Yes. The Bible says it is possible to have assurance of one's salvation. John the evangelist wrote:

I write these things to you who believe in the name of the Son of God, so that you may know that you have eternal life (1 John 5:13).

Note that John says that those who have believed in Christ can know that they have eternal life. Therefore if you have believed in Jesus as your Savior then you are saved. The Bible says so.

Faith In His Word

The main reason we can be assured of our salvation is because God's Word clearly teaches it. We do not have to hope that we will someday be saved or continue to pray that we might be saved. If we have trusted Jesus Christ God says we are saved and His Word is sufficient. The Scripture says:

Forever, O Lord, Your Word is settled in heaven (Psalm 119:89).

Trusting God at His Word can give us the assurance of salvation. Therefore we are encouraged to draw near to Him.

Let us draw near with a true heart in full assurance of faith, having our hearts sprinkled from an evil conscience and our bodies washed with pure water (Hebrews 10:22).

Peace

One of the ways God assures us of our salvation is the peace He gives:

And the peace of God, which surpasses all understanding, will guard your hearts and minds through Jesus Christ (Philippians 4:7).

                             
New Life

In addition those who have trusted Christ enter into a new life.

And such were some of you. But you were washed, but you were sanctified, but you were justified in the name of the Lord Jesus and by the Spirit of our God (1 Corinthians 6:11).

Our actions and attitudes change when we trust Christ.
                         
The Mark Of A True Believer

Apart from the clear teaching of Scripture, there are other ways in which a person can know they are saved.

1.Loves Other Believers

Those who have trusted Christ will love other believers.

We know that we have passed from death to life, because we love our brothers. Anyone who does not love remains in death (1 John 3:14).

2.Confirmation Of The Holy Spirit

The Bible says the Holy Spirit gives confirmation to us that we are His children.

The Spirit himself testifies with our spirit that we are God's children (Romans 8:16).

The Apostle John wrote:

Anyone who believes in the Son of God has this testimony in their heart. Anyone who does not believe God has made them out to be a liar, because they have not believed the testimony God has given about His Son (1 John 5:10).

3.Hatred Of Sin

A hatred of sin is a mark of the Christian.

For sin shall not be your master, because you are not under law, but under grace (Romans 6:14).

4.Loves The Good

Those who have trusted Jesus Christ as Savior will love that which is good. The Bible encourages us to hate evil and love the good. The prophet Amos said:

Hate evil, love good; maintain justice in the courts. Perhaps the LORD God Almighty will have mercy on the remnant of Joseph (Amos 5:15).

             
Summary

The Bible says that a person can know that they are saved. The Scripture also promises assurance to those who trust Christ as Savior.