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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
The discrete difference equations may then be solved iteratively to calculate a price for the option. [4] The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example the Black–Scholes PDE. Once in this form, a ...
This specific example is the Schrödinger equation for a particle in a potential (), but the structure is more general. In many practical partial differential equations, one has a term that involves derivatives (such as a kinetic energy contribution), and a multiplication with a function (for example, a potential).
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.
Suppose we are given the stochastic differential equation = + , where B t is a Wiener process and the functions , are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution X t {\displaystyle X_{t}} directly in terms of B t . {\displaystyle B_{t}.}
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation