Stochastic calculus

Itô and friends

Calculus that works, in a certain sense, for random objects, of certain types. Very popular for stochastic differential equations. This is a popular and well-explored tool; notes here are not supposed to be tutorial; I simply want to maintain a list of useful definitions because the literature gets messy and ambiguous sometimes.

Itô Integral


Itô’s lemma

Specifically, let \(X=\left(X^{1}, \ldots, X^{n}\right)\) be a tuple of semimartingales and let \(f: \mathbb{R}^{n} \rightarrow\mathbb{R}\) have continuous second order partial derivatives. Then \(f(X)\) is also a semimartingale and the following formula holds:

\[\begin{aligned} f\left(X_{t}\right) - f\left(X_{0}\right) &= +\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \mathrm{d} X_{s}^{i} \\ &\quad +\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s-}\right) \mathrm{d}\left[X^{i}, X^{j}\right]_{s}^{c} \\ &\quad +\sum_{0<s \leq t}\left(f\left(X_{s}\right)-f\left(X_{s-}\right)-\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \Delta X_{s}^{i}\right) \end{aligned}\]

Here the bracket term is the quadratic variation, \[ [X,Y] := XY-\int X_{s-} \mathrm{d}Y(s)-\int Y_{s-} \mathrm{d}X(s) \]

For a continuous semimartingale, the jump terms are null, and the left limits are equal to the function itself \[\begin{aligned} f\left(X_{t}\right) - f\left(X_{0}\right) &= +\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s}\right) \mathrm{d} X_{s}^{i} \\ &+\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s}\right) \mathrm{d}\left[X^{i}, X^{j}\right]_{s}^{c} \end{aligned}\]

Some authors assume that in Itô calculus the driving noise is not a general semimartingale but a Brownian motion, which is a continuous driving noise. Others use the term Itô calculus to describe a more general version. Sometimes the distinction is made between an Itô diffusion, with a Brownian driving term and a Lévy SDE, which has no implication that the driving term is Brownian. However, this is generally messy and particularly in tutorials generated nby the applied finance people it can be quite hard to work out which set of definitions they are using.

Stratonovich Integral

Doss-Sussman transform

Reducing a stochastic integral to a deterministic one, i.e. replacing the Itô integral with a Lebesgue one. (Sussmann 1978; Karatzas and Ruf 2016).

Assumptions: \(\sigma \in C^{1,1}(\mathbb{R}), \sigma, \sigma^{\prime} \in L^{\infty}, b \in C^{0,1}\) \[ \mathrm{d} X(t)=b(X(s)) \mathrm{d} s+\frac{1}{2} \sigma(X(s)) \sigma^{\prime}(X(s)) \mathrm{d} s+\sigma(X(s)) \mathrm{d} W_{s} \] has a unique (strong) solution \(X=u(W, Y)\) for some \(u \in C^{2}(\mathbb{R})\) and \[ \mathrm{d} Y(t)=f(W(t), Y(t)) \mathrm{d} t \] for some \(f \in C^{0,1}\).

Rogers and Williams (1987) section V.28.

Paley-Wiener integral

There is a narrower, but lazier, version of the Itô integral. Jonathan Mattingly introduces it in Paley-Wiener-Zygmund Integral Assuming \(f\) continuous with continuous first derivative and \(f(1)\)=0.

We define the stochastic integral \(\int_{0}^{1} f(t) * \mathrm{d} W(t)\) for these functions by the standard Rieman integral, \[ \int_{0}^{1} f(t) * \mathrm{d} W(t)=-\int_{0}^{1} f^{\prime}(t) W(t) \mathrm{d} t \] Then \[ \mathbf{E}\left[\left(\int_{0}^{1} f(t) * \mathrm{d} W(t)\right)^{2}\right]=\int_{0}^{1} f^{2}(t) \mathrm{d} t. \] Paley, Wiener, and Zygmund then used this isometry to extend the integral to \(f\in L^{2}[0,1]\) as the limit of approximating continuous functions.

What does this get us in terms of SDEs?


Applebaum, David, and Markus Riedle. 2010. “Cylindrical Levy Processes in Banach Spaces.” Proceedings of the London Mathematical Society 101 (3): 697–726.
Arfken, George B., and Hans-Jurgen Weber. 2005. Mathematical Methods for Physicists. 6th ed. Boston: Elsevier.
Arfken, George B., Hans-Jurgen Weber, and Frank E. Harris. 2013. Mathematical Methods for Physicists: A Comprehensive Guide. 7th ed. Amsterdam ; Boston: Elsevier.
Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. “Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences 13 (3).
Arnold, LUDWIG, and WOLFGANG Kliemann. 1983. “Qualitative Theory of Stochastic Systems.” In Probabilistic Analysis and Related Topics, edited by A. T. Bharucha-reid, 1–79. Academic Press.
Baudoin, Fabrice. 2014. Diffusion Processes and Stochastic Calculus. EMS Textbooks in Mathematics. Zurich, Switzerland: European Mathematical Society.
Baudoin, Fabrice, and Alice Vatamanelu. n.d. “Stochastic Calculus,” 114.
Bertoin, Jean, Marc Yor, and others. 2001. “On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab 6 (95): 106.
Coulaud, Benjamin, and Frédéric JP Richard. 2018. “A Consistent Framework for a Statistical Analysis of Surfaces Based on Generalized Stochastic Processes.”
Davis, Mark H. A., Xin Guo, and Guoliang Wu. 2009. “Impulse Control of Multidimensional Jump Diffusions.” December 16, 2009.
Goldys, Beniamin, and Szymon Peszat. 2021. “On Linear Stochastic Flows,” May.
Hanson, Floyd B. 2007. “Stochastic Processes and Control for Jump-Diffusions.” SSRN Scholarly Paper ID 1023497. Rochester, NY: Social Science Research Network.
Hassler, Uwe. 2016. Stochastic Processes and Calculus. Springer Texts in Business and Economics. Cham: Springer International Publishing.
Holden, Helge, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang. 1996. Stochastic Partial Differential Equations. Boston, MA: Birkhäuser Boston.
Inchiosa, M. E., and A. R. Bulsara. 1996. “Signal Detection Statistics of Stochastic Resonators.” Physical Review E 53 (3): R2021–24.
Jacod, Jean, and Albert N. Shiryaev. 1987. “The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.” In Limit Theorems for Stochastic Processes, edited by Jean Jacod and Albert N. Shiryaev, 1–63. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.
Karatzas, Ioannis, and Johannes Ruf. 2016. “Pathwise Solvability of Stochastic Integral Equations with Generalized Drift and Non-Smooth Dispersion Functions.” Annales de l’Institut Henri Poincaré, Probabilités Et Statistiques 52 (2): 915–38.
Karczewska, Anna. 2007. “Convolution Type Stochastic Volterra Equations.” December 28, 2007.
Kelly, David. 2016. “Rough Path Recursions and Diffusion Approximations.” The Annals of Applied Probability 26 (1).
Kelly, David, and Ian Melbourne. 2014a. “Smooth Approximation of Stochastic Differential Equations,” March.
———. 2014b. “Deterministic Homogenization for Fast-Slow Systems with Chaotic Noise,” September.
Klebaner, Fima C. 1999. Introduction to Stochastic Calculus With Applications. Imperial College Press.
Kloeden, P. E., and E. Platen. 1991. “Stratonovich and Ito Stochastic Taylor Expansions.” Mathematische Nachrichten 151 (1): 33–50.
Kloeden, P. E., E. Platen, and I. W. Wright. 1992. “The Approximation of Multiple Stochastic Integrals.” Stochastic Analysis and Applications 10 (4): 431–41.
Kloeden, Peter E., and Eckhard Platen. 1992. “Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations, edited by Peter E. Kloeden and Eckhard Platen, 161–226. Applications of Mathematics. Berlin, Heidelberg: Springer.
Korzeniowski, Andrzej. 1989. “On Diffusions That Cannot Escape from a Convex Set.” Statistics & Probability Letters 8 (3): 229–34.
Kushner, Harold J, and Giovanni DiMasi. 1978. “Approximations for Functionals and Optimal Control Problems on Jump Diffusion Processes.” Journal of Mathematical Analysis and Applications 63 (3): 772–800.
Liu, Xiao, Kyongmin Yeo, and Siyuan Lu. 2020. “Statistical Modeling for Spatio-Temporal Data From Stochastic Convection-Diffusion Processes.” Journal of the American Statistical Association 0 (0): 1–18.
Matheron, G. 1973. “The Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68.
Meidan, R. 1980. “On the Connection Between Ordinary and Generalized Stochastic Processes.” Journal of Mathematical Analysis and Applications 76 (1): 124–33.
Mikosch, Thomas, and Rimas Norvaiša. 2000. “Stochastic Integral Equations Without Probability.” Bernoulli 6 (3): 401–34.
Papanicolaou, Andrew. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” January 2, 2019.
Privault, Nicolas. n.d. Notes on Stochastic Finance.
Protter, Philip. 2005. Stochastic Integration and Differential Equations. Springer.
Pugachev, V. S., and I. N. Sinitsyn. 2001. Stochastic Systems: Theory and Applications. River Edge, NJ: World Scientific.
Revuz, Daniel, and Marc Yor. 2004. Continuous Martingales and Brownian Motion. Springer Science & Business Media.
Rogers, L. C. G., and D. Williams. 2000. Diffusions, Markov Processes, and Martingales. 2nd ed. Cambridge Mathematical Library. Cambridge, U.K. ; New York: Cambridge University Press.
Rogers, L. C. G., and David Williams. 1987. Diffusions, Markov Processes and Martingales 2. Cambridge University Press.
Rößler, Andreas. 2004. “Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes.” Stochastic Analysis and Applications 22 (6): 1553–76.
Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24.
Sussmann, Hector J. 1978. “On the Gap Between Deterministic and Stochastic Ordinary Differential Equations.” The Annals of Probability 6 (1): 19–41.
Tautu, Petre. 2014. Stochastic Spatial Processes. Springer.
Xiu, Dongbin. 2010. Numerical Methods for Stochastic Computations: A Spectral Method Approach. USA: Princeton University Press.
Yaglom, A. M. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics. New York, NY: Springer Science & Business Media.
Øksendal, Bernt. 1985. Stochastic Differential Equations: An Introduction With Applications. Springer.

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