37.06.03 · probability / 06-brownian-motion-stochastic-calculus

Brownian Local Time and Tanaka's Formula

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Anchor (Master): Revuz–Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. VI §§1–2 (local times of continuous semimartingales, Tanaka, the occupation-density formula, joint continuity), §VI.2 (Trotter's theorem); Le Gall Ch. 7; Karatzas–Shreve §3.6–3.7

Intuition Beginner

A Brownian path is so jittery that asking "how long did it spend exactly at the level $a$ ?" sounds hopeless: it touches $a$ , leaves, comes back, and the set of times it sits precisely at $a$ has zero total length. Yet the path crosses $a$ over and over, and near $a$ it lingers far more than chance alone would suggest. Local time is the device that measures this lingering. It is a number, growing in time, that records how much the path has clustered around the level $a$ — a clock that ticks only while the path hugs that one height.

Here is the trick that makes the number meaningful. Fix a thin window around $a$ , say everything within a tiny distance of $a$ , and measure the ordinary clock time the path spends inside that window. Divide by the width of the window. As the window shrinks, this ratio settles down to a definite limit. That limit is the local time at $a$ , the density of time-at-level. It is large where the path visits often and zero where the path never reaches.

Why bother? Because the absolute value of a Brownian path, $∣ B_{t} - a ∣$ , is the kind of bent function that ordinary calculus cannot differentiate at the corner. When you try to apply the change-of-variables rule anyway, an extra rising term appears out of the corner — and that extra term is exactly the local time. Tanaka's formula is the precise statement of this, and it turns a stubborn non-smoothness into a clean new object you can actually compute with.

Visual Beginner

Picture a horizontal line at height $a$ drawn across the page, and a jagged Brownian path weaving above and below it from left to right. Beneath, a second curve only ever rises, and it climbs precisely during the stretches when the path above is pressed close to the line $a$ ; whenever the path is far from $a$ , this lower curve is flat.

Read the two curves together. The lower curve grows only while the path overhead is jammed up against the level $a$ , and it grows faster the more violently the path rattles back and forth across $a$ . It never falls, because time spent near $a$ only accumulates. The height of the lower curve at any moment is the local time at $a$ collected so far. Make the shaded band thinner and divide the time-in-band by the band width; the resulting number stops depending on the width once the band is thin enough, and that stable number is what the lower curve is plotting.

Worked example Beginner

Let us measure local time at level $a = 0$ by the window recipe, using made-up but illustrative numbers for a path watched over the time interval from $0$ to $1$ . Take a window of half-width $0.1$ around $0$ , that is all heights between $- 0.1$ and $0.1$ , so the window width is $0.2$ . Suppose that, watching the path, you clock a total of $0.08$ units of time during which the path sits inside this window.

Divide the time-in-window by the window width: $0.08 \div 0.2 = 0.4$ . This is your first estimate of the local time at $0$ up to time $1$ .

Now shrink the window. Use half-width $0.05$ , so width $0.1$ . Because the path lingers near $0$ , halving the window does not halve the time inside it — you might clock $0.045$ units of time inside the narrower window. Divide: $0.045 \div 0.1 = 0.45$ . Shrink once more to half-width $0.025$ , width $0.05$ , and suppose you clock $0.0235$ units of time inside; divide: $0.0235 \div 0.05 = 0.47$ .

The three estimates are $0.4$ , $0.45$ , $0.47$ — they are settling toward a fixed value near $0.5$ rather than running off to zero or to infinity. What this tells us: the ratio of time-in-window to window-width has a genuine limit, and that limit is the local time at $0$ . Compare what happens at a level the path barely reaches, say $a = 3$ : there the time-in-window is essentially $0$ for every window, so every ratio is $0$ and the local time at $3$ is $0$ . Local time is positive exactly at the levels the path actually frequents.

Check your understanding Beginner

Formal definition Intermediate+

Work on a filtered probability space $(Ω, F, (F_{t})_{t \geq 0}, P)$ satisfying the usual conditions, carrying a standard Brownian motion $B$ as constructed in 02.15.01, and use the continuous-semimartingale and quadratic-variation framework of 37.06.01: a continuous semimartingale is $X = X_{0} + M + V$ with $M$ a continuous local martingale and $V$ a continuous finite-variation process, and $⟨ X ⟩ = ⟨ M ⟩$ is the quadratic variation. For Brownian motion $⟨ B ⟩_{t} = t$ .

For a real number $a$ , the occupation measure of $B$ up to time $t$ is the (random) Borel measure $μ_{t}^{B}$ on $R$ defined by $μ_{t}^{B} (A) = \int_{0}^{t} 1_{A} (B_{s}) d s$ for Borel $A$ ; it records how the path's time is distributed across levels. The local time of $B$ at level $a$ up to time $t$ , written $L_{t}^{a}$ , is the Radon–Nikodym density of $μ_{t}^{B}$ with respect to Lebesgue measure $d a$ , normalised so that the occupation-times formula

\int_{0}^{t} f (B_{s}) d s = \int_{R} f (a) L_{t}^{a} d a

holds for every bounded Borel $f$ . Equivalently, for almost every $a$ ,

L_{t}^{a} = ε ↓ 0 lim \frac{1}{2 ε} \int_{0}^{t} 1_{{∣ B_{s} - a ∣ < ε}} d s,

the density of time-at-level made precise as a window limit. One shows $a \mapsto L_{t}^{a}$ may be taken jointly continuous in $(t, a)$ (Trotter), and for fixed $a$ the map $t \mapsto L_{t}^{a}$ is continuous, nondecreasing, with $L_{0}^{a} = 0$ ; it increases only on the closed set ${s : B_{s} = a}$ .

The signum convention used throughout is the left-continuous version $sgn (x) = 1$ for $x > 0$ and $sgn (x) = - 1$ for $x \leq 0$ ; the value at $0$ is irrelevant to the stochastic integrals below because Brownian motion spends Lebesgue-null time at any fixed level. With this convention Tanaka's formula at level $a$ reads

∣ B_{t} - a ∣ = ∣ B_{0} - a ∣ + \int_{0}^{t} sgn (B_{s} - a) d B_{s} + L_{t}^{a},

and the one-sided variants are

(B_{t} - a)^{+} = (B_{0} - a)^{+} + \int_{0}^{t} 1_{{B_{s} > a}} d B_{s} + \frac{1}{2} L_{t}^{a}, (B_{t} - a)^{-} = (B_{0} - a)^{-} - \int_{0}^{t} 1_{{B_{s} \leq a}} d B_{s} + \frac{1}{2} L_{t}^{a} .

Adding the two one-sided forms recovers the symmetric one, since $∣ x ∣ = x^{+} + x^{-}$ ; subtracting them recovers the identity $B_{t} - a = (B_{0} - a) + \int_{0}^{t} d B_{s}$ . The formalisation follows Le Gall ^{[Le Gall 2016]} §7.1–7.2 and Revuz–Yor ^{[Revuz 1999]} Ch. VI §1.

Counterexamples to common slips

$L_{t}^{a}$ is not the amount of time $B$ spends at $a$ : that time is zero. It is a density, a derivative of occupation against level, and is generically positive even though the level set has measure zero.
The map $a \mapsto L_{t}^{a}$ is continuous, but the map $a \mapsto L_{t}^{a}$ has a subtle one-sided behaviour for general semimartingales; for Brownian motion it is jointly continuous, yet the symmetric Tanaka local time differs by a factor of $2$ from the one-sided local times — keep track of which normalisation a source uses.
The integrand $sgn (B_{s} - a)$ is not the derivative of $∣ x - a ∣$ in the classical sense at $x = a$ ; the corner is exactly where the extra term $L_{t}^{a}$ is born, and dropping it makes the formula false.

Key theorem with proof Intermediate+

The central statement is Tanaka's formula, the precise form of the Itô rule for the non-smooth function $x \mapsto ∣ x - a ∣$ .

Theorem (Tanaka's formula). Let $B$ be a standard Brownian motion and $a \in R$ . There is a continuous, nondecreasing, adapted process $(L_{t}^{a})_{t \geq 0}$ with $L_{0}^{a} = 0$ , increasing only on ${s : B_{s} = a}$ , such that

∣ B_{t} - a ∣ = ∣ B_{0} - a ∣ + \int_{0}^{t} sgn (B_{s} - a) d B_{s} + L_{t}^{a}, t \geq 0,

and this $L_{t}^{a}$ is the occupation density: $\int_{0}^{t} f (B_{s}) d s = \int_{R} f (a) L_{t}^{a} d a$ for bounded Borel $f$ .

Proof. Fix $a = 0$ without loss of generality (translate $B$ ). The function $x \mapsto ∣ x ∣$ is not $C^{2}$ , so smooth it. For $ε > 0$ let $f_{ε}$ be the even $C^{2}$ function with $f_{ε} (x) = ∣ x ∣$ for $∣ x ∣ \geq ε$ and $f_{ε} (x) = \frac{1}{2 ε} x^{2} + \frac{ε}{2}$ for $∣ x ∣ \leq ε$ , so $f_{ε}^{'} (x) = sgn (x)$ for $∣ x ∣ \geq ε$ , $f_{ε}^{'} (x) = x / ε$ for $∣ x ∣ \leq ε$ , and $f_{ε}^{''} (x) = \frac{1}{ε} 1_{{∣ x ∣ < ε}}$ . Itô's formula from 02.15.02 applied to $f_{ε} (B_{t})$ gives

f_{ε} (B_{t}) = f_{ε} (B_{0}) + \int_{0}^{t} f_{ε}^{'} (B_{s}) d B_{s} + \frac{1}{2} \int_{0}^{t} f_{ε}^{''} (B_{s}) d ⟨ B ⟩_{s},

and since $⟨ B ⟩_{s} = s$ the last term is $\frac{1}{2 ε} \int_{0}^{t} 1_{{∣ B_{s} ∣ < ε}} d s$ .

As $ε ↓ 0$ , the left side $f_{ε} (B_{t}) \to ∣ B_{t} ∣$ pointwise and $f_{ε} (B_{0}) \to ∣ B_{0} ∣$ . For the stochastic integral, $f_{ε}^{'} (B_{s}) \to sgn (B_{s})$ for every $s$ with $B_{s} \neq = 0$ , hence for $d s d P$ -almost every $(s, ω)$ because $\int_{0}^{t} 1_{{B_{s} = 0}} d s = 0$ ; moreover $∣ f_{ε}^{'} ∣ \leq 1$ , so the Itô isometry of 02.15.02 gives $E [(\int_{0}^{t} (f_{ε}^{'} (B_{s}) - sgn (B_{s})) d B_{s})^{2}] = E \int_{0}^{t} (f_{ε}^{'} (B_{s}) - sgn (B_{s}))^{2} d s \to 0$ by dominated convergence. Thus $\int_{0}^{t} f_{ε}^{'} (B_{s}) d B_{s} \to \int_{0}^{t} sgn (B_{s}) d B_{s}$ in $L^{2}$ .

Because every other term converges, the correction term converges too: define

L_{t}^{0} := ε ↓ 0 lim \frac{1}{2 ε} \int_{0}^{t} 1_{{∣ B_{s} ∣ < ε}} d s = ∣ B_{t} ∣ - ∣ B_{0} ∣ - \int_{0}^{t} sgn (B_{s}) d B_{s},

the limit existing in $L^{2}$ by the displayed convergences. The right-hand expression shows $L^{0}$ is adapted and continuous in $t$ . It is nondecreasing because each approximant $\frac{1}{2 ε} \int_{0}^{t} 1_{{∣ B_{s} ∣ < ε}} d s$ is nondecreasing in $t$ and the monotone property passes to the limit. The window-average form is exactly the occupation density at $0$ : for bounded Borel $f$ , Fubini and the same window limit give $\int_{0}^{t} f (B_{s}) d s = \int_{R} f (a) L_{t}^{a} d a$ , since $\frac{1}{2 ε} \int_{a - ε}^{a + ε} (\cdot) d a$ recovers $f$ at Lebesgue-almost every $a$ . Finally $L^{0}$ increases only on ${s : B_{s} = 0}$ : on any interval where $∣ B_{s} ∣ \geq δ > 0$ the approximants are eventually constant, so their limit does not grow there. $□$

Bridge. Tanaka's formula is the foundational reason local time exists as a genuine stochastic-calculus object rather than a heuristic: it builds toward the Itô–Tanaka formula for arbitrary convex $f$ , where the second derivative becomes a measure and $L^{a}$ is integrated against it, and the construction appears again in the excursion theory and the Ray–Knight description of $a \mapsto L_{t}^{a}$ as a Markov process in the space variable. The corner term forced out of $∣ x - a ∣$ is exactly the occupation density the window recipe of the Beginner tier was estimating, so the smoothing argument is dual to the density definition: one builds $L^{a}$ from the change-of-variables defect, the other from the Radon–Nikodym derivative of the occupation measure, and they coincide. This generalises the smooth Itô rule of 02.15.02 to functions with a kink, and the central insight is that the non-differentiability of $∣ x ∣$ at one point is not an obstruction but the very mechanism that creates a new continuous increasing process; putting these together, the bracket theory of 37.06.01 supplies the $d ⟨ B ⟩_{s} = d s$ that drives the second-order term, and local time is what that term becomes when the second derivative collapses to a point mass.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the process $M_{t} := \int_{0}^{t} sgn (B_{s}) d B_{s}$ is a continuous local martingale with $⟨ M ⟩_{t} = t$ , and conclude $M$ is itself a standard Brownian motion.

Hint

The integrand has $sgn (B_{s})^{2} = 1$ for $s$ with $B_{s} \neq = 0$ ; use the bracket formula $⟨ \int H d B ⟩_{t} = \int_{0}^{t} H_{s}^{2} d ⟨ B ⟩_{s}$ from 37.06.01, then Lévy's characterisation.

Answer

$M$ is a stochastic integral of a bounded adapted integrand against $B$ , hence a continuous local martingale with $M_{0} = 0$ . Its bracket is $⟨ M ⟩_{t} = \int_{0}^{t} sgn (B_{s})^{2} d ⟨ B ⟩_{s} = \int_{0}^{t} sgn (B_{s})^{2} d s = t$ , because $sgn (B_{s})^{2} = 1$ except on ${B_{s} = 0}$ , a set of Lebesgue measure zero in $s$ . A continuous local martingale with bracket $t$ is a standard Brownian motion by Lévy's characterisation 02.15.01. Rubric: full credit for the bracket computation, the null-set remark, and the Lévy step.

Exercise 4 (medium, symbolic).

Derive the one-sided Tanaka formula $(B_{t})^{+} = (B_{0})^{+} + \int_{0}^{t} 1_{{B_{s} > 0}} d B_{s} + \frac{1}{2} L_{t}^{0}$ from the symmetric formula and the identity $B_{t} = B_{0} + \int_{0}^{t} d B_{s}$ .

Hint

Use $x^{+} = \frac{1}{2} (∣ x ∣ + x)$ and combine the symmetric Tanaka formula with the plain decomposition $B_{t} = B_{0} + \int_{0}^{t} d B_{s}$ .

Answer

Since $x^{+} = \frac{1}{2} (∣ x ∣ + x)$ , take half the sum of the symmetric Tanaka formula $∣ B_{t} ∣ = ∣ B_{0} ∣ + \int_{0}^{t} sgn (B_{s}) d B_{s} + L_{t}^{0}$ and the identity $B_{t} = B_{0} + \int_{0}^{t} d B_{s}$ . This gives $(B_{t})^{+} = (B_{0})^{+} + \frac{1}{2} \int_{0}^{t} (sgn (B_{s}) + 1) d B_{s} + \frac{1}{2} L_{t}^{0}$ . Now $\frac{1}{2} (sgn (x) + 1) = 1_{{x > 0}}$ off the null set ${x = 0}$ , so the integral is $\int_{0}^{t} 1_{{B_{s} > 0}} d B_{s}$ , yielding the claim. Rubric: full credit for the $x^{+} = \frac{1}{2} (∣ x ∣ + x)$ averaging and the simplification of $\frac{1}{2} (sgn + 1)$ to the indicator.

Exercise 5 (medium, short-answer).

Explain why the occupation-times formula forces $a \mapsto L_{t}^{a}$ to be (Lebesgue-almost everywhere) the only candidate density, and what additional input upgrades "almost every $a$ " to "every $a$ ".

Hint

A Radon–Nikodym density is unique up to a Lebesgue-null set; a continuous version is unique outright.

Answer

The occupation measure $μ_{t}^{B} (A) = \int_{0}^{t} 1_{A} (B_{s}) d s$ is absolutely continuous with respect to Lebesgue measure (it has a density because Brownian occupation has no atoms), and its Radon–Nikodym derivative $a \mapsto L_{t}^{a}$ is determined uniquely up to a Lebesgue-null set. Two densities agreeing almost everywhere need not agree everywhere, so to speak of $L_{t}^{a}$ at a fixed $a$ one selects the version that is jointly continuous in $(t, a)$ ; Trotter's theorem provides exactly such a continuous version, and a continuous density is unique, pinning $L_{t}^{a}$ at every $a$ . Rubric: full credit for the Radon–Nikodym uniqueness-a.e. point and naming joint continuity (Trotter) as the upgrade to every $a$ .

Exercise 6 (hard, short-answer).

Let $S_{t} = max_{0 \leq s \leq t} B_{s}$ be the running maximum. State Lévy's theorem identifying the law of the pair $(S - B, S)$ and explain its connection to Tanaka's formula at level $0$ .

Hint

Lévy's theorem says $(S - B, S) = d (∣ B ∣, L^{0})$ for a Brownian motion. Compare $S$ with the local time of $∣ B ∣$ at $0$ .

Answer

Lévy's theorem states that the two-dimensional process $(S_{t} - B_{t}, S_{t})_{t \geq 0}$ has the same law as $(∣ β_{t} ∣, L_{t}^{0} (β))_{t \geq 0}$ for a standard Brownian motion $β$ , where $L^{0} (β)$ is the local time of $β$ at $0$ . Equivalently, reflected Brownian motion $∣ β ∣$ and its local time at $0$ are jointly distributed as the reflected-at-its-maximum process $S - B$ and the maximum $S$ itself. The link to Tanaka is that $∣ β_{t} ∣ = \int_{0}^{t} sgn (β_{s}) d β_{s} + L_{t}^{0}$ exhibits $∣ β ∣$ as a Brownian motion (the stochastic integral, by Exercise 3) plus the increasing process $L^{0}$ , and the Skorokhod-type decomposition matches this to $S - B = - (Brownian motion) + S$ with $S$ increasing only when $S = B$ , exactly as $L^{0}$ increases only when $β = 0$ . Rubric: full credit for the identity in law $(S - B, S) = d (∣ β ∣, L^{0})$ and the Tanaka/Skorokhod matching of the increasing parts.

Exercise 7 (hard, short-answer).

Use the Itô–Tanaka formula to compute the decomposition of $f (B_{t})$ for the convex function $f (x) = (x - a)^{+} + (x - b)^{+}$ with $a < b$ , identifying the increasing part in terms of local times.

Hint

For convex $f$ , $f^{''}$ is a measure; here it is a sum of point masses. The Itô–Tanaka formula integrates $L^{x}$ against $\frac{1}{2} f^{''} (d x)$ .

Answer

The convex function $f (x) = (x - a)^{+} + (x - b)^{+}$ has left-derivative $f_{-}^{'} (x) = 1_{{x > a}} + 1_{{x > b}}$ and second derivative the measure $f^{''} (d x) = δ_{a} (d x) + δ_{b} (d x)$ . The Itô–Tanaka formula $f (B_{t}) = f (B_{0}) + \int_{0}^{t} f_{-}^{'} (B_{s}) d B_{s} + \frac{1}{2} \int_{R} L_{t}^{x} f^{''} (d x)$ then gives $f (B_{t}) = f (B_{0}) + \int_{0}^{t} (1_{{B_{s} > a}} + 1_{{B_{s} > b}}) d B_{s} + \frac{1}{2} (L_{t}^{a} + L_{t}^{b})$ . The increasing part is $\frac{1}{2} (L_{t}^{a} + L_{t}^{b})$ , one half the sum of the local times at the two kinks. Rubric: full credit for $f^{''} (d x) = δ_{a} + δ_{b}$ , the indicator integrand, and the increasing part $\frac{1}{2} (L_{t}^{a} + L_{t}^{b})$ .

Advanced results Master

The local time and Tanaka's formula generalise from $∣ x - a ∣$ to arbitrary convex functions and to all continuous semimartingales, and they encode deep structural facts about Brownian paths.

Itô–Tanaka formula. Let $f : R \to R$ be the difference of two convex functions, so its left-derivative $f_{-}^{'}$ has locally bounded variation and its second derivative $f^{''}$ is a signed Radon measure. For a continuous semimartingale $X$ with local time $(L_{t}^{a})_{a \in R}$ ,

f (X_{t}) = f (X_{0}) + \int_{0}^{t} f_{-}^{'} (X_{s}) d X_{s} + \frac{1}{2} \int_{R} L_{t}^{a} f^{''} (d a) .

This is the change-of-variables rule with the second-order term written as the local time integrated against the second-derivative measure; the classical Itô formula is the special case $f \in C^{2}$ , where $f^{''} (d a) = f^{''} (a) d a$ and the occupation-times formula collapses the spatial integral back to $\frac{1}{2} \int_{0}^{t} f^{''} (X_{s}) d ⟨ X ⟩_{s}$ . Tanaka's formula is the case $f (x) = ∣ x - a ∣$ , whose second derivative is $2 δ_{a}$ .

Local time of a continuous semimartingale. For $X = X_{0} + M + V$ a continuous semimartingale, $∣ X_{t} - a ∣ = ∣ X_{0} - a ∣ + \int_{0}^{t} sgn (X_{s} - a) d X_{s} + L_{t}^{a}$ defines a local time satisfying the occupation-density formula $\int_{0}^{t} f (X_{s}) d ⟨ X ⟩_{s} = \int_{R} f (a) L_{t}^{a} d a$ against the bracket measure $d ⟨ X ⟩_{s}$ . The map $a \mapsto L_{t}^{a}$ is càdlàg, and $L_{t}^{a} - L_{t}^{a -} = 2 \int_{0}^{t} 1_{{X_{s} = a}} d V_{s}$ , so for a continuous local martingale ( $V = 0$ ) the local time is jointly continuous.

Joint continuity (Trotter). For Brownian motion the map $(t, a) \mapsto L_{t}^{a}$ admits a version that is jointly continuous, and in $a$ it is Hölder continuous of every order below $\frac{1}{2}$ . This is Trotter's theorem; it is what allows one to speak of $L_{t}^{a}$ at a single fixed level $a$ rather than only for almost every $a$ , and it is the entry point to the Ray–Knight theorems describing $a \mapsto L_{t}^{a}$ as a squared-Bessel process.

Lévy's characterisation of reflected Brownian motion. The reflected process $∣ B ∣$ and its local time $L^{0}$ at $0$ satisfy $(∣ B_{t} ∣, L_{t}^{0})_{t \geq 0} = d (S_{t} - B_{t}, S_{t})_{t \geq 0}$ , where $S_{t} = max_{s \leq t} B_{s}$ . In particular $∣ B ∣$ is itself a Brownian motion (by Tanaka, $∣ B ∣ = \tilde{B} + L^{0}$ with $\tilde{B} = \int sgn (B) d B$ a Brownian motion and $L^{0}$ the Skorokhod reflection term), and $L^{0}$ is the unique continuous increasing process making $∣ B ∣ - L^{0}$ a Brownian motion and growing only on ${∣ B ∣ = 0}$ .

Downcrossing representation (Lévy). Local time at $0$ is the renormalised count of downcrossings: if $D_{ε} (t)$ is the number of downcrossings of the interval $[0, ε]$ by $B$ on $[0, t]$ , then $ε D_{ε} (t) \to \frac{1}{2} L_{t}^{0}$ almost surely (uniformly on compacts) as $ε ↓ 0$ . This recovers local time combinatorially, without any stochastic integral, and exhibits $L^{0}$ as a genuine geometric feature of the path.

Synthesis. Putting these together, the central insight is that one continuous increasing process — the local time $L^{a}$ — simultaneously measures occupation density, repairs the Itô formula at kinks, reflects Brownian motion at a barrier, and counts crossings, and the theory shows these four faces to be one object. The Itô–Tanaka formula is exactly the statement that the second-order term in stochastic calculus is the pairing of local time against the second-derivative measure, so it generalises the smooth Itô rule of 02.15.02 to all convex functions; the occupation-times formula is dual to this, expressing the same $L^{a}$ as the Radon–Nikodym density of occupation against the bracket measure $d ⟨ X ⟩$ of 37.06.01. The foundational reason the whole edifice closes is Tanaka's smoothing argument, and putting these together with Lévy's reflection identity shows that reflected Brownian motion and the Skorokhod problem are the same data as $(\tilde{B}, L^{0})$ ; this appears again in excursion theory, the Ray–Knight theorems, and the construction of Brownian motion with sticky or reflecting boundaries. The bridge is that local time converts every pointwise singularity of a path functional into a smooth increasing clock, making the non-smooth calculus of Brownian motion as tractable as the smooth one.

Full proof set Master

Tanaka's formula and the occupation-density identity are proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (Itô–Tanaka formula for convex $f$ ). Let $f$ be convex on $R$ with left-derivative $f_{-}^{'}$ and second-derivative measure $f^{''}$ , and let $X$ be a continuous semimartingale with local times $L^{a}$ . Then $f (X_{t}) = f (X_{0}) + \int_{0}^{t} f_{-}^{'} (X_{s}) d X_{s} + \frac{1}{2} \int_{R} L_{t}^{a} f^{''} (d a)$ .

Proof. A convex $f$ is the increasing limit of mollifications $f_{n} = f * ρ_{n}$ with $ρ_{n}$ a smooth approximate identity; each $f_{n} \in C^{2}$ , $f_{n}^{'} \to f_{-}^{'}$ pointwise at continuity points of $f_{-}^{'}$ (hence $d a$ -a.e.), and $f_{n}^{''} d a \to f^{''}$ weakly as measures. Itô's formula 02.15.02 for the $C^{2}$ function $f_{n}$ reads $f_{n} (X_{t}) = f_{n} (X_{0}) + \int_{0}^{t} f_{n}^{'} (X_{s}) d X_{s} + \frac{1}{2} \int_{0}^{t} f_{n}^{''} (X_{s}) d ⟨ X ⟩_{s}$ , and the last term equals $\frac{1}{2} \int_{R} L_{t}^{a} f_{n}^{''} (a) d a$ by the occupation-density formula. Pass to the limit: $f_{n} (X_{t}) \to f (X_{t})$ and $f_{n} (X_{0}) \to f (X_{0})$ by uniform convergence on compacts; the stochastic integral converges because $f_{n}^{'} (X_{s}) \to f_{-}^{'} (X_{s})$ boundedly on compacts and the integrands are uniformly bounded over the localising stopping times, so the Itô isometry against $d ⟨ X ⟩$ gives $L^{2}$ -convergence; and $\int_{R} L_{t}^{a} f_{n}^{''} (a) d a \to \int_{R} L_{t}^{a} f^{''} (d a)$ because $a \mapsto L_{t}^{a}$ is bounded with compact support (in $a$ ) and continuous, so it is a legitimate test function against the weak convergence $f_{n}^{''} d a \to f^{''}$ . The three limits assemble to the claimed identity. The difference-of-convex case follows by linearity. $□$

Proposition (occupation-density form of the bracket). For a continuous semimartingale $X$ with local times $L^{a}$ and bounded Borel $g$ , $\int_{0}^{t} g (X_{s}) d ⟨ X ⟩_{s} = \int_{R} g (a) L_{t}^{a} d a$ .

Proof. Apply the Itô–Tanaka formula to $f_{a} (x) = ∣ x - a ∣$ , whose second-derivative measure is $2 δ_{a}$ : this gives $L_{t}^{a} = ∣ X_{t} - a ∣ - ∣ X_{0} - a ∣ - \int_{0}^{t} sgn (X_{s} - a) d X_{s}$ . For $g \geq 0$ first take $f (x) = \int_{R} g (a) ∣ x - a ∣ d a$ when this is finite and convex, with $f^{''} (d a) = 2 g (a) d a$ ; the Itô–Tanaka formula gives $\frac{1}{2} f^{''}$ -term $= \int_{R} g (a) L_{t}^{a} d a$ , while the $C^{2}$ Itô formula computes the same second-order term as $\int_{0}^{t} g (X_{s}) d ⟨ X ⟩_{s}$ because $f^{''} (x) = 2 g (x) d x$ . Equating the two expressions for the bounded-variation part of $f (X_{t})$ yields the identity for nonnegative $g$ ; a general bounded $g$ splits into positive and negative parts. $□$

Proposition (Lévy: $∣ B ∣$ is Brownian and $L^{0}$ is its reflection term). With $\tilde{B}_{t} := \int_{0}^{t} sgn (B_{s}) d B_{s}$ , the process $\tilde{B}$ is a standard Brownian motion, $∣ B_{t} ∣ = \tilde{B}_{t} + L_{t}^{0}$ , and $L^{0}$ is the unique continuous nondecreasing process, increasing only on ${∣ B ∣ = 0}$ , that keeps $∣ B ∣ \geq 0$ .

Proof. That $\tilde{B}$ is a standard Brownian motion is the content of Exercise 3: $\tilde{B}$ is a continuous local martingale with $⟨ \tilde{B} ⟩_{t} = \int_{0}^{t} sgn (B_{s})^{2} d s = t$ , so Lévy's characterisation 02.15.01 applies. Tanaka's formula at $a = 0$ gives $∣ B_{t} ∣ = \tilde{B}_{t} + L_{t}^{0}$ directly. Uniqueness is the Skorokhod lemma: if $Y_{t} = \tilde{B}_{t} + K_{t} \geq 0$ with $K$ continuous nondecreasing, $K_{0} = 0$ , increasing only on ${Y = 0}$ , then $K_{t} = max_{s \leq t} (- \tilde{B}_{s})^{+}$ is forced, since $\int_{0}^{t} 1_{{Y_{s} > 0}} d K_{s} = 0$ and an integration shows $K$ must equal the running maximum of $- \tilde{B}$ clipped at $0$ . Hence $L^{0} = K$ is unique, and the pair $(∣ B ∣, L^{0}) = (\tilde{B} + K, K) = d (S - \tilde{B}, S)$ with $S$ the maximum of $\tilde{B}$ , which is Lévy's identity in law. $□$

Proposition (downcrossing limit). Let $D_{ε} (t)$ count the downcrossings of $[0, ε]$ by $B$ up to time $t$ . Then $ε D_{ε} (t) \to \frac{1}{2} L_{t}^{0}$ almost surely, uniformly on compact $t$ -intervals, as $ε ↓ 0$ .

Proof. Each downcrossing of $[0, ε]$ corresponds to an excursion of $B$ reaching height $\geq ε$ on the positive side between two visits to $0$ . By the one-sided Tanaka formula $(B_{t})^{+} = (B_{0})^{+} + \int_{0}^{t} 1_{{B_{s} > 0}} d B_{s} + \frac{1}{2} L_{t}^{0}$ , the increasing part $\frac{1}{2} L^{0}$ counts the accumulated boundary pushes at $0$ from above. Discretising the excursions of $B^{+}$ above level $ε$ , the number of upcrossing–downcrossing pairs of $[0, ε]$ multiplied by $ε$ approximates the total positive local time at $0$ ; the excursion theory of Itô (or directly the convergence of the Tanaka approximants $\frac{1}{2 ε} \int_{0}^{t} 1_{{0 < B_{s} < ε}} d s$ ) gives $ε D_{ε} (t) \to \frac{1}{2} L_{t}^{0}$ in $L^{2}$ , and a Borel–Cantelli argument along $ε = 2^{- n}$ together with monotonicity in $ε$ upgrades this to almost-sure uniform convergence on compacts. $□$

Connections Master

The occupation-density identity rests entirely on the bracket theory of 37.06.01: local time is the Radon–Nikodym density of occupation against the quadratic-variation measure $d ⟨ X ⟩_{s}$ , and the Tanaka decomposition $∣ X - a ∣ = ∣ X_{0} - a ∣ + \int sgn (X - a) d X + L^{a}$ is a semimartingale decomposition into the continuous local martingale and increasing parts that unit makes available. Without the existence and uniqueness of $⟨ X ⟩$ proved there, the second-order term that becomes $L^{a}$ has no carrier.

Tanaka's formula is the kink-extension of the smooth Itô formula of 02.15.02: that unit proves $f (B_{t}) = f (B_{0}) + \int f^{'} (B_{s}) d B_{s} + \frac{1}{2} \int f^{''} (B_{s}) d s$ for $f \in C^{2}$ , and the present unit removes the smoothness hypothesis by replacing $f^{''} (x) d x$ with the second-derivative measure $f^{''} (d a)$ paired against $L^{a}$ . The smoothing proof here invokes exactly the Itô isometry that unit constructs, so this material is the direct continuation of its change-of-variables theorem.

Brownian motion 02.15.01 supplies the Lévy characterisation used to identify $\tilde{B} = \int sgn (B) d B$ as a Brownian motion and the reflection/strong-Markov apparatus behind the identity $(∣ B ∣, L^{0}) = d (S - B, S)$ ; the running maximum $S$ and the reflection principle proved there are precisely the objects Lévy's theorem matches to reflected Brownian motion and its local time. The downcrossing representation is a pathwise statement about the Brownian trajectories that unit constructs.

The local-time machinery here is the foundation for the excursion-theoretic and additive-functional developments catalogued downstream: the Ray–Knight theorems describe $a \mapsto L_{t}^{a}$ as a squared-Bessel diffusion and feed the Brownian-motion-on-a-manifold constructions 03.02.45, while the Stratonovich correction of 02.15.05 and the time-change reductions of stochastic differential equations both use the occupation-density formula to convert temporal integrals against $d ⟨ X ⟩$ into spatial integrals against $L^{a}$ .

Historical & philosophical context Master

Local time originates with Paul Lévy, who in his 1948 monograph Processus stochastiques et mouvement brownien ^{[Lévy 1948]} introduced the measure of time a Brownian path spends near a level and gave the downcrossing construction at $0$ , together with the identity in law between reflected Brownian motion paired with its local time and the process $(S - B, S)$ . Harry Trotter established in 1958 ^{[Trotter 1958]} that the field $(t, a) \mapsto L_{t}^{a}$ has a jointly continuous version, the result that makes local time a function of the level and not merely an almost-everywhere density. The change-of-variables identity for $∣ B - a ∣$ was given by Hiroshi Tanaka in 1963 ^{[Tanaka 1963]} as a note on continuous additive functionals; it recasts Lévy's occupation density as the defect in the Itô formula at the corner of the absolute value, and its extension to differences of convex functions, with the second derivative read as a measure, became the Itô–Tanaka formula.

The conceptual content is that the non-differentiability of a path functional is a source of structure rather than an obstruction. Where ordinary calculus stalls at the corner of $∣ x ∣$ , stochastic calculus produces a new continuous increasing process; the occupation-times formula then identifies this process as a density, the Itô–Tanaka formula as the carrier of the second-order term, and Lévy's theorem as the reflection mechanism. Itô's excursion theory and the Ray–Knight theorems of the 1960s built on Tanaka's identity to describe the local-time field as a Markov process in the spatial variable, completing the picture in which a single increasing process organises the fine structure of the Brownian path.

Bibliography Master

@book{LeGall2016,
  author    = {Le Gall, Jean-Fran\c{c}ois},
  title     = {Brownian Motion, Martingales, and Stochastic Calculus},
  series    = {Graduate Texts in Mathematics},
  number    = {274},
  publisher = {Springer},
  year      = {2016}
}

@book{RevuzYor1999,
  author    = {Revuz, Daniel and Yor, Marc},
  title     = {Continuous Martingales and Brownian Motion},
  series    = {Grundlehren der mathematischen Wissenschaften},
  number    = {293},
  edition   = {3rd},
  publisher = {Springer-Verlag},
  year      = {1999}
}

@book{KaratzasShreve1991,
  author    = {Karatzas, Ioannis and Shreve, Steven E.},
  title     = {Brownian Motion and Stochastic Calculus},
  series    = {Graduate Texts in Mathematics},
  number    = {113},
  edition   = {2nd},
  publisher = {Springer-Verlag},
  year      = {1991}
}

@article{Tanaka1963,
  author  = {Tanaka, Hiroshi},
  title   = {Note on continuous additive functionals of the 1-dimensional Brownian path},
  journal = {Zeitschrift f\"ur Wahrscheinlichkeitstheorie und verwandte Gebiete},
  volume  = {1},
  pages   = {251--257},
  year    = {1963}
}

@article{Trotter1958,
  author  = {Trotter, Hale F.},
  title   = {A property of Brownian motion paths},
  journal = {Illinois Journal of Mathematics},
  volume  = {2},
  pages   = {425--433},
  year    = {1958}
}

@book{Levy1948,
  author    = {L\'{e}vy, Paul},
  title     = {Processus stochastiques et mouvement brownien},
  publisher = {Gauthier-Villars},
  year      = {1948}
}

Prerequisites

37.06.01
02.15.01
02.15.02

Tier anchors

beginner: Le Gall, Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016), Ch. 7 (occupation times and local time, the density picture); Mörters–Peres, Brownian Motion (Cambridge, 2010), Ch. 6 (local time intuition)
intermediate: Le Gall, Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016), §§7.1–7.2 (Tanaka's formula, the Itô–Tanaka formula, occupation-times formula); Karatzas–Shreve, Brownian Motion and Stochastic Calculus, §3.6–3.7 (local time, Tanaka)
master: Revuz–Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. VI §§1–2 (local times of continuous semimartingales, Tanaka, the occupation-density formula, joint continuity), §VI.2 (Trotter's theorem); Le Gall Ch. 7; Karatzas–Shreve §3.6–3.7

References

Le Gall — Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016) · Ch. 7 §7.1 (local time and the occupation-times formula Thm 7.2), §7.2 (Tanaka's formula, the Itô–Tanaka formula, generalised Itô formula for convex functions)
Revuz, Yor — Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999) · Ch. VI §1 (Tanaka's formula Thm VI.1.2, local time of a continuous semimartingale, the occupation-density formula Cor VI.1.6), §2 (joint continuity Thm VI.1.7, the Trotter theorem, the downcrossing representation Thm VI.1.10)
Karatzas, Shreve — Brownian Motion and Stochastic Calculus (Springer GTM 113, 2nd ed. 1991) · §3.6 (local time, Tanaka's formula Thm 3.6.8), §3.7 (the Lévy theorem on the equivalence of (|B|, L⁰) and (reflected BM, its local time), the downcrossing representation)
Tanaka — Note on continuous additive functionals of the 1-dimensional Brownian path (Z. Wahrsch. 1, 1963) · pp. 251–257; the formula now bearing his name, expressing |B_t − a| via a stochastic integral plus local time
Trotter — A property of Brownian motion paths (Illinois J. Math. 2, 1958) · pp. 425–433; the joint continuity of (t, a) ↦ L^a_t for Brownian local time
Lévy — Processus stochastiques et mouvement brownien (Gauthier-Villars, 1948) · the downcrossing construction of local time at 0 and the identity in law between (|B|, L⁰) and (reflected Brownian motion, its local time)

Estimated time

beginner: 18m
intermediate: 48m
master: 90m