Astochastic differential equation (SDE) is adifferential equation in which one or more of the terms is astochastic process,^[1] resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used tomodel various behaviours of stochastic models such asstock prices,^[2]random growth models^[3] or physical systems that are subjected tothermal fluctuations.

SDEs have a random differential that is in the most basic case randomwhite noise calculated as the distributional derivative of aBrownian motion or more generally asemimartingale. However, other types of random behaviour are possible, such asjump processes likeLévy processes^[4] or semimartingales with jumps.

Stochastic differential equations are in general neither differential equations norrandom differential equations. Random differential equations are conjugate to stochastic differential equations. Stochastic differential equations can also be extended todifferential manifolds.^[5][6][7][8]

Stochastic differential equations originated in the theory ofBrownian motion, in the work ofAlbert Einstein andMarian Smoluchowski in 1905, althoughLouis Bachelier was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known asBachelier model. Some of these early examples were linear stochastic differential equations, also calledLangevin equations after French physicistLangevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematicianKiyosi Itô, who introduced the concept ofstochastic integral and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicistStratonovich, leading to a calculus similar to ordinary calculus.

The most common form of SDEs in the literature is anordinary differential equation with the right hand side perturbed by a term dependent on awhite noise variable. In most cases, SDEs are understood as continuous time limit of the correspondingstochastic difference equations. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral.^[1][3] Such a mathematical definition was first proposed byKiyosi Itô in the 1940s, leading to what is known today as theItô calculus.Another construction was later proposed by Russian physicistStratonovich, leading to what is known as theStratonovich integral.TheItô integral andStratonovich integral are related, but different, objects and the choice between them depends on the application considered. TheItô calculus is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time.The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion onmanifolds, although it is possible and in some cases preferable to model random motion on manifolds through Itô SDEs,^[6] for example when trying to optimally approximate SDEs on submanifolds.^[9]

An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is theSmoluchowski equation or theFokker–Planck equation, an equation describing the time evolution ofprobability distribution functions. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept ofstochastic evolution operator.

In physical science, there is an ambiguity in the usage of the term"Langevin SDEs". While Langevin SDEs can be of amore general form, this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure,^[10] leading to a N=2 supersymmetric model closely related tosupersymmetric quantum mechanics. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e.,(overdamped) Langevin SDEs are never chaotic.

Brownian motion or theWiener process was discovered to be exceptionally complex mathematically. TheWiener process is almost surely nowhere differentiable;^[1][3] thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, theItô stochastic calculus and theStratonovich stochastic calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003)^[3] and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again.^[1][3] Still, one must be careful which calculus to use when the SDE is initially written down.

Numerical methods for solving stochastic differential equations^[11] include theEuler–Maruyama method,Milstein method,Runge–Kutta method (SDE), Rosenbrock method,^[12] and methods based on different representations of iterated stochastic integrals.^[13][14]

In physics, SDEs have wide applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of thedynamical systems theory to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.

There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:

${\displaystyle \frac{\mathrm{d}x(t)}{\mathrm{d}t}=F(x(t))+\sum _{\alpha =1}^n{g}_{\alpha }(x(t)){\xi }^{\alpha }(t),}$

where${\displaystyle x\in X}$ is the position in the system in itsphase (or state) space,${\displaystyle X}$, assumed to be a differentiable manifold, the${\displaystyle F\in TX}$ is a flow vector field representing deterministic law of evolution, and${\displaystyle g_{\alpha }\in TX}$ is a set of vector fields that define the coupling of the system to Gaussian white noise,${\displaystyle {\xi }^{\alpha }}$. If${\displaystyle X}$ is a linear space and${\displaystyle g}$ are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. For additive noise, the Itô and Stratonovich forms of the SDE generate the same solution, and it is not important which definition is used to solve the SDE. For multiplicative noise SDEs the Itô and Stratonovich forms of the SDE are different, and care should be used in mapping between them.^[15]

For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.^[16] Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of astochastic difference equation. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is auniquely defined mathematical object that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.

In physics, the main method of solution is to find theprobability distribution function as a function of time using the equivalentFokker–Planck equation (FPE). The Fokker–Planck equation is a deterministicpartial differential equation. It tells how the probability distribution function evolves in time similarly to how theSchrödinger equation gives the time evolution of the quantum wave function or thediffusion equation gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained byMonte Carlo simulation. Other techniques include thepath integration that draws on the analogy between statistical physics andquantum mechanics (for example, the Fokker-Planck equation can be transformed into theSchrödinger equation by rescaling a few variables) or by writing downordinary differential equations for the statisticalmoments of the probability distribution function.^{[citation needed]}

The notation used inprobability theory (and in many applications of probability theory, for instance in signal processing with thefiltering problem and inmathematical finance) is slightly different. It is also the notation used in publications onnumerical methods for solving stochastic differential equations. This notation makes the exotic nature of the random function of time${\displaystyle {\xi }^{\alpha }}$ in the physics formulation more explicit. In strict mathematical terms,${\displaystyle {\xi }^{\alpha }}$ cannot be chosen as an ordinary function, but only as ageneralized function. The mathematical formulation treats this complication with less ambiguity than the physics formulation.

A typical equation is of the form

${\displaystyle \mathrm{d}{X}_t=\mu (X_t,t)\mathrm{d}t+\sigma (X_t,t)\mathrm{d}{B}_t,}$

where${\displaystyle B}$ denotes aWiener process (standard Brownian motion).This equation should be interpreted as an informal way of expressing the correspondingintegral equation

${\displaystyle X_{t+s}-X_t={\int }_t^{t+s}\mu (X_u,u)\mathrm{d}u+{\int }_t^{t+s}\sigma (X_u,u)\mathrm{d}{B}_u.}$

The equation above characterizes the behavior of thecontinuous timestochastic processX_t as the sum of an ordinaryLebesgue integral and anItô integral. Aheuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of lengthδ the stochastic processX_t changes its value by an amount that isnormally distributed withexpectationμ(X_t, t) δ andvarianceσ(X_t, t)² δ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The functionμ is referred to as the drift coefficient, whileσ is called the diffusion coefficient. The stochastic processX_t is called adiffusion process, and satisfies theMarkov property.^[1]

The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution^[1] Both require the existence of a processX_t that solves the integral equation version of the SDE. The difference between the two lies in the underlyingprobability space (${\displaystyle \mathrm{\Omega },\mathcal{F},P}$). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. TheYamada–Watanabe theorem makes a connection between the two.

An important example is the equation forgeometric Brownian motion

${\displaystyle \mathrm{d}{X}_t=\mu X_t\mathrm{d}t+\sigma X_t\mathrm{d}{B}_t.}$

which is the equation for the dynamics of the price of astock in theBlack–Scholes options pricing model^[2] of financial mathematics.

Generalizing the geometric Brownian motion, it is also possible to define SDEs admitting strong solutions and whose distribution is a convex combination of densities coming from different geometric Brownian motions or Black Scholes models, obtaining a single SDE whose solutions is distributed as a mixture dynamics of lognormal distributions of different Black Scholes models.^{[2][17][18][19]} This leads to models that can deal with thevolatility smile in financial mathematics.

The simpler SDE calledarithmetic Brownian motion^[3]

${\displaystyle \mathrm{d}{X}_t=\mu \mathrm{d}t+\sigma \mathrm{d}{B}_t}$

was used by Louis Bachelier as the first model for stock prices in 1900, known today asBachelier model.

There are also more general stochastic differential equations where the coefficientsμ andσ depend not only on the present value of the processX_t, but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process,X, is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values ofX, the defining equation is called a stochastic delay differential equation.

A generalization of stochastic differential equations with the Fisk-Stratonovich integral to semimartingales with jumps are the SDEs ofMarcus type. The Marcus integral is an extension of McShane's stochastic calculus.^[20]

An application instochastic finance derives from the usage of the equation forOrnstein–Uhlenbeck process

${\displaystyle \mathrm{d}{R}_t=\mu R_t\mathrm{d}t+{\sigma }_t\mathrm{d}{B}_t.}$

which is the equation for the dynamics of the return of the price of astock under the hypothesis that returns display aLog-normal distribution.Under this hypothesis, the methodologies developed by Marcello Minenna determines prediction interval able to identify abnormal return that could hidemarket abuse phenomena.^[21][22]

More generally one can extend the theory of stochastic calculus ontodifferential manifolds and for this purpose one uses the Fisk-Stratonovich integral. Consider a manifold${\displaystyle M}$, some finite-dimensional vector space${\displaystyle E}$, a filtered probability space${\displaystyle (\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbb{R}}_{+}},P)}$ with${\displaystyle ({\mathcal{F}}_t{)}_{t\in {\mathbb{R}}_{+}}}$ satisfying theusual conditions and let${\displaystyle \hat{M}=M\cup \{\mathrm{\infty }\}}$ be theone-point compactification and${\displaystyle x_0}$ be${\displaystyle {\mathcal{F}}_0}$-measurable. Astochastic differential equation on${\displaystyle M}$ written

${\displaystyle \mathrm{d}X=A(X)\circ dZ}$

is a pair${\displaystyle (A,Z)}$, such that

• ${\displaystyle Z}$ is a continuous${\displaystyle E}$-valued semimartingale,
• ${\displaystyle A:M\times E\to TM,(x,e)\mapsto A(x)e}$ is a homomorphism ofvector bundles over${\displaystyle M}$.

For each${\displaystyle x\in M}$ the map${\displaystyle A(x):E\to T_{x}M}$ is linear and${\displaystyle A(\cdot )e\in \mathrm{\Gamma }(TM)}$ for each${\displaystyle e\in E}$.

A solution to the SDE on${\displaystyle M}$ with initial condition${\displaystyle X_0=x_0}$ is a continuous${\displaystyle \{{\mathcal{F}}_t\}}$-adapted${\displaystyle M}$-valued process up to life time${\displaystyle \zeta }$, s.t. for each test function${\displaystyle f\in C_c^{\mathrm{\infty }}(M)}$ the process${\displaystyle f(X)}$ is a real-valued semimartingale and for each stopping time${\displaystyle \tau }$ with the equation

${\displaystyle f(X_{\tau })=f(x_0)+{\int }_0^{\tau }(\mathrm{d}f{)}_{X}A(X)\circ \mathrm{d}Z}$

holds${\displaystyle P}$-almost surely, where${\displaystyle (df{)}_X:T_{x}M\to T_{f(x)}M}$ is thedifferential at${\displaystyle X}$. It is amaximal solution if the life time is maximal, i.e.,

${\displaystyle P}$-almost surely. It follows from the fact that${\displaystyle f(X)}$ for each test function${\displaystyle f\in C_c^{\mathrm{\infty }}(M)}$ is a semimartingale, that${\displaystyle X}$ is asemimartingale on${\displaystyle M}$. Given a maximal solution we can extend the time of${\displaystyle X}$ onto full${\displaystyle {\mathbb{R}}_{+}}$ and after a continuation of${\displaystyle f}$ on${\displaystyle \hat{M}}$ we get

${\displaystyle f(X_t)=f(X_0)+{\int }_0^t(\mathrm{d}f{)}_{X}A(X)\circ \mathrm{d}Z,t\ge 0}$

up to indistinguishable processes.^[23]Although Stratonovich SDEs are the natural choice for SDEs on manifolds, given that they satisfy the chain rule and that their drift and diffusion coefficients behave as vector fields under changes of coordinates, there are cases where Ito calculus on manifolds is preferable. A theory of Ito calculus on manifolds was first developed byLaurent Schwartz through the concept of Schwartz morphism,^[6] see also the related 2-jet interpretation of Ito SDEs on manifolds based on the jet-bundle.^[8] This interpretation is helpful when trying to optimally approximate the solution of an SDE given on a large space with the solutions of an SDE given on a submanifold of that space,^[9] in that a Stratonovich based projection does not result to be optimal. This has been applied to thefiltering problem, leading to optimal projection filters.^[9]

Usually the solution of an SDE requires a probabilistic setting, as the integral implicit in the solution is a stochastic integral. If it were possible to deal with the differential equation path by path, one would not need to define a stochastic integral and one could develop a theory independently of probability theory. This points to considering the SDE

${\displaystyle \mathrm{d}{X}_t(\omega )=\mu (X_t(\omega ),t)\mathrm{d}t+\sigma (X_t(\omega ),t)\mathrm{d}{B}_t(\omega )}$

as a single deterministic differential equation for every${\displaystyle \omega \in \mathrm{\Omega }}$, where${\displaystyle \mathrm{\Omega }}$ is the sample space in the given probability space (${\displaystyle \mathrm{\Omega },\mathcal{F},P}$). However, a direct path-wise interpretation of the SDE is not possible, as the Brownian motion paths have unbounded variation and are nowhere differentiable with probability one, so that there is no naive way to give meaning to terms like${\displaystyle \mathrm{d}{B}_t(\omega )}$, precluding also a naive path-wise definition of the stochastic integral as an integral against every single${\displaystyle \mathrm{d}{B}_t(\omega )}$. However, motivated by the Wong-Zakai result^[24] for limits of solutions of SDEs with regular noise and usingrough paths theory, while adding a chosen definition of iterated integrals of Brownian motion, it is possible to define a deterministic rough integral for every single${\displaystyle \omega \in \mathrm{\Omega }}$ that coincides for example with the Ito integral with probability one for a particular choice of the iterated Brownian integral.^[24] Other definitions of the iterated integral lead to deterministic pathwise equivalents of different stochastic integrals, like the Stratonovich integral. This has been used for example in financial mathematics to price options without probability.^[25]

As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values inn-dimensionalEuclidean spaceRⁿ and driven by anm-dimensional Brownian motionB; the proof may be found in Øksendal (2003, §5.2).^[3]

LetT > 0, and let

${\displaystyle \mu :{\mathbb{R}}^n\times [0,T]\to {\mathbb{R}}^n;}$

${\displaystyle \sigma :{\mathbb{R}}^n\times [0,T]\to {\mathbb{R}}^{n\times m};}$

bemeasurable functions for which there exist constantsC andD such that

${\displaystyle |\mu (x,t)|+|\sigma (x,t)|\le C(1+|x|);}$

${\displaystyle |\mu (x,t)-\mu (y,t)|+|\sigma (x,t)-\sigma (y,t)|\le D|x-y|;}$

for allt ∈ [0, T] and allx andy ∈ Rⁿ, where

${\displaystyle |\sigma {|}^2=\sum _{i,j=1}^n|{\sigma }_{ij}{|}^2.}$

LetZ be a random variable that is independent of theσ-algebra generated byB_s,s ≥ 0, and with finitesecond moment:

Then the stochastic differential equation/initial value problem

${\displaystyle \mathrm{d}{X}_t=\mu (X_t,t)\mathrm{d}t+\sigma (X_t,t)\mathrm{d}{B}_t\text{ for }t\in [0,T];}$

${\displaystyle X_0=Z;}$

has a P-almost surely uniquet-continuous solution (t, ω) ↦ X_t(ω) such thatX isadapted to thefiltrationF_t^Z generated byZ andB_s,s ≤ t, and

The stochastic differential equation above is only a special case of a more general form

${\displaystyle \mathrm{d}{Y}_t=\alpha (t,Y_t)\mathrm{d}{X}_t}$

where

• ${\displaystyle X}$ is a continuous semimartingale in${\displaystyle {\mathbb{R}}^n}$ and${\displaystyle Y}$ is a continuous semimartingal in${\displaystyle {\mathbb{R}}^d}$
• ${\displaystyle \alpha :{\mathbb{R}}_{+}\times U\to \mathrm{Lin}({\mathbb{R}}^n;{\mathbb{R}}^d)}$ is a map from some open nonempty set${\displaystyle U\subset {\mathbb{R}}^d}$, where${\displaystyle \mathrm{Lin}({\mathbb{R}}^n;{\mathbb{R}}^d)}$ is the space of all linear maps from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle {\mathbb{R}}^d}$.

More generally one can also look at stochastic differential equations onmanifolds.

Whether the solution of this equation explodes depends on the choice of${\displaystyle \alpha }$. Suppose${\displaystyle \alpha }$ satisfies some local Lipschitz condition, i.e., for${\displaystyle t\ge 0}$ and some compact set${\displaystyle K\subset U}$ and some constant${\displaystyle L(t,K)}$ the condition

${\displaystyle |\alpha (s,y)-\alpha (s,x)|\le L(t,K)|y-x|,x,y\in K,0\le s\le t,}$

where${\displaystyle |\cdot |}$ is the Euclidean norm. This condition guarantees the existence and uniqueness of a so-calledmaximal solution.

Suppose${\displaystyle \alpha }$ is continuous and satisfies the above local Lipschitz condition and let${\displaystyle F:\mathrm{\Omega }\to U}$ be some initial condition, meaning it is a measurable function with respect to the initial σ-algebra. Let${\displaystyle \zeta :\mathrm{\Omega }\to {\overline{\mathbb{R}}}_{+}}$ be apredictable stopping time with${\displaystyle \zeta >0}$ almost surely. A${\displaystyle U}$-valued semimartingale is called amaximal solution of

${\displaystyle d{Y}_t=\alpha (t,Y_t)d{X}_t,Y_0=F}$

withlife time${\displaystyle \zeta }$ if

• for one (and hence all) announcing${\displaystyle {\zeta }_n\nearrow \zeta }$ the stopped process${\displaystyle Y^{{\zeta }_n}}$ is a solution to thestopped stochastic differential equation

${\displaystyle \mathrm{d}Y=\alpha (t,Y)\mathrm{d}{X}^{{\zeta }_n}}$

• on the set we have almost surely that${\displaystyle Y_t\to \mathrm{\partial }U}$ with${\displaystyle t\to \zeta }$.^[26]

${\displaystyle \zeta }$ is also a so-calledexplosion time.

Explicitly solvable SDEs include:^[11]

${\displaystyle \mathrm{d}{X}_t=(a(t)X_t+c(t))\mathrm{d}t+(b(t)X_t+d(t))\mathrm{d}{W}_t}$

${\displaystyle X_t={\mathrm{\Phi }}_{t,t_0}\left(X_{t_0}+{\int }_{t_0}^t{\mathrm{\Phi }}_{s,t_0}^{-1}(c(s)-b(s)d(s))\mathrm{d}s+{\int }_{t_0}^t{\mathrm{\Phi }}_{s,t_0}^{-1}d(s)\mathrm{d}{W}_s\right)}$

where

${\displaystyle {\mathrm{\Phi }}_{t,t_0}=\exp \left({\int }_{t_0}^t\left(a(s)-\frac{b^2(s)}{2}\right)\mathrm{d}s+{\int }_{t_0}^{t}b(s)\mathrm{d}{W}_s\right)}$

${\displaystyle \mathrm{d}{X}_t=\frac{1}{2}f(X_t)f^{\prime }(X_t)\mathrm{d}t+f(X_t)\mathrm{d}{W}_t}$

for a given differentiable function${\displaystyle f}$ is equivalent to the Stratonovich SDE

${\displaystyle \mathrm{d}{X}_t=f(X_t)\circ W_t}$

which has a general solution

${\displaystyle X_t=h^{-1}(W_t+h(X_0))}$

where

${\displaystyle h(x)={\int }^x\frac{\mathrm{d}s}{f(s)}}$

${\displaystyle \mathrm{d}{X}_t=\left(\alpha f(X_t)+\frac{1}{2}f(X_t)f^{\prime }(X_t)\right)\mathrm{d}t+f(X_t)\mathrm{d}{W}_t}$

for a given differentiable function${\displaystyle f}$ is equivalent to the Stratonovich SDE

${\displaystyle \mathrm{d}{X}_t=\alpha f(X_t)\mathrm{d}t+f(X_t)\circ W_t}$

which is reducible to

${\displaystyle \mathrm{d}{Y}_t=\alpha \mathrm{d}t+\mathrm{d}{W}_t}$

where${\displaystyle Y_t=h(X_t)}$ where${\displaystyle h}$ is defined as before.Its general solution is

${\displaystyle X_t=h^{-1}(\alpha t+W_t+h(X_0))}$

In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on thedifferential forms on thephase/state space of the model. In this formulation of stochastic dynamics, all SDEs possess topologicalsupersymmetry which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines aschaos.

• Backward stochastic differential equation
• Langevin dynamics
• Local volatility
• Stochastic process
• Stochastic volatility
• Stochastic partial differential equations
• Diffusion process
• Stochastic difference equation
• Supersymmetric theory of stochastic dynamics

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• Higham, Desmond J. (January 2001). "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations".SIAM Review.43 (3):525–546.Bibcode:2001SIAMR..43..525H.CiteSeerX 10.1.1.137.6375.doi:10.1137/S0036144500378302.
• Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM,ISBN 978-1-611976-42-7 (2021).

• Stochastic differential equations
• Differential equations
• Stochastic processes
• Mathematical finance

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