02.15.05 · analysis / stochastic-analysis

The Stratonovich Integral and Stratonovich Calculus

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Anchor (Master): Hsu 2002 *Stochastic Analysis on Manifolds* (AMS, GSM 38) Ch. 1-2; Ikeda-Watanabe 1989 *Stochastic Differential Equations and Diffusion Processes* (North-Holland, 2nd ed.) Ch. III-V

Intuition Beginner

Imagine you are adding up tiny pushes from a jittery, unpredictable wind to figure out how far a leaf drifts. To turn the pushes into a total you have to decide when to read the wind's strength during each tiny step: at the start of the step, at the end, or in the middle. For an ordinary smooth breeze the choice would not matter, because over a tiny interval the wind barely changes. Random noise is rougher than that, so the choice genuinely changes the answer.

The Stratonovich integral makes one specific choice: read the strength at the middle of each step and split the difference. That single decision has a beautiful payoff. The familiar rules of calculus you already trust — the chain rule, the product rule — keep working with no extra correction terms. You compute as if the noise were a smooth wind, and the bookkeeping comes out right.

There is a competing choice, the Itô integral, which reads the wind at the start of each step. Itô is the natural choice for gambling and finance, where you must bet before you see the outcome. Stratonovich is the natural choice for physics and geometry, where you want your equations to look the same no matter what coordinates you use to describe space.

Visual Beginner

The picture shows a rough random path and zooms in on one short step. Three dots mark where you could read the random signal's strength: the left end, the middle, and the right end. Itô uses the left dot. Stratonovich uses the middle dot — really the average of the two endpoints. For a smooth curve all three dots sit at almost the same height, so the three rules agree. For the rough random path the dots sit at visibly different heights, and that gap is the reason the two calculi give different answers.

Worked example Beginner

Take the simplest interesting case: a random signal $B$ that starts at $0$ , and we want to total up the quantity "current height times the next push", written informally as the running sum of $B$ times its own increments. Suppose over a unit of time the signal happens to move from height $0$ up to height $2$ , and we slice the time into many equal steps.

The Itô total (read height at the start of each step) and the Stratonovich total (read height at the middle) differ by a fixed amount. For this self-referential case the Stratonovich answer minus the Itô answer equals one half of the total "wiggle" the path accumulated. Over one unit of time a standard random signal accumulates exactly $1$ unit of wiggle on average, so the gap is $\frac{1}{2} \times 1 = 0.5$ .

Concretely, the Itô rule gives $\frac{1}{2} (2^{2}) - \frac{1}{2} (1) = 2 - 0.5 = 1.5$ . The Stratonovich rule gives the clean $\frac{1}{2} (2^{2}) = 2$ , exactly what the ordinary calculus formula "integral of $x d x$ is $\frac{1}{2} x^{2}$ " would predict.

What this tells us: the Stratonovich answer is the one that obeys the schoolbook rule with no correction. That convenience is the whole point — the price is paid elsewhere, in how you set up the problem.

Check your understanding Beginner

Formal definition Intermediate+

Fix a filtered probability space $(Ω, F, (F_{t})_{t \geq 0}, P)$ satisfying the usual conditions and a one-dimensional Brownian motion $B$ adapted to it. Let $H$ be a continuous $F_{t}$ -adapted process, and more generally let $X, Y$ be continuous semimartingales. We use the convention that $[X, Y]_{t}$ denotes the quadratic covariation, the limit in probability of $\sum_{k} (X_{t_{k + 1}} - X_{t_{k}}) (Y_{t_{k + 1}} - Y_{t_{k}})$ along refining partitions of $[0, t]$ ; in particular $[B, B]_{t} = t$ .

The Stratonovich integral of $H$ against $X$ is defined as the limit in probability of the symmetric (midpoint) Riemann sums

\int_{0}^{t} H_{s} \circ d X_{s} = ∣Π∣ \to 0 lim k \sum \frac{H _{t_{k}} + H _{t_{k + 1}}}{2} (X_{t_{k + 1}} - X_{t_{k}}),

where $Π = {0 = t_{0} < \dots < t_{n} = t}$ ranges over partitions with mesh $∣Π∣ \to 0$ . The averaging of the integrand at the two endpoints is the defining feature; the ring symbol $\circ$ distinguishes it from the Itô integral $\int_{0}^{t} H_{s} d X_{s}$ , whose sums use the left endpoint $H_{t_{k}}$ .

When $H = f (X)$ for a $C^{1}$ function $f$ — the case that arises in solving SDEs — the symmetric sum can be rewritten, and the limit relates the Stratonovich integral to the Itô integral by the Itô-Stratonovich conversion formula

\int_{0}^{t} H_{s} \circ d X_{s} = \int_{0}^{t} H_{s} d X_{s} + \frac{1}{2} [H, X]_{t} .

The extra term $\frac{1}{2} [H, X]_{t}$ is the correction term: it is the difference between reading the integrand at the midpoint and reading it at the left endpoint, and it is exactly half the covariation of the integrand process with the integrator. For $X = B$ and $H = f (B)$ one has $[H, B]_{t} = \int_{0}^{t} f^{'} (B_{s}) d s$ , so the correction is $\frac{1}{2} \int_{0}^{t} f^{'} (B_{s}) d s$ , an ordinary Lebesgue integral. The Stratonovich integral is not a martingale in general — the correction term has nonzero drift — which is the price paid for the chain rule below.

A Stratonovich stochastic differential equation in $R^{n}$ is written $$ dX_t ;=; V_0(X_t),dt ;+; \sum_{i=1}^d V_i(X_t) \circ dB^i_t, $$ with $V_{0}, V_{1}, \dots, V_{d}$ smooth vector fields on $R^{n}$ and $B = (B^{1}, \dots, B^{d})$ a $d$ -dimensional Brownian motion. The fields $V_{i}$ are the diffusion vector fields and $V_{0}$ is the drift vector field.

Key theorem with proof Intermediate+

Theorem (Stratonovich chain rule). Let $X$ be a continuous semimartingale and $f \in C^{2} (R)$ . Then $f (X)$ is a continuous semimartingale and, in Stratonovich form, $$ df(X_t) ;=; f'(X_t)\circ dX_t . $$ There is no second-derivative term: the rule is the ordinary Newton-Leibniz chain rule, in contrast to Itô's formula $df (X_{t}) = f^{'} (X_{t}) d X_{t} + \frac{1}{2} f^{''} (X_{t}) d [X, X]_{t}$ .

Proof. Start from Itô's formula, which holds for any continuous semimartingale ^{[oksendal Ch. 3]}: $$ df(X_t) = f'(X_t),dX_t + \tfrac{1}{2}f''(X_t),d[X,X]_t . $$ Apply the conversion formula to the right-hand Stratonovich object we wish to evaluate. With integrand $H = f^{'} (X)$ and integrator $X$ , $$ \int_0^t f'(X_s)\circ dX_s = \int_0^t f'(X_s),dX_s + \tfrac{1}{2},[f'(X), X]_t . $$ The covariation $[f^{'} (X), X]_{t}$ is computed by Itô's formula applied to the integrand process: $d (f^{'} (X)) = f^{''} (X) d X + \frac{1}{2} f^{'''} (X) d [X, X]$ when $f \in C^{3}$ , and only the $f^{''} (X) d X$ part contributes to a bracket with $X$ , since the finite-variation piece has zero covariation with the continuous-martingale part. Therefore $$ [f'(X), X]_t = \int_0^t f''(X_s),d[X,X]_s . $$ For $f \in C^{2}$ the same identity holds by a standard mollification argument, approximating $f$ by $C^{3}$ functions and passing to the limit, since both sides depend continuously on $f$ in $C^{2}$ on compacts. Substituting, $$ \int_0^t f'(X_s)\circ dX_s = \int_0^t f'(X_s),dX_s + \tfrac{1}{2}\int_0^t f''(X_s),d[X,X]_s . $$ The right-hand side is precisely the Itô-formula expansion of $f (X_{t}) - f (X_{0})$ . Hence $\int_{0}^{t} f^{'} (X_{s}) \circ d X_{s} = f (X_{t}) - f (X_{0})$ , which in differential form reads $df (X_{t}) = f^{'} (X_{t}) \circ d X_{t}$ . The Itô second-derivative term has been absorbed into the symmetric integral's correction, with the two contributions equal and combining to cancel. $□$

Bridge. This cancellation builds toward the entire reason Stratonovich calculus is the right language for stochastic differential equations on manifolds, a thread that appears again in 03.02.45. The foundational reason is functorial: a chain rule with no second-derivative term is a chain rule that commutes with composition by diffeomorphisms exactly as ordinary calculus does, so a Stratonovich SDE pushed forward through a change of coordinates $φ$ becomes another Stratonovich SDE whose vector fields are the pushed-forward fields $φ_{*} V_{i}$ . This is exactly the transformation law of a vector field, the same law that governs the Levi-Civita data of 03.02.27; the Itô form fails it because the correction $\frac{1}{2} f^{''}$ does not transform tensorially. The conversion formula is dual to Itô's formula in the precise sense that each is obtained from the other by adding or subtracting one half the covariation bracket, and putting these together gives the central insight of the subject: the bridge from analysis to geometry is the symmetric integral, because only it lets "solve an SDE" mean the same thing in every chart. The construction generalises verbatim from $R^{n}$ to any smooth manifold, which is the program the next tier carries out.

Exercises Intermediate+

Exercise (medium, short-answer). Explain, in two or three sentences, why two engineers modelling the same physical system with smooth noise approximations will agree on the limiting equation if they interpret it in the Stratonovich sense but may disagree if they use Itô. Reference the Wong-Zakai theorem.

Hint

Smooth approximations obey ordinary calculus; what limit theorem says their solutions converge?

Answer

Rubric: a correct answer states that piecewise-smooth (or mollified) approximations $B^{ε}$ to Brownian motion produce ODE solutions that obey the ordinary chain rule, and by the Wong-Zakai theorem these solutions converge to the solution of the Stratonovich SDE with the same coefficients. Itô interpretation would require an extra correction drift not present in the physical ODE, so it is the Stratonovich reading that is approximation-stable.

Exercise (hard, symbolic). For the $d$-dimensional Stratonovich SDE $dX = V_0\,dt + \sum_{i=1}^d V_i \circ dB^i$ on $\mathbb{R}^n$, derive the second-order differential generator $L$ acting on test functions $g \in C^\infty_c(\mathbb{R}^n)$.

Hint

Convert to Itô form first: the Stratonovich-to-Itô drift correction is $\frac{1}{2} \sum_{i} V_{i} V_{i}$ acting as a first-order operator (the field $V_{i}$ applied to the components of $V_{i}$ ). Then read off the generator from the Itô SDE.

Answer

$L = V_{0} + \frac{1}{2} \sum_{i = 1}^{d} V_{i}^{2}$ , where each $V_{i} = \sum_{j} V_{i}^{j} \partial_{j}$ is the first-order directional-derivative operator and $V_{i}^{2} = V_{i} (V_{i} \cdot)$ is its square (a second-order operator). The $\frac{1}{2} \sum_{i} V_{i}^{2}$ is exactly the Hörmander sum-of-squares form; the cross term from squaring supplies both the principal symbol $\frac{1}{2} \sum_{i} V_{i}^{j} V_{i}^{k} \partial_{j} \partial_{k}$ and the Stratonovich correction to the drift.

Exercise (hard, short-answer). A Stratonovich SDE on $\mathbb{R}^n$ is transported to a manifold $M$ by a diffeomorphism $\varphi$. Argue that the transported diffusion fields are $\varphi_* V_i$ and explain why the same statement is false for the Itô form.

Hint

Apply the Stratonovich chain rule to $Y = φ (X)$ componentwise. Then ask whether the Itô correction $\frac{1}{2} \sum V_{i}^{2}$ is a tensor.

Answer

Rubric: because $d Y = φ^{'} (X) \circ d X$ has no second-order term, substituting the SDE for $d X$ gives $d Y = (φ_{*} V_{0}) (Y) d t + \sum_{i} (φ_{*} V_{i}) (Y) \circ d B^{i}$ , so each field transforms by the pushforward — the transformation law of a vector field. For the Itô form the correction term $\frac{1}{2} \sum_{i} \partial^{2}$ involves second derivatives that pick up extra Christoffel-type terms under change of coordinates and so do not transform tensorially; only after adding a connection-dependent drift does an invariant Itô formulation exist.

Advanced results Master

The conversion formula and the chain rule together make the Stratonovich integral the calculus in which stochastic flows are coordinate-free. On a smooth manifold $M$ with smooth vector fields $V_{0}, V_{1}, \dots, V_{d} \in Γ (T M)$ , the Stratonovich SDE $d X_{t} = V_{0} (X_{t}) d t + \sum_{i = 1}^{d} V_{i} (X_{t}) \circ d B_{t}^{i}$ has an intrinsic meaning: a continuous $M$ -valued semimartingale $X$ solves it when, for every $g \in C^{\infty} (M)$ , the real semimartingale $g (X_{t})$ satisfies $d g (X_{t}) = (V_{0} g) (X_{t}) d t + \sum_{i} (V_{i} g) (X_{t}) \circ d B_{t}^{i}$ . The chain-rule absence of a second-derivative term is what makes this test consistent across overlapping charts: composing with a transition map is composing with a diffeomorphism, and the symmetric integral commutes with that composition.

Generator and Hörmander form. The conversion of the Stratonovich SDE to Itô form adds the drift $\frac{1}{2} \sum_{i} \nabla_{V_{i}} V_{i}$ (or $\frac{1}{2} \sum_{i} V_{i}^{j} \partial_{j} V_{i}$ in coordinates), and the diffusion process $X$ is Markov with infinitesimal generator $$ L ;=; V_0 ;+; \tfrac{1}{2}\sum_{i=1}^d V_i^2 , $$ where each $V_{i}$ is read as a first-order operator $g \mapsto V_{i} g$ and $V_{i}^{2} = V_{i} \circ V_{i}$ . This is the Hörmander sum-of-squares operator. Its principal symbol is $\frac{1}{2} \sum_{i} (V_{i})^{\otimes 2}$ , degenerate wherever the fields fail to span; the operator is nonetheless hypoelliptic when the Lie algebra generated by $V_{0}, V_{1}, \dots, V_{d}$ spans $T_{x} M$ at every $x$ (Hörmander's bracket-generating condition). The Stratonovich form is what exhibits $L$ in this manifestly geometric square; the Itô form would present the same $L$ with its drift and second-order parts artificially separated and chart-dependent.

Wong-Zakai approximation. Let $B^{ε}$ be smooth approximations to $B$ — piecewise-linear interpolation on a mesh of width $ε$ , or mollification $B * ρ_{ε}$ . Let $X^{ε}$ solve the ordinary (random) ODE $\dot{X}^{ε} = V_{0} (X^{ε}) + \sum_{i} V_{i} (X^{ε}) \dot{B}^{i, ε}$ , path by path. The Wong-Zakai theorem asserts $X^{ε} \to X$ in probability, uniformly on compact time intervals, where $X$ solves the Stratonovich SDE with the same coefficients ^{[Wong-Zakai 1965]}. The limit is Stratonovich, not Itô, precisely because each $X^{ε}$ obeys the ordinary chain rule, and that property survives the limit only for the symmetric integral. In higher dimensions with non-commuting fields ( $[V_{i}, V_{j}] \neq = 0$ ) the smoothing scheme can contribute an additional Lévy-area drift term; the canonical piecewise-linear scheme yields the pure Stratonovich limit, and rough-path theory later organises these correction terms systematically.

Synthesis. The symmetric integral is the foundational reason the three faces of a stochastic flow — its defining SDE, its generator, and its smooth-noise approximation — line up in a single coordinate-free package. The Stratonovich chain rule is exactly the ordinary chain rule, and that is the central insight from which everything else follows: it is what makes the generator appear in the Hörmander square $V_{0} + \frac{1}{2} \sum V_{i}^{2}$ , what makes the Wong-Zakai limit land on the Stratonovich equation rather than an Itô one, and what makes the diffusion fields transform as honest vector fields under change of chart. This is dual to the Itô picture, where the same generator is recovered but its second-order part is glued to a chart-dependent drift; the bridge between the two is the conversion formula, one half a covariation bracket. Putting these together, Stratonovich calculus generalises ordinary calculus to rough integrators while preserving its functoriality, and it is this functoriality — not mere notational convenience — that makes it the calculus of Brownian motion on manifolds, the construction carried out in 03.02.45.

Full proof set Master

Proposition (Stratonovich-to-Itô conversion is well defined and unique). Let $H$ be a continuous semimartingale and $X$ a continuous semimartingale on a common filtered space. The midpoint Riemann sums defining $\int_{0}^{t} H_{s} \circ d X_{s}$ converge in probability, and the limit equals $\int_{0}^{t} H_{s} d X_{s} + \frac{1}{2} [H, X]_{t}$ .

Proof. Write a partition $Π = {t_{k}}$ of $[0, t]$ and decompose the symmetric sum: $$ \sum_k \frac{H_{t_k} + H_{t_{k+1}}}{2},\Delta_k X = \sum_k H_{t_k},\Delta_k X + \frac{1}{2}\sum_k (H_{t_{k+1}} - H_{t_k}),\Delta_k X, $$ where $Δ_{k} X = X_{t_{k + 1}} - X_{t_{k}}$ . The first sum is the left-endpoint (Itô) Riemann sum, which converges in probability to the Itô integral $\int_{0}^{t} H_{s} d X_{s}$ as $∣Π∣ \to 0$ , by the construction of the Itô integral for continuous semimartingales ^{[oksendal Ch. 3]}. The second sum is $$ \frac{1}{2}\sum_k (\Delta_k H)(\Delta_k X), $$ which is one half the cross-variation Riemann sum and converges in probability to $\frac{1}{2} [H, X]_{t}$ by the definition of the quadratic covariation as the limit of such products along refining partitions. Both limits exist independently of the choice of refining sequence because each is itself an in-probability limit of an established object, so their sum exists and equals $\int_{0}^{t} H_{s} d X_{s} + \frac{1}{2} [H, X]_{t}$ . Uniqueness of the limit follows from uniqueness of limits in probability. $□$

Proposition (coordinate invariance of the Stratonovich SDE). Let $φ : U \to \tilde{U}$ be a diffeomorphism between open subsets of $R^{n}$ , and let $X$ solve $d X = V_{0} (X) d t + \sum_{i} V_{i} (X) \circ d B^{i}$ with values in $U$ . Then $Y = φ (X)$ solves $d Y = (φ_{*} V_{0}) (Y) d t + \sum_{i} (φ_{*} V_{i}) (Y) \circ d B^{i}$ , where $φ_{*} V = (D φ \cdot V) \circ φ^{- 1}$ is the pushforward field.

Proof. Apply the Stratonovich chain rule componentwise to $Y^{a} = φ^{a} (X)$ for each coordinate $a$ . By the chain rule with no second-order term, $$ dY^a = \sum_j \partial_j \varphi^a(X)\circ dX^j = \sum_j \partial_j\varphi^a(X)\Bigl(V_0^j(X),dt + \sum_i V_i^j(X)\circ dB^i\Bigr). $$ Grouping the $d t$ and $\circ d B^{i}$ terms, the $d t$ coefficient is $\sum_{j} \partial_{j} φ^{a} (X) V_{0}^{j} (X) = (D φ (X) V_{0} (X))^{a} = (φ_{*} V_{0})^{a} (Y)$ , using $X = φ^{- 1} (Y)$ . Identically, the $\circ d B^{i}$ coefficient is $(D φ (X) V_{i} (X))^{a} = (φ_{*} V_{i})^{a} (Y)$ . The validity of moving the smooth coefficient $D φ (X)$ through the symmetric integral is precisely the content of the chain rule, which carries no Itô correction. Hence $Y$ solves the stated Stratonovich SDE with pushed-forward fields. The same computation with Itô integrals would leave a residual $\frac{1}{2} \sum_{i, j, k} \partial_{j k} φ^{a} V_{i}^{j} V_{i}^{k} d t$ term from Itô's formula, which is the obstruction to tensoriality of the Itô form. $□$

Connections Master

The multivariable chain rule 02.05.03 is the deterministic statement that this unit restores in the stochastic setting. The ordinary chain rule $d (f \circ g) = (f^{'} \circ g) d g$ is exactly the form the Stratonovich chain rule recovers, with the random integrator $\circ d X$ in place of the smooth $d g$ . The entire payoff of the symmetric integral is that it returns calculus to this familiar law, so the second proposition's pushforward computation reads line for line like the deterministic change-of-variables in 02.05.03.

The Itô integral and Itô's formula 02.15.02 is the co-equal counterpart and the route through which the Stratonovich integral is even defined. The conversion formula $\int H \circ d X = \int H d X + \frac{1}{2} [H, X]$ expresses one in terms of the other, and the proof of the Stratonovich chain rule runs entirely through Itô's formula. Itô is the martingale-preserving calculus of choice for filtration-respecting models such as option pricing, where the integrand must be evaluated before the increment is revealed; Stratonovich trades the martingale property for the ordinary chain rule. Each is the natural setting for problems the other handles awkwardly, and the bracket term is the exact dictionary between them.

The Levi-Civita connection and vector fields on manifolds 03.02.27 supply the geometric scaffolding the Master tier stands on. The pushforward law $φ_{*} V_{i}$ established here is the same transformation law that governs the connection coefficients there, and the Stratonovich-to-Itô drift correction $\frac{1}{2} \sum_{i} \nabla_{V_{i}} V_{i}$ is written with that connection's covariant derivative. Coordinate invariance of the SDE is what lets the flow be assembled from local charts into a process on the manifold, exactly as a vector field is assembled from local trivialisations.

Brownian motion on a manifold 03.02.45 is the immediate downstream consumer. Constructing $M$ -valued Brownian motion means solving a Stratonovich SDE whose diffusion fields are an orthonormal frame for the metric, lifted to the orthonormal frame bundle (the Eells-Elworthy-Malliavin construction); only the symmetric integral makes the chart-by-chart solutions agree on overlaps, so this unit is the analytic prerequisite for that geometric object.

The heat equation and heat kernel 02.13.03 is the analytic shadow of the generator constructed here. When the Hörmander operator $L = V_{0} + \frac{1}{2} \sum_{i} V_{i}^{2}$ is the Laplace-Beltrami operator, the transition density of the Stratonovich diffusion is the heat kernel, and the Feynman-Kac representation expresses heat-equation solutions as expectations along the diffusion. The hypoellipticity of bracket-generating $L$ is the probabilistic source of smoothing for the associated parabolic equation, whose Gaussian transition density extends the normal increments of Brownian motion ^{[quantum-well Continuous probability distributions]}.

Historical & philosophical context Master

The symmetric integral was introduced by Ruslan Stratonovich in 1966 in the context of nonlinear filtering and optimal control, where preserving the ordinary rules of calculus simplified the manipulation of stochastic systems arising in radio engineering ^{[Stratonovich 1966]}. Donald Fisk independently developed an equivalent symmetric stochastic integral in the same period, so the construction is sometimes called the Fisk-Stratonovich integral. The choice between Stratonovich's symmetric convention and Itô's non-anticipating convention — Kiyosi Itô's integral dating to 1944 — is not a matter of correctness but of which structural property one wishes to keep: Itô keeps the martingale property and adaptedness, Stratonovich keeps the chain rule.

The approximation theorem that fixes the Stratonovich convention as the physically natural one is due to Eugene Wong and Moshe Zakai in 1965, who showed that ODE solutions driven by smoothed noise converge to the Stratonovich and not the Itô solution ^{[Wong-Zakai 1965]}. This resolved a genuine ambiguity in modelling: a physicist writing down an equation with white-noise forcing, understood as the limit of band-limited noise, means the Stratonovich equation. The geometric consequence — that Stratonovich SDEs are the coordinate-invariant ones — was developed into a full theory of stochastic differential geometry by Itô himself, Paul Malliavin, and David Elworthy through the 1970s and 1980s, and the connection to Lars Hörmander's 1967 hypoellipticity theorem made the sum-of-squares generator the meeting point of probability and partial differential equations.

Bibliography Master

@article{stratonovich1966,
  author  = {Stratonovich, R. L.},
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  pages   = {362--371},
  year    = {1966},
  doi     = {10.1137/0304028}
}

@article{wongzakai1965,
  author  = {Wong, Eugene and Zakai, Moshe},
  title   = {On the Convergence of Ordinary Integrals to Stochastic Integrals},
  journal = {The Annals of Mathematical Statistics},
  volume  = {36},
  number  = {5},
  pages   = {1560--1564},
  year    = {1965},
  doi     = {10.1214/aoms/1177699916}
}

@article{ito1944,
  author  = {It\^o, Kiyosi},
  title   = {Stochastic Integral},
  journal = {Proceedings of the Imperial Academy (Tokyo)},
  volume  = {20},
  number  = {8},
  pages   = {519--524},
  year    = {1944},
  doi     = {10.3792/pia/1195572786}
}

@article{hormander1967,
  author  = {H\"ormander, Lars},
  title   = {Hypoelliptic Second Order Differential Equations},
  journal = {Acta Mathematica},
  volume  = {119},
  pages   = {147--171},
  year    = {1967},
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}

@book{ikedawatanabe1989,
  author    = {Ikeda, Nobuyuki and Watanabe, Shinzo},
  title     = {Stochastic Differential Equations and Diffusion Processes},
  edition   = {2nd},
  publisher = {North-Holland},
  year      = {1989}
}

@book{hsu2002,
  author    = {Hsu, Elton P.},
  title     = {Stochastic Analysis on Manifolds},
  series    = {Graduate Studies in Mathematics},
  volume    = {38},
  publisher = {American Mathematical Society},
  year      = {2002}
}