Itō’s Formula, Applications, and Stratonovich Integration

December 8, 2023 · probability theorystochastic processes

Work on a filtered probability space satisfying the usual conditions. \(B\) denotes Brownian motion, and stochastic integrands are predictable with the stated integrability. A local martingale need not be a true martingale: the moment conditions in each claim are essential. See Brownian motion and martingale inequalities.

The ordinary chain rule comes from the first-order term in a Taylor expansion. For stochastic integrals, the second-order term may survive summation. To see where the difference enters, first consider a smooth path. Suppose \(f: \mathbb R \to \mathbb R\) is smooth, and \(x : \mathbb R \to \mathbb R\) is a continuously differentiable path. Heuristically, we ought to have, for small \(\Delta t > 0\),

\[\begin{aligned} f(x_{t+\Delta t}) = f(x_t + (x_{t+\Delta t} - x_t)) \approx f(x_t) + f'(x_t) (x_{t+\Delta t} - x_t) \approx f(x_t) + f'(x_t) \dif x_t ~. \end{aligned}\]

This suggests, by summation, the formula

\[\begin{aligned} f(x_t) = f(x_0) + \int_0^t f'(x_s) \dif x_s ~. \end{aligned}\]

For a Brownian motion, since the quadratic variation is not zero, \(\langle B \rangle_t = t\), so a Taylor expansion needs to be of order two. As such, heuristically,

\[\begin{aligned} f(B_{t+\Delta t}) &= f(B_t + (B_{t+\Delta t} - B_t)) \\ &\approx f(B_t) + f'(B_t) (B_{t+\Delta t} - B_t) + \frac{f''(B_t)}{2} (B_{t+\Delta t} - B_t)^2 \\ &= f(B_t) + f'(B_t) \dif B_t + \frac{f''(B_t)}{2} \dif t ~. \end{aligned}\]

Again, by summation, we have the formula

\[\begin{aligned} f(B_t) = f(B_0) + \int_0^t f'(B_s) \dif B_s + \frac{1}{2} \int_0^t f''(B_s) \dif s ~. \end{aligned}\]

The Taylor expansion explains the extra term, but a proof must also justify the limit of the remainder. We establish the product rule first and then use polynomial approximation to obtain the change-of-variables formula.

To see the validity, we start with integration by parts of continuous semimartingales. Suppose \(\{X_t\}_{t \geq 0}\) and \(\{Y_t\}_{t \geq 0}\) are two continuous semimartingales. The product rule isolates the covariation term that ordinary integration by parts does not have; see the continuous-semimartingale notes for the underlying integral and covariation properties. Then, the process \(\{X_t Y_t\}_{t \geq 0}\) is a continuous semimartingale, and we have

\[\begin{aligned} X_t Y_t = X_0 Y_0 + \int_0^t X_s \dif Y_s + \int_0^t Y_s \dif X_s + \langle X,Y \rangle_t \:\:, \quad \forall t \geq 0 ~. \end{aligned}\]

To see this, bilinearity implies we can assume \(X=Y\) without loss of generality. Let us also recentre and assume \(X_0=0\). For any \(t \geq 0\) and every sequence \(\Delta_n[0,t]\) such that \(|\Delta_n[0,t]| \xrightarrow{n\to\infty} 0\), we have

\[\begin{aligned} \sum_{1 \leq k \leq n} \left( X_{t_k^n} - X_{t_{k-1}^n} \right)^2 = X_t^2 - 2 \sum_{1 \leq k \leq n} X_{t_{k-1}^n} \left( X_{t_k^n} - X_{t_{k-1}^n} \right) ~. \end{aligned}\]

Taking \(n \to \infty\) gives the desired statement. \(\blacksquare\)

Itō’s formula then follows for this case. Specifically, if \(\{X_t\}_{t \geq 0}\) is a continuous, adapted semimartingale, and \(f: \mathbb R \to \mathbb R\) is a twice continuously differentiable function, then the process \(\{f(X_t)\}_{t \geq 0}\) is a semimartingale, and the following change of variable formula holds:

\[\begin{aligned} f(X_t) = f(X_0) + \int_0^t f'(X_s) \dif X_s + \frac12 \int_0^t f''(X_s) \dif \, \langle X \rangle_s ~. \end{aligned}\tag{1}\]

To see this holds, suppose \(\{X_t\}_{t \geq 0}\) is bounded. Then, let \(\mathcal A\) be the set of twice continuously differentiable functions \(f\) such that (1) holds. Then, \(\mathcal A\) is a vector space; we need to show that this is an algebra. If this holds, then \(\mathcal A\) contains the identity function, so \(\mathcal A\) contains all polynomial functions. All functions that are twice continuously differentiable (not necessarily having the expansion (1)) is contained in \(\mathcal A\); to see this, note that because \(\{X_t\}_{t \geq 0}\) is bounded by assumption, it is possible to find a sequence of polynomials \(P_n\) such that on compact sets, \(P_n\) converges uniformly to \(f\), \(P'_n\) converges uniformly to \(f'\), and \(P''_n\) converges uniformly to \(f''\). Passing to the limit in the formula for the polynomials gives the formula for the original function.

To see that \(\mathcal A\) is an algebra, we note that for \(f,g \in \mathcal A\), applying the integration by parts formula to semimartingales \(\{f(X_t)\}_{t \geq 0}\) and \(\{g(X_t)\}_{t \geq 0}\) gives

\[\begin{aligned} f(X_t) g(X_t) = f(X_0) g(X_0) + \int_0^t f(X_s) \dif g(X_s) + \int_0^t g(X_s) \dif f(X_s) + \langle f(X), g(X) \rangle_t ~. \end{aligned}\tag{2}\]

Since \(f,g \in \mathcal A\), it must be that

\[\begin{aligned} \int_0^t f(X_s) \dif g(X_s) &= \int_0^t f(X_s) g'(X_s) \dif X_s + \frac12 \int_0^t f(X_s) g''(X_s) \dif \, \langle X \rangle_s ~, \\ \int_0^t g(X_s) \dif f(X_s) &= \int_0^t g(X_s) f'(X_s) \dif X_s + \frac12 \int_0^t g(X_s) f''(X_s) \dif \, \langle X \rangle_s ~, \\ \langle f(X), g(X) \rangle_t &= \int_0^t f'(X_s) g'(X_s) \dif \, \langle X \rangle_s ~. \end{aligned}\]

Therefore,

\[\begin{gathered} (2) = f(X_0) g(X_0) + \int_0^t f(X_s) g'(X_s) \dif X_s + \frac12 \int_0^t f(X_s) g''(X_s) \dif \, \langle X \rangle_s \\ + \int_0^t g(X_s) f'(X_s) \dif X_s + \frac12 \int_0^t g(X_s) f''(X_s) \dif \, \langle X \rangle_s + \int_0^t f'(X_s) g'(X_s) \dif \, \langle X \rangle_s \\ = f(X_0) g(X_0) + \int_0^t (fg)'(X_s) \dif X_s + \frac12 \int_0^t (fg)''(X_s) \dif \, \langle X \rangle_s ~. \end{gathered}\]

Thus \(fg\in\mathcal A\). To justify the polynomial approximation, uniformly approximate \(f''\) on the compact range by polynomials \(Q_n\), and integrate twice, choosing \(P_n(0)=f(0)\), \(P_n'(0)=f'(0)\). Then \(P_n,P_n',P_n''\) converge uniformly there. Localise \(X=X_0+M+A\) so that its range, \(\langle M\rangle\), and the variation of \(A\) on the time interval are bounded. The Itō isometry gives convergence of the martingale integrals; the variation bound gives convergence of the other integrals. For an unbounded initial value, first multiply by the \(\mathcal F_0\)-measurable indicators \(\{|X_0|\leq m\}\). On each of these events, continuous-time hitting times of \(|X_t|+\langle M\rangle_t+\operatorname{Var}_{[0,t]}(A)\), together with deterministic time cutoffs, provide the required bounds and tend to infinity. Let the bounds, and then \(m\), increase. This proves (1) simultaneously at all times by continuity.

Remark 1. The change of variables formula above is contained in Bass (2011, Theorem 11.1); the proof therein is substantially more complicated without the integration by parts formula (which they deduce as a consequence of Itō’s formula). Bass (2011, sec. 17.2 and 17.3) contains a presentation of Itō’s formula in a slightly different setting: consider \(X_t = M_t + A_t\), where \(M_t\) is a square-integrable martingale, and \(A_t\) is a process of bounded variation and whose total variation is integrable.

Remark 2. Let \(X=M+A\) and \(Y=N+C\) be continuous semimartingales, and let deterministic partitions of \([0,T]\) have mesh tending to zero. Their sums of increment products converge in probability to \(\langle X,Y\rangle_T=\langle M,N\rangle_T\). For \(X=Y\), expand the square. Center the martingale part by replacing \(M\) with \(M-M_0\) and absorbing \(M_0\) into the finite-variation part. Neither increments nor covariation change. The martingale square sum then converges to \(\langle M\rangle_T\), as proved in the quadratic-variation theorem. The remaining terms satisfy

\[\left|\sum_i\Delta M_i\Delta A_i\right|\leq\max_i|\Delta M_i|\operatorname{Var}_{[0,T]}(A),\qquad \sum_i(\Delta A_i)^2\leq\max_i|\Delta A_i|\operatorname{Var}_{[0,T]}(A).\]

Both bounds tend to zero pathwise, using continuity of \(M\) and \(A\), respectively. Polarisation gives the assertion for two different processes. Only variation on \([0,T]\) is needed.

Remark 3. Itō’s formula is often applied to a function \(F\) that is twice continuously differentiable and defined (only) on an open subset \(U \subset \mathbb R^n\). See Le Gall (2016, 117) for a discussion for Itō’s formula applied to this case.

1. Application of Itō’s formula

Itō’s formula recognises Brownian motion from its quadratic variation. Let \(M\) be a continuous local martingale with \(M_0=0\) and \(\langle M\rangle_t=t\). For \(\theta\in\mathbb R\), the complex-valued process \(Z_t=\exp(i\theta M_t+\theta^2t/2)\) is a local martingale by Itō’s formula, with real and imaginary parts understood separately. On each finite time interval its modulus is deterministically bounded, so it is a true martingale. Thus

\[\mathbb{E}[e^{i\theta(M_t-M_s)}\mid\mathcal F_s] =e^{-\theta^2(t-s)/2},\qquad0\leq s\leq t.\]

The conditional characteristic function identifies the increment as \(\mathcal{N}(0,t-s)\), independent of \(\mathcal F_s\). Together with continuity this proves Lévy’s characterization.

A continuous local martingale with unbounded quadratic variation can therefore be viewed on a Brownian clock. Put \(A=\langle M\rangle\), assume \(A_\infty=\infty\), and define the continuous-time inverse

\[\tau(u)=\inf\{t\geq0:A_t>u\},\qquad B_u=M_{\tau(u)},\qquad\mathcal G_u=\mathcal F_{\tau(u)}.\]

Each \(\tau(u)\) is a finite stopping time; right-continuity of the filtration handles the strict inequality in its definition. Continuity and divergence of \(A\) imply \(A_{\tau(u)}=u\). If \(A\) is strictly increasing, its inverse is continuous. In general it may have flat intervals, so one extra fact is needed: \(M\) is constant on every interval where \(A\) is constant.

Here is a proof of that fact. Fix rational \(a<b\) and stop \(M_t-M_a\), for \(a\leq t\leq b\), when \(A_t-A_a\) first reaches \(\delta>0\). The localised isometry and the maximal inequality bound the probability that its absolute maximum exceeds \(\varepsilon\) by \(\delta/\varepsilon^2\). On \(\{A_b=A_a\}\) no such bracket stop occurs. Let \(\delta\downarrow0\), then intersect over rational \(a,b,\varepsilon\). Continuity extends constancy to every flat interval, including its endpoints. Consequently \(u\mapsto M_{\tau(u)}\) is continuous even across jumps of the inverse, and \(M_t=B_{A_t}\).

For completeness, the stopping argument does give the required integrability. A zero-start local martingale stopped where its bracket is bounded by \(v\) is an \(L^2\)-bounded martingale. To see this, first localise to bounded martingales. For two nested localising times, the isometry expresses the squared \(L^2\)-distance of the stopped terminal values as the expected bracket increment. Dominated convergence, using the bound \(v\), makes these values Cauchy in \(L^2\). Their limit closes the stopped martingale, and its second moment is the expected stopped bracket. Apply this to \(M^{\tau(v)}\). Optional sampling at \(\tau(u)\), \(u\leq v\), now proves that \(B\) is a martingale on \([0,v]\). The same argument for \(M^2-A\), whose stopped family is uniformly integrable, proves that \(B_u^2-u\) is a martingale. Hence \(\langle B\rangle_u=u\), and Lévy’s characterization proves the Brownian time-change theorem.

The integral form of the square-increment approximation is also useful. For a continuous local martingale \(M\), fix \(T<\infty\) and a deterministic partition \(0=t_0<\cdots<t_m=T\). Applying Itō’s formula on each subinterval gives

\[\sum_{j=1}^m(M_{t_j}-M_{t_{j-1}})^2-\langle M\rangle_T =2\int_0^T(M_s-M_{\pi(s)})\,dM_s,\]

where \(\pi(s)=t_{j-1}\) for \(t_{j-1}<s\leq t_j\). After stopping at \(\inf\{t\geq0:|M_t-M_0|\geq K\}\wedge K\), the centered process is bounded and its expected bracket on \([0,T]\) is finite. The isometry gives

\[\mathbb{E}\left|2\int_0^T(M_s-M_{\pi(s)})\,dM_s\right|^2 =4\mathbb{E}\int_0^T|M_s-M_{\pi(s)}|^2\,d\langle M\rangle_s\longrightarrow0.\]

Continuity gives pointwise convergence of the integrand, and boundedness provides an integrable dominating measure. Removing the stop proves convergence in probability for the original martingale. Using partitions that end exactly at \(T\) avoids an extra increment beyond the terminal time.

Remark 4. The stopped approximation just proved converges in \(L^2\). Removing the stopping yields convergence in probability; \(L^2\) convergence for the unstopped process needs an additional moment bound. Square integrability alone does not supply fourth moments of its increments.

2. Stratonovich integrals

The Stratonovich integral incorporates half the covariation correction into the definition of the integral. This restores the ordinary form of the chain rule, which is useful when changing coordinates. If \(X\) and \(Y\) are continuous semimartingales, we can define the Stratonovich integral as

\[\begin{aligned} \int_0^t X_s \circ \dif Y_s = \int_0^t X_s \dif Y_s + \frac12 \langle X, Y \rangle_t ~. \end{aligned}\]

The resulting chain rule takes the following form. Suppose \(f\) is three-times continuously differentiable, and \(X\) is a continuous semimartingale. Then,

\[\begin{aligned} f(X_t) = f(X_0) + \int_0^t f'(X_s) \circ \dif X_s ~. \end{aligned}\]

To see this, Itō’s formula applied to the function \(f\) reveals that we ought to show

\[\begin{aligned} \langle f'(X), X \rangle_t = \int_0^t f''(X_s) \dif \, \langle X \rangle_s ~. \end{aligned}\]

But note that because \(f'\) is twice continuously differentiable, we can apply Itō’s formula to see that

\[\begin{aligned} f'(X_t) = f'(X_0) + \int_0^t f''(X_s) \dif X_s + \frac12 \int_0^t f'''(X_s) \dif \, \langle X \rangle_s ~, \end{aligned}\]

and because quadratic variation processes are nondecreasing, they have finite variations. As such,

\[\begin{aligned} \langle f'(X), X \rangle_t = 0 + \int_0^t f''(X_s) \dif \, \langle X,X \rangle_s + \frac12 \int_0^t f'''(X_s) \dif \, \langle \langle X \rangle_s, X \rangle_s = \int_0^t f''(X_s) \dif \, \langle X \rangle_s ~, \end{aligned}\]

as desired. \(\blacksquare\)

As a result of this construction, if \(X\) and \(Y\) are continuous semimartingales, we can apply the change of variables with \(f(x) = x^2\) to \(X+Y\) and \(X-Y\), from which we can obtain an analogue to the polarization identity:

\[\begin{aligned} (X_t + Y_t)^2 &= (X_0 + Y_0)^2 + 2 \int_0^t (X_s + Y_s) \circ \dif \, (X_s + Y_s) ~, \\ (X_t - Y_t)^2 &= (X_0 - Y_0)^2 + 2 \int_0^t (X_s - Y_s) \circ \dif \, (X_s - Y_s) ~, \end{aligned}\]

implying

\[\begin{aligned} X_t Y_t = X_0 Y_0 + \int_0^t X_s \circ \dif Y_s + \int_0^t Y_s \circ \dif X_s ~. \end{aligned}\]

This is the product formula for Stratonovich integrals. The same correction also appears in symmetric approximating sums. If \(X\) and \(H\) are continuous semimartingales, and \(t_0 > 0\), then

\[\begin{aligned} \sum_{0 \leq k \leq 2^n-1} \frac{H_{k t_0 2^{-n}} + H_{(k+1) t_0 2^{-n}}}{2} (X_{(k+1)t_0 2^{-n}} - X_{k t_0 2^{-n}}) \xrightarrow{\mathop{\mathrm{\mathbf{P}\!\,}}} \int_0^{t_0} H_s \circ \dif X_s ~. \end{aligned}\]

To see this, observe that

\[\begin{gathered} \sum_{0 \leq k \leq 2^n-1} \frac{H_{k t_0 2^{-n}} + H_{(k+1) t_0 2^{-n}}}{2} (X_{(k+1)t_0 2^{-n}} - X_{k t_0 2^{-n}}) \\ = \sum_{0 \leq k \leq 2^n-1} H_{k t_0 2^{-n}} (X_{(k+1)t_0 2^{-n}} - X_{k t_0 2^{-n}}) + \frac12 \sum_{0 \leq k \leq 2^n-1} (H_{(k+1) t_0 2^{-n}} - H_{k t_0 2^{-n}}) (X_{(k+1)t_0 2^{-n}} - X_{k t_0 2^{-n}}) ~. \end{gathered}\]

Clearly, the first sum converges to \(\int_0^{t_0} H_s \dif X_s\) in probability. It remains to show that the second term converges to \(\frac12 \langle H,X \rangle_{t_0}\) in probability. This is the increment-product convergence for continuous semimartingales proved in Remark 2. \(\blacksquare\)

Bass, Richard F. 2011. Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics 33. Cambridge ; New York: Cambridge University Press.

Le Gall, Jean-François. 2016. Brownian Motion, Martingales, and Stochastic Calculus. 1st ed. 2016. Graduate Texts in Mathematics 274. Cham: Springer International Publishing : Imprint: Springer. https://doi.org/10.1007/978-3-319-31089-3.

Reading: Le Gall (2016), Brownian Motion, Martingales, and Stochastic Calculus.

← All posts