# Analytical solutions for stochastic differential equations via Martingale processes

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## Abstract

In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

### Keywords

Martingale process Itô formula Change of variable Differentiable process Analytical solution## Introduction

In this paper, we resolve to represent analytical methods for stochastic differential equations, specially reputed and famous equations in pricing and investment rate models, based on Martingale processes with various examples about them which we have found in a couple of papers like [2, 5, 6, 7]. There are two main reasons for this approach. Firstly, the each solutions of these kind of equations are Martingale processes or analytic function of Martingale Processes. Thus, due to drift-free property, it will be caused computational error less than numerical computations with existing classic methods. Secondly, for each Martingale process (especially differentiable process), there exists a spectral expansion of two-dimensional Hermite polynomials with constant coefficients [8]. Therefore, it could be made higher the strong order of convergence with increasing the number of polynomials in this expansion. Equations are just obtained with diffusion part or drift free, by making Martingale process from other process. This method can be done by Itô product formula on initial process and an appropriate Martingale process. Another suitable method to convert SDEs into ODEs that we try is to omit the diffusion part of the stochastic equation.

This article is organized as follows. In Sect. 2, it is verified the making of Martingales processes by exponential Martingale process. In Sect. 3, we solve equations as a function of Martingales with prominent analytical solution, by applying change of appropriate variables method on drift-free SDEs. In Sect. 5, some analytical and numerical examples of expressed methods are demonstrated. Finally, the conclusions and remarks are brought in last section.

## Change of measure and Martingale process

**Theorem 1**

*Suppose that stochastic processes*\(X_{t}\)

*verify in differential equation:*

*and let*\(\lambda (t):=-\mu (X_{t},t)/\sigma (X_{t},t).\)

*Therefore,*\(X\mathcal {Z}_t^{\lambda }\)

*is a Martingale process.*

*Proof*

## Change of variable method

**Case 1**

**Case 2**

**Case 3**

**Case 4**

**Theorem 2**

*The stochastic differential equations in*(20)

*given by continuous functions*\(f:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\)

*and*\(C:\mathbb {R}\rightarrow \mathbb {R}\)

*can be written as:*

*where*\(\mathcal {Z}_t^c(t)\)

*is an exponential Martingale process.*

## Examples

*Example 1*

*Example 2*

*Example 3*

*Example 4*

*Example 5*

*Example 6*

## Conclusions and remarks

In this paper, a couple of analytical solutions of some determined set of stochastic differential equations was indicated via making the Martingale process from a stochastic process. Converting stochastic differential equations to ordinary ones as another suitable method was posed. Indeed, it is tried to omit diffusion part of stochastic equation by applying Martingale processes. In addition, change of variable method on SDEs related to Martingale processes was discussed. Last of all with some examples, we analyzed and obtained its exact solutions and in some cases their solutions compared with other numerical methods.

### References

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## Copyright information

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