }_k{C}_k]$ (Shreve I Ch 9).

2.2 Expectation = the Lebesgue integral & the standard machine (Shreve II §1.4–1.5)

The integral is built on the $y$-axis partition (Lebesgue, not Riemann): indicator $\to$ simple $\to$ nonnegative $\to$ general. Computation via density: $\mathbb{E}[h(X)]=\int h(x)f_X(x)dx$ (Shreve I Thm 3.32); the density is the RN derivative of the law w.r.t. Lebesgue measure.

2.3 The convergence theorems (Shreve II Thms 1.4.5/1.4.9)

• Monotone Convergence (MCT): $0\le X_n\uparrow X$ a.s. $\Rightarrow$ $\mathbb{E}[X_n]\uparrow \mathbb{E}[X]$.
• Fatou’s Lemma: $X_n\ge 0$ $\Rightarrow$ $\mathbb{E}[\mathrm{liminf}{X}_n]\le \mathrm{liminf}\mathbb{E}[X_n]$.
• Dominated Convergence (DCT): $X_n\to X$ a.s. and $|X_n|\le Y$ integrable $\Rightarrow$ $\mathbb{E}[X_n]\to \mathbb{E}[X]$.

These justify passing limits through expectations everywhere in Ch 3 - the stopped exponential martingale $Z(t\wedge {\tau }_m)$ with dominated/monotone convergence yields $P\{{\tau }_m<\infty \}=1$ and $\mathbb{E}[{\tau }_m]=\infty$ (Shreve II Thm 3.6.2, Remark 3.6.3). The SLLN/CLT that run Monte Carlo are themselves consequences (Glasserman §1.1).

2.4 Conditional expectation as $L^2$ orthogonal projection (Shreve II Def 2.3.1)

$\mathbb{E}[X\mid \mathcal{G}]$ minimizes $\mathbb{E}[(X-V{)}^2]$ over $\mathcal{G}$-measurable $V$: the residual is orthogonal to every $\mathcal{G}$-measurable variable, $\mathbb{E}[(X-\mathbb{E}[X\mid \mathcal{G}])\cdot V]=0$. This is the measure-theoretic statement of “best predictor,” and the root of the ESL regression function $\mathbb{E}[Y\mid X=x]$ (ESL eq. 2.13) and its MSE decomposition.

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3. Computational Implementation - verify RN derivative, the projection, and CLT

Check the discrete RN derivative values, the $L^2$ orthogonality of the conditional-expectation residual, and the CLT standard error of an MC estimator. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. Missing the absolute-continuity direction. Radon–Nikodym builds $\tilde{\mathbb{P}}$ from a density $Z\ge 0$ with $\mathbb{E}Z=1$; equivalence is the extra $Z>0$ a.s. assumption (Shreve II Thm 1.6.1). Assuming equivalence when only absolute continuity holds, or vice-versa, silently breaks the inverse density $1/Z$.
2. Exchanging limit and expectation without a convergence theorem. $\lim \mathbb{E}[X_n]=\mathbb{E}[\lim X_n]$ is false in general; only under MCT (monotone), Fatou (inequality), or DCT (dominated) does it hold (Shreve II §1.4). The stopped-martingale first-passage computation relies exactly on this.
3. Requiring densities where there are none. Not every law has a density - ${\mu }_X\ll \text{Leb}$ is exactly the condition (Shreve I Ch 11). The $Y=ZX$ pair has no joint density despite each marginal being normal; forcing one misrepresents the dependence.
4. Treating the MC average as deterministic. $\frac{1}{n}\sum f(U_i)$ is a random variable whose error is $O({\sigma }_f/\sqrt{n})$; without SLLN/CLT you cannot justify either convergence or the error bar (Glasserman §1.1).
5. Confusing measurability with integrability. A random variable can be $\mathcal{F}$-measurable yet non-integrable; expectation and conditional expectation are defined only on (suitable) integrable objects. Unbounded payoffs need the standard machine’s limit step.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Glasserman, Monte Carlo Methods in Financial Engineering
• Casella & Berger, Statistical Inference

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6. Connected Graph Bridges

• Back: 05 · Martingales · Index Hub
• Forward topic-folder: Stochastic Calculus (Girsanov as the continuous Radon–Nikodym, martingale representation) · Monte Carlo (SLLN/CLT in practice)
• Theory: Black–Scholes–Merton (risk-neutral pricing is a RN measure change) · No-Arbitrage & Binomial Trees